Duality for Schemes

Introduction

This chapter studies relative duality for morphisms of schemes and the dualizing complex on a scheme. A reference is [RD].

Dualizing complexes for Noetherian rings were defined and studied in Dualizing Complexes, Section 0A7A ff. In this chapter we continue this by studying dualizing complexes on schemes, see Section 0A85.

The bulk of this chapter is devoted to studying the right adjoint of pushforward in the setting of derived categories of sheaves of modules with quasi-coherent cohomology sheaves. See Sections 0A9D, 0E4G, 0AA5, 0BZF, 0AWG, 0B6N, 0A74, 0AWZ, 0AA9, 0B4A, and 0BQV. Here we follow the papers [Neeman-Grothendieck], [LN], [Lipman-notes], and [Neeman-improvement].

We discuss the important and useful upper shriek functors \(f^!\) for separated morphisms of finite type between Noetherian schemes in Sections 0A9Y, 0ATZ, and 0BZX culminating in the overview Section 0AU3.

In Section 0AU5 we explain alternative theory of duality and dualizing complexes when working over a fixed locally Noetherian base endowed with a dualizing complex (this section corresponds to a remark in Hartshorne’s book).

In the remaining sections we give a few applications.

This chapter is continued by the chapter on duality on algebraic spaces, see Duality for Spaces, Section 0E4W.

Dualizing complexes on schemes

We define a dualizing complex on a locally Noetherian scheme to be a complex which affine locally comes from a dualizing complex on the corresponding ring. This is not completely standard but agrees with all definitions in the literature on Noetherian schemes of finite dimension.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(K\) be an object of \(D(\mathcal{O}_X)\). The following are equivalent

1. For every affine open \(U = \Spec(A) \subset X\) there exists a dualizing complex \(\omega_A^\bullet\) for \(A\) such that \(K|_U\) is isomorphic to the image of \(\omega_A^\bullet\) by the functor \(\widetilde{} : D(A) \to D(\mathcal{O}_U)\).

2. There is an affine open covering \(X = \bigcup U_i\), \(U_i = \Spec(A_i)\) such that for each \(i\) there exists a dualizing complex \(\omega_i^\bullet\) for \(A_i\) such that \(K|_{U_i}\) is isomorphic to the image of \(\omega_i^\bullet\) by the functor \(\widetilde{} : D(A_i) \to D(\mathcal{O}_{U_i})\).

Proof

Assume (2) and let \(U = \Spec(A)\) be an affine open of \(X\). Since condition (2) implies that \(K\) is in \(D_\QCoh(\mathcal{O}_X)\) we find an object \(\omega_A^\bullet\) in \(D(A)\) whose associated complex of quasi-coherent sheaves is isomorphic to \(K|_U\), see Derived Categories of Schemes, Lemma 06Z0. We will show that \(\omega_A^\bullet\) is a dualizing complex for \(A\) which will finish the proof.

Since \(X = \bigcup U_i\) is an open covering, we can find a standard open covering \(U = D(f_1) \cup \ldots \cup D(f_m)\) such that each \(D(f_j)\) is a standard open in one of the affine opens \(U_i\), see Schemes, Lemma 01IW. Say \(D(f_j) = D(g_j)\) for \(g_j \in A_{i_j}\). Then \(A_{f_j} \cong (A_{i_j})_{g_j}\) and we have \[(\omega_A^\bullet)_{f_j} \cong (\omega_i^\bullet)_{g_j}\] in the derived category by Derived Categories of Schemes, Lemma 06Z0. By Dualizing Complexes, Lemma 0A7G we find that the complex \((\omega_A^\bullet)_{f_j}\) is a dualizing complex over \(A_{f_j}\) for \(j = 1, \ldots, m\). This implies that \(\omega_A^\bullet\) is dualizing by Dualizing Complexes, Lemma 0A7H.

Definition

Let \(X\) be a locally Noetherian scheme. An object \(K\) of \(D(\mathcal{O}_X)\) is called a dualizing complex if \(K\) satisfies the equivalent conditions of Lemma 0A86.

Please see remarks made at the beginning of this section.

Lemma

Let \(A\) be a Noetherian ring and let \(X = \Spec(A)\). Let \(K, L\) be objects of \(D(A)\). If \(K \in D_{\textit{Coh}}(A)\) and \(L\) has finite injective dimension, then \[R\SheafHom_{\mathcal{O}_X}(\widetilde{K}, \widetilde{L}) = \widetilde{R\Hom_A(K, L)}\] in \(D(\mathcal{O}_X)\).

Proof

We may assume that \(L\) is given by a finite complex \(I^\bullet\) of injective \(A\)-modules. By induction on the length of \(I^\bullet\) and compatibility of the constructions with distinguished triangles, we reduce to the case that \(L = I[0]\) where \(I\) is an injective \(A\)-module. In this case, Derived Categories of Schemes, Lemma 0A6H, tells us that the \(n\)th cohomology sheaf of \(R\SheafHom_{\mathcal{O}_X}(\widetilde{K}, \widetilde{L})\) is the sheaf associated to the presheaf \[D(f) \longmapsto \Ext^n_{A_f}(K \otimes_A A_f, I \otimes_A A_f)\] Since \(A\) is Noetherian, the \(A_f\)-module \(I \otimes_A A_f\) is injective (Dualizing Complexes, Lemma 0A6I). Hence we see that \[\begin{align*} \Ext^n_{A_f}(K \otimes_A A_f, I \otimes_A A_f) & = \Hom_{A_f}(H^{-n}(K \otimes_A A_f), I \otimes_A A_f) \\ & = \Hom_{A_f}(H^{-n}(K) \otimes_A A_f, I \otimes_A A_f) \\ & = \Hom_A(H^{-n}(K), I) \otimes_A A_f \end{align*}\] The last equality because \(H^{-n}(K)\) is a finite \(A\)-module, see Algebra, Lemma 0583. This proves that the canonical map \[\widetilde{R\Hom_A(K, L)} \longrightarrow R\SheafHom_{\mathcal{O}_X}(\widetilde{K}, \widetilde{L})\] is a quasi-isomorphism in this case and the proof is done.

Lemma

Let \(X\) be a Noetherian scheme. Let \(K, L, M \in D_\QCoh(\mathcal{O}_X)\). Then the map \[R\SheafHom(L, M) \otimes_{\mathcal{O}_X}^\mathbf{L} K \longrightarrow R\SheafHom(R\SheafHom(K, L), M)\] of Cohomology, Lemma 0A8U is an isomorphism in the following two cases

1. \(K \in D^-_{\textit{Coh}}(\mathcal{O}_X)\), \(L \in D^+_{\textit{Coh}}(\mathcal{O}_X)\), and \(M\) affine locally has finite injective dimension (see proof), or

2. \(K\) and \(L\) are in \(D_{\textit{Coh}}(\mathcal{O}_X)\), the object \(R\SheafHom(L, M)\) has finite tor dimension, and \(L\) and \(M\) affine locally have finite injective dimension (in particular \(L\) and \(M\) are bounded).

Proof

Proof of (1). We say \(M\) has affine locally finite injective dimension if \(X\) has an open covering by affines \(U = \Spec(A)\) such that the object of \(D(A)\) corresponding to \(M|_U\) (Derived Categories of Schemes, Lemma 06Z0) has finite injective dimension¹. To prove the lemma we may replace \(X\) by \(U\), i.e., we may assume \(X = \Spec(A)\) for some Noetherian ring \(A\). Observe that \(R\SheafHom(K, L)\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) by Derived Categories of Schemes, Lemma 0D0C. Moreover, the formation of the left and right hand side of the arrow commutes with the functor \(D(A) \to D_\QCoh(\mathcal{O}_X)\) by Lemma 0A88 and Derived Categories of Schemes, Lemma 0A6H (to be sure this uses the assumptions on \(K\), \(L\), \(M\) and what we just proved about \(R\SheafHom(K, L)\)). Then finally the arrow is an isomorphism by More on Algebra, Lemmas 0A68 part (2).

Proof of (2). We argue as above. A small change is that here we get \(R\SheafHom(K, L)\) in \(D_{\textit{Coh}}(\mathcal{O}_X)\) because affine locally (which is allowable by Lemma 0A88) we may appeal to Dualizing Complexes, Lemma 0G4H. Then we finally conclude by More on Algebra, Lemma 0A69.

Lemma

Let \(K\) be a dualizing complex on a locally Noetherian scheme \(X\). Then \(K\) is an object of \(D_{\textit{Coh}}(\mathcal{O}_X)\) and \(D = R\SheafHom_{\mathcal{O}_X}(-, K)\) induces an anti-equivalence \[D : D_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_X)\] which comes equipped with a canonical isomorphism \(\text{id} \to D \circ D\). If \(X\) is quasi-compact, then \(D\) exchanges \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) and \(D^-_{\textit{Coh}}(\mathcal{O}_X)\) and induces an anti-equivalence \(D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

Let \(U \subset X\) be an affine open. Say \(U = \Spec(A)\) and let \(\omega_A^\bullet\) be a dualizing complex for \(A\) corresponding to \(K|_U\) as in Lemma 0A86. By Lemma 0A88 the diagram \[\xymatrix{ D_{\textit{Coh}}(A) \ar[r] \ar[d]_{R\Hom_A(-, \omega_A^\bullet)} & D_{\textit{Coh}}(\mathcal{O}_U) \ar[d]^{R\SheafHom_{\mathcal{O}_X}(-, K|_U)} \\ D_{\textit{Coh}}(A) \ar[r] & D(\mathcal{O}_U) }\] commutes. We conclude that \(D\) sends \(D_{\textit{Coh}}(\mathcal{O}_X)\) into \(D_{\textit{Coh}}(\mathcal{O}_X)\). Moreover, the canonical map \[L \longrightarrow R\SheafHom_{\mathcal{O}_X}(K, K) \otimes_{\mathcal{O}_X}^\mathbf{L} L \longrightarrow R\SheafHom_{\mathcal{O}_X}(R\SheafHom_{\mathcal{O}_X}(L, K), K)\] (using Cohomology, Lemma 0A8U for the second arrow) is an isomorphism for all \(L\) because this is true on affines by Dualizing Complexes, Lemma 0A7C² and we have already seen on affines that we recover what happens in algebra. The statement on boundedness properties of the functor \(D\) in the quasi-compact case also follows from the corresponding statements of Dualizing Complexes, Lemma 0A7C.

Let \(X\) be a locally ringed space. Recall that an object \(L\) of \(D(\mathcal{O}_X)\) is invertible if it is an invertible object for the symmetric monoidal structure on \(D(\mathcal{O}_X)\) given by derived tensor product. In Cohomology, Lemma 0FPG we have seen this means \(L\) is perfect and there is an open covering \(X = \bigcup U_i\) such that \(L|_{U_i} \cong \mathcal{O}_{U_i}[-n_i]\) for some integers \(n_i\). In this case, the function \[x \mapsto n_x,\quad \text{where }n_x\text{ is the unique integer such that } H^{n_x}(L_x) \not = 0\] is locally constant on \(X\). In particular, we have \(L = \bigoplus H^n(L)[-n]\) which gives a well defined complex of \(\mathcal{O}_X\)-modules (with zero differentials) representing \(L\).

Lemma

Let \(X\) be a locally Noetherian scheme. If \(K\) and \(K'\) are dualizing complexes on \(X\), then \(K'\) is isomorphic to \(K \otimes_{\mathcal{O}_X}^\mathbf{L} L\) for some invertible object \(L\) of \(D(\mathcal{O}_X)\).

Proof

Set \[L = R\SheafHom_{\mathcal{O}_X}(K, K')\] This is an invertible object of \(D(\mathcal{O}_X)\), because affine locally this is true, see Dualizing Complexes, Lemma 0A7F and its proof. The evaluation map \(L \otimes_{\mathcal{O}_X}^\mathbf{L} K \to K'\) is an isomorphism for the same reason.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\omega_X^\bullet\) be a dualizing complex on \(X\). Then \(X\) is universally catenary and the function \(X \to \mathbf{Z}\) defined by \[x \longmapsto \delta(x)\text{ such that } \omega_{X, x}^\bullet[-\delta(x)] \text{ is a normalized dualizing complex over } \mathcal{O}_{X, x}\] is a dimension function.

Proof

Immediate from the affine case Dualizing Complexes, Lemma 0A7Z and the definitions.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\omega_X^\bullet\) be a dualizing complex on \(X\) with associated dimension function \(\delta\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Set \(\mathcal{E}^i = \SheafExt^{-i}_{\mathcal{O}_X}(\mathcal{F}, \omega_X^\bullet)\). Then \(\mathcal{E}^i\) is a coherent \(\mathcal{O}_X\)-module and for \(x \in X\) we have

1. \(\mathcal{E}^i_x\) is nonzero only for \(\delta(x) \leq i \leq \delta(x) + \dim(\text{Supp}(\mathcal{F}_x))\),

2. \(\dim(\text{Supp}(\mathcal{E}^{i + \delta(x)}_x)) \leq i\),

3. \(\text{depth}(\mathcal{F}_x)\) is the smallest integer \(i \geq 0\) such that \(\mathcal{E}_x^{i + \delta(x)} \not = 0\), and

4. we have \(x \in \text{Supp}(\bigoplus_{j \leq i} \mathcal{E}^j) \Leftrightarrow \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \delta(x) \leq i\).

Proof

Lemma 0A89 tells us that \(\mathcal{E}^i\) is coherent. Choosing an affine neighbourhood of \(x\) and using Derived Categories of Schemes, Lemma 0A6H and More on Algebra, Lemma 0A6A part (3) we have \[\mathcal{E}^i_x = \SheafExt^{-i}_{\mathcal{O}_X}(\mathcal{F}, \omega_X^\bullet)_x = \Ext^{-i}_{\mathcal{O}_{X, x}}(\mathcal{F}_x, \omega_{X, x}^\bullet) = \Ext^{\delta(x) - i}_{\mathcal{O}_{X, x}}(\mathcal{F}_x, \omega_{X, x}^\bullet[-\delta(x)])\] By construction of \(\delta\) in Lemma 0AWF this reduces parts (1), (2), and (3) to Dualizing Complexes, Lemma 0A7U. Part (4) is a formal consequence of (3) and (1).

Right adjoint of pushforward

References for this section and the following are [Neeman-Grothendieck], [LN], [Lipman-notes], and [Neeman-improvement].

Let \(f : X \to Y\) be a morphism of schemes. In this section we consider the right adjoint to the functor \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\). In the literature, if this functor exists, then it is sometimes denoted \(f^{\times}\). This notation is not universally accepted and we refrain from using it. We will not use the notation \(f^!\) for such a functor, as this would clash (for general morphisms \(f\)) with the notation in [RD].

Lemma

Let \(f : X \to Y\) be a morphism between quasi-separated and quasi-compact schemes. The functor \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) has a right adjoint.

Proof

We will prove a right adjoint exists by verifying the hypotheses of Derived Categories, Proposition 0A8G. First off, the category \(D_\QCoh(\mathcal{O}_X)\) has direct sums, see Derived Categories of Schemes, Lemma 08DT. The category \(D_\QCoh(\mathcal{O}_X)\) is compactly generated by Derived Categories of Schemes, Theorem 09IS. Since \(X\) and \(Y\) are quasi-compact and quasi-separated, so is \(f\), see Schemes, Lemmas 01KV and 03GI. Hence the functor \(Rf_*\) commutes with direct sums, see Derived Categories of Schemes, Lemma 08DZ. This finishes the proof.

Example

Let \(A \to B\) be a ring map. Let \(Y = \Spec(A)\) and \(X = \Spec(B)\) and \(f : X \to Y\) the morphism corresponding to \(A \to B\). Then \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) corresponds to restriction \(D(B) \to D(A)\) via the equivalences \(D(B) \to D_\QCoh(\mathcal{O}_X)\) and \(D(A) \to D_\QCoh(\mathcal{O}_Y)\). Hence the right adjoint corresponds to the functor \(K \longmapsto R\Hom(B, K)\) of Dualizing Complexes, Section 0A6Z.

Example

If \(f : X \to Y\) is a separated finite type morphism of Noetherian schemes, then the right adjoint of \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) does not map \(D_{\textit{Coh}}(\mathcal{O}_Y)\) into \(D_{\textit{Coh}}(\mathcal{O}_X)\). Namely, let \(k\) be a field and consider the morphism \(f : \mathbf{A}^1_k \to \Spec(k)\). By Example 0A9F this corresponds to the question of whether \(R\Hom(B, -)\) maps \(D_{\textit{Coh}}(A)\) into \(D_{\textit{Coh}}(B)\) where \(A = k\) and \(B = k[x]\). This is not true because \[R\Hom(k[x], k) = \left(\prod\nolimits_{n \geq 0} k\right)[0]\] which is not a finite \(k[x]\)-module. Hence \(a(\mathcal{O}_Y)\) does not have coherent cohomology sheaves.

Example

If \(f : X \to Y\) is a proper or even finite morphism of Noetherian schemes, then the right adjoint of \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) does not map \(D_\QCoh^-(\mathcal{O}_Y)\) into \(D_\QCoh^-(\mathcal{O}_X)\). Namely, let \(k\) be a field, let \(k[\epsilon]\) be the dual numbers over \(k\), let \(X = \Spec(k)\), and let \(Y = \Spec(k[\epsilon])\). Then \(\Ext^i_{k[\epsilon]}(k, k)\) is nonzero for all \(i \geq 0\). Hence \(a(\mathcal{O}_Y)\) is not bounded above by Example 0A9F.

Lemma

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)\) be the right adjoint to \(Rf_*\) of Lemma 0A9E. Then \(a\) maps \(D^+_\QCoh(\mathcal{O}_Y)\) into \(D^+_\QCoh(\mathcal{O}_X)\). In fact, there exists an integer \(N\) such that \(H^i(K) = 0\) for \(i \leq c\) implies \(H^i(a(K)) = 0\) for \(i \leq c - N\).

Proof

By Derived Categories of Schemes, Lemma 08D5 the functor \(Rf_*\) has finite cohomological dimension. In other words, there exist an integer \(N\) such that \(H^i(Rf_*L) = 0\) for \(i \geq N + c\) if \(H^i(L) = 0\) for \(i \geq c\). Say \(K \in D^+_\QCoh(\mathcal{O}_Y)\) has \(H^i(K) = 0\) for \(i \leq c\). Then \[\Hom_{D(\mathcal{O}_X)}(\tau_{\leq c - N}a(K), a(K)) = \Hom_{D(\mathcal{O}_Y)}(Rf_*\tau_{\leq c - N}a(K), K) = 0\] by what we said above. Clearly, this implies that \(H^i(a(K)) = 0\) for \(i \leq c - N\).

Let \(f : X \to Y\) be a morphism of quasi-separated and quasi-compact schemes. Let \(a\) denote the right adjoint to \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\). For every \(K \in D_\QCoh(\mathcal{O}_Y)\) and \(L \in D_\QCoh(\mathcal{O}_X)\) we obtain a canonical map [0B6H]\[\begin{equation} Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \longrightarrow R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K) \end{equation}\] Namely, this map is constructed as the composition \[Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*L, Rf_*a(K)) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)\] where the first arrow is Cohomology, Remark 0B69 and the second arrow is the counit \(Rf_*a(K) \to K\) of the adjunction.

Lemma

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(a\) be the right adjoint to \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\). Let \(L \in D_\QCoh(\mathcal{O}_X)\) and \(K \in D_\QCoh(\mathcal{O}_Y)\). Then the map (0B6H) \[Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \longrightarrow R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)\] becomes an isomorphism after applying the functor \(DQ_Y : D(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_Y)\) discussed in Derived Categories of Schemes, Section 0CQZ.

Proof

The statement makes sense as \(DQ_Y\) exists by Derived Categories of Schemes, Lemma 0CR0. Since \(DQ_Y\) is the right adjoint to the inclusion functor \(D_\QCoh(\mathcal{O}_Y) \to D(\mathcal{O}_Y)\) to prove the lemma we have to show that for any \(M \in D_\QCoh(\mathcal{O}_Y)\) the map (0B6H) induces an bijection \[\Hom_Y(M, Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K))) \longrightarrow \Hom_Y(M, R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K))\] To see this we use the following string of equalities \[\begin{align*} \Hom_Y(M, Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K))) & = \Hom_X(Lf^*M, R\SheafHom_{\mathcal{O}_X}(L, a(K))) \\ & = \Hom_X(Lf^*M \otimes_{\mathcal{O}_X}^\mathbf{L} L, a(K)) \\ & = \Hom_Y(Rf_*(Lf^*M \otimes_{\mathcal{O}_X}^\mathbf{L} L), K) \\ & = \Hom_Y(M \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L, K) \\ & = \Hom_Y(M, R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)) \end{align*}\] The first equality holds by Cohomology, Lemma 079W. The second equality by Cohomology, Lemma 08DJ. The third equality by construction of \(a\). The fourth equality by Derived Categories of Schemes, Lemma 08EU (this is the important step). The fifth by Cohomology, Lemma 08DJ.

Example

The statement of Lemma 0A9Q is not true without applying the “coherator” \(DQ_Y\). Indeed, suppose \(Y = \Spec(R)\) and \(X = \mathbf{A}^1_R\). Take \(L = \mathcal{O}_X\) and \(K = \mathcal{O}_Y\). The left hand side of the arrow is in \(D_\QCoh(\mathcal{O}_Y)\) but the right hand side of the arrow is isomorphic to \(\prod_{n \geq 0} \mathcal{O}_Y\) which is not quasi-coherent.

Remark

In the situation of Lemma 0A9Q we have \[DQ_Y(Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K))) = Rf_* DQ_X(R\SheafHom_{\mathcal{O}_X}(L, a(K)))\] by Derived Categories of Schemes, Lemma 0CR1. Thus if \(R\SheafHom_{\mathcal{O}_X}(L, a(K)) \in D_\QCoh(\mathcal{O}_X)\), then we can “erase” the \(DQ_Y\) on the left hand side of the arrow. On the other hand, if we know that \(R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K) \in D_\QCoh(\mathcal{O}_Y)\), then we can “erase” the \(DQ_Y\) from the right hand side of the arrow. If both are true then we see that (0B6H) is an isomorphism. Combining this with Derived Categories of Schemes, Lemma 0A6H we see that \(Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)\) is an isomorphism if

1. \(L\) and \(Rf_*L\) are perfect, or

2. \(K\) is bounded below and \(L\) and \(Rf_*L\) are pseudo-coherent.

For (2) we use that \(a(K)\) is bounded below if \(K\) is bounded below, see Lemma 0A9I.

Example

Let \(f : X \to Y\) be a proper morphism of Noetherian schemes, \(L \in D^-_{\textit{Coh}}(X)\) and \(K \in D^+_{\QCoh}(\mathcal{O}_Y)\). Then the map \(Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)\) is an isomorphism. Namely, the complexes \(L\) and \(Rf_*L\) are pseudo-coherent by Derived Categories of Schemes, Lemmas 08E8 and 08E2 and the discussion in Remark 0GEV applies.

Lemma

Let \(f : X \to Y\) be a morphism of quasi-separated and quasi-compact schemes. For all \(L \in D_\QCoh(\mathcal{O}_X)\) and \(K \in D_\QCoh(\mathcal{O}_Y)\) (0B6H) induces an isomorphism \(R\Hom_X(L, a(K)) \to R\Hom_Y(Rf_*L, K)\) of global derived homs.

Proof

By the construction in Cohomology, Section 0B6A we have \[R\Hom_X(L, a(K)) = R\Gamma(X, R\SheafHom_{\mathcal{O}_X}(L, a(K))) = R\Gamma(Y, Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)))\] and \[R\Hom_Y(Rf_*L, K) = R\Gamma(Y, R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K))\] Thus the lemma is a consequence of Lemma 0A9Q. Namely, a map \(E \to E'\) in \(D(\mathcal{O}_Y)\) which induces an isomorphism \(DQ_Y(E) \to DQ_Y(E')\) induces a quasi-isomorphism \(R\Gamma(Y, E) \to R\Gamma(Y, E')\). Indeed we have \(H^i(Y, E) = \Ext^i_Y(\mathcal{O}_Y, E) = \Hom(\mathcal{O}_Y[-i], E) = \Hom(\mathcal{O}_Y[-i], DQ_Y(E))\) because \(\mathcal{O}_Y[-i]\) is in \(D_\QCoh(\mathcal{O}_Y)\) and \(DQ_Y\) is the right adjoint to the inclusion functor \(D_\QCoh(\mathcal{O}_Y) \to D(\mathcal{O}_Y)\).

Right adjoint of pushforward and restriction to opens

In this section we study the question to what extent the right adjoint of pushforward commutes with restriction to open subschemes. This is a base change question, so let’s first discuss this more generally.

We often want to know whether the right adjoints to pushforward commutes with base change. Thus we consider a cartesian square [0A9J]\[\begin{equation} \vcenter{ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } } \end{equation}\] of quasi-compact and quasi-separated schemes. Denote \[a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X) \quad\text{and}\quad a' : D_\QCoh(\mathcal{O}_{Y'}) \to D_\QCoh(\mathcal{O}_{X'})\] the right adjoints to \(Rf_*\) and \(Rf'_*\) (Lemma 0A9E). Consider the base change map of Cohomology, Remark 08HY. It induces a transformation of functors \[Lg^* \circ Rf_* \longrightarrow Rf'_* \circ L(g')^*\] on derived categories of sheaves with quasi-coherent cohomology. Hence a transformation between the right adjoints in the opposite direction \[a \circ Rg_* \longleftarrow Rg'_* \circ a'\]

Lemma

In diagram (0A9J) assume that \(g\) is flat or more generally that \(f\) and \(g\) are Tor independent. Then \(a \circ Rg_* \leftarrow Rg'_* \circ a'\) is an isomorphism.

Proof

In this case the base change map \(Lg^* \circ Rf_* K \longrightarrow Rf'_* \circ L(g')^*K\) is an isomorphism for every \(K\) in \(D_\QCoh(\mathcal{O}_X)\) by Derived Categories of Schemes, Lemma 08IB. Thus the corresponding transformation between adjoint functors is an isomorphism as well.

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(V \subset Y\) be a quasi-compact open subscheme and set \(U = f^{-1}(V)\). This gives a cartesian square \[\xymatrix{ U \ar[r]_{j'} \ar[d]_{f|_U} & X \ar[d]^f \\ V \ar[r]^j & Y }\] as in (0A9J). By Lemma 0A9K the map \(\xi : a \circ Rj_* \leftarrow Rj'_* \circ a'\) is an isomorphism where \(a\) and \(a'\) are the right adjoints to \(Rf_*\) and \(R(f|_U)_*\). We obtain a transformation of functors \(D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_U)\) [0A9L]\[\begin{equation} (j')^* \circ a \to (j')^* \circ a \circ Rj_* \circ j^* \xrightarrow{\xi^{-1}} (j')^* \circ Rj'_* \circ a' \circ j^* \to a' \circ j^* \end{equation}\] where the first arrow comes from \(\text{id} \to Rj_* \circ j^*\) and the final arrow from the isomorphism \((j')^* \circ Rj'_* \to \text{id}\). In particular, we see that (0A9L) is an isomorphism when evaluated on \(K\) if and only if \(a(K)|_U \to a(Rj_*(K|_V))|_U\) is an isomorphism.

Example

There is a finite morphism \(f : X \to Y\) of Noetherian schemes such that (0A9L) is not an isomorphism when evaluated on some \(K \in D_{\textit{Coh}}(\mathcal{O}_Y)\). Namely, let \(X = \Spec(B) \to Y = \Spec(A)\) with \(A = k[x, \epsilon]\) where \(k\) is a field and \(\epsilon^2 = 0\) and \(B = k[x] = A/(\epsilon)\). For \(n \in \mathbf{N}\) set \(M_n = A/(\epsilon, x^n)\). Observe that \[\Ext^i_A(B, M_n) = M_n,\quad i \geq 0\] because \(B\) has the free periodic resolution \(\ldots \to A \to A \to A\) with maps given by multiplication by \(\epsilon\). Consider the object \(K = \bigoplus M_n[n] = \prod M_n[n]\) of \(D_{\textit{Coh}}(A)\) (equality in \(D(A)\) by Derived Categories, Lemmas 0A5L and 07KC). Then we see that \(a(K)\) corresponds to \(R\Hom(B, K)\) by Example 0A9F and \[H^0(R\Hom(B, K)) = \Ext^0_A(B, K) = \prod\nolimits_{n \geq 1} \Ext^n_A(B. M_n) = \prod\nolimits_{n \geq 1} M_n\] by the above. But this module has elements which are not annihilated by any power of \(x\), whereas the complex \(K\) does have every element of its cohomology annihilated by a power of \(x\). In other words, for the map (0A9L) with \(V = D(x)\) and \(U = D(x)\) and the complex \(K\) cannot be an isomorphism because \((j')^*(a(K))\) is nonzero and \(a'(j^*K)\) is zero.

Lemma

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(a\) be the right adjoint to \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\). Let \(V \subset Y\) be quasi-compact open with inverse image \(U \subset X\).

1. For every \(Q \in D_\QCoh^+(\mathcal{O}_Y)\) supported on \(Y \setminus V\) the image \(a(Q)\) is supported on \(X \setminus U\) if and only if (0A9L) is an isomorphism on all \(K\) in \(D_\QCoh^+(\mathcal{O}_Y)\).

2. For every \(Q \in D_\QCoh(\mathcal{O}_Y)\) supported on \(Y \setminus V\) the image \(a(Q)\) is supported on \(X \setminus U\) if and only if (0A9L) is an isomorphism on all \(K\) in \(D_\QCoh(\mathcal{O}_Y)\).

3. If \(a\) commutes with direct sums, then the equivalent conditions of (1) imply the equivalent conditions of (2).

Proof

Proof of (1). Let \(K \in D_\QCoh^+(\mathcal{O}_Y)\). Choose a distinguished triangle \[K \to Rj_*K|_V \to Q \to K[1]\] Observe that \(Q\) is in \(D_\QCoh^+(\mathcal{O}_Y)\) (Derived Categories of Schemes, Lemma 08D5) and is supported on \(Y \setminus V\) (Derived Categories of Schemes, Definition 08DA). Applying \(a\) we obtain a distinguished triangle \[a(K) \to a(Rj_*K|_V) \to a(Q) \to a(K)[1]\] on \(X\). If \(a(Q)\) is supported on \(X \setminus U\), then restricting to \(U\) the map \(a(K)|_U \to a(Rj_*K|_V)|_U\) is an isomorphism, i.e., (0A9L) is an isomorphism on \(K\). The converse is immediate.

The proof of (2) is exactly the same as the proof of (1).

Proof of (3). Assume the equivalent conditions of (1) hold. Set \(T = Y \setminus V\). We will use the notation \(D_{\QCoh, T}(\mathcal{O}_Y)\) and \(D_{\QCoh, f^{-1}(T)}(\mathcal{O}_X)\) to denote complexes whose cohomology sheaves are supported on \(T\) and \(f^{-1}(T)\). Since \(a\) commutes with direct sums, the strictly full, saturated, triangulated subcategory \(\mathcal{D}\) with objects \[\{Q \in D_{\QCoh, T}(\mathcal{O}_Y) \mid a(Q) \in D_{\QCoh, f^{-1}(T)}(\mathcal{O}_X)\}\] is preserved by direct sums and hence derived colimits. On the other hand, the category \(D_{\QCoh, T}(\mathcal{O}_Y)\) is generated by a perfect object \(E\) (see Derived Categories of Schemes, Lemma 0A9A). By assumption we see that \(E \in \mathcal{D}\). By Derived Categories, Lemma 09SN every object \(Q\) of \(D_{\QCoh, T}(\mathcal{O}_Y)\) is a derived colimit of a system \(Q_1 \to Q_2 \to Q_3 \to \ldots\) such that the cones of the transition maps are direct sums of shifts of \(E\). Arguing by induction we see that \(Q_n \in \mathcal{D}\) for all \(n\) and finally that \(Q\) is in \(\mathcal{D}\). Thus the equivalent conditions of (2) hold.

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a proper morphism. If³

1. \(f\) is flat and of finite presentation, or

2. \(Y\) is Noetherian

then the equivalent conditions of Lemma 0A9N part (1) hold for all quasi-compact opens \(V\) of \(Y\).

Proof

Let \(Q \in D^+_\QCoh(\mathcal{O}_Y)\) be supported on \(Y \setminus V\). To get a contradiction, assume that \(a(Q)\) is not supported on \(X \setminus U\). Then we can find a perfect complex \(P_U\) on \(U\) and a nonzero map \(P_U \to a(Q)|_U\) (follows from Derived Categories of Schemes, Theorem 09IS). Then using Derived Categories of Schemes, Lemma 09IM we may assume there is a perfect complex \(P\) on \(X\) and a map \(P \to a(Q)\) whose restriction to \(U\) is nonzero. By definition of \(a\) this map is adjoint to a map \(Rf_*P \to Q\).

The complex \(Rf_*P\) is pseudo-coherent. In case (1) this follows from Derived Categories of Schemes, Lemma 0CSD. In case (2) this follows from Derived Categories of Schemes, Lemmas 08E2 and 08E8. Thus we may apply Derived Categories of Schemes, Lemma 0A9C and get a map \(I \to \mathcal{O}_Y\) of perfect complexes whose restriction to \(V\) is an isomorphism such that the composition \(I \otimes^\mathbf{L}_{\mathcal{O}_Y} Rf_*P \to Rf_*P \to Q\) is zero. By Derived Categories of Schemes, Lemma 08EU we have \(I \otimes^\mathbf{L}_{\mathcal{O}_Y} Rf_*P = Rf_*(Lf^*I \otimes^\mathbf{L}_{\mathcal{O}_X} P)\). We conclude that the composition \[Lf^*I \otimes^\mathbf{L}_{\mathcal{O}_X} P \to P \to a(Q)\] is zero. However, the restriction to \(U\) is the map \(P|_U \to a(Q)|_U\) which we assumed to be nonzero. This contradiction finishes the proof.

Right adjoint of pushforward and base change, I

The map (0A9L) is a special case of a base change map. Namely, suppose that we have a cartesian diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] of quasi-compact and quasi-separated schemes, i.e., a diagram as in (0A9J). Assume \(f\) and \(g\) are Tor independent. Then we can consider the morphism of functors \(D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_{X'})\) given by the composition [0AA6]\[\begin{equation} L(g')^* \circ a \to L(g')^* \circ a \circ Rg_* \circ Lg^* \leftarrow L(g')^* \circ Rg'_* \circ a' \circ Lg^* \to a' \circ Lg^* \end{equation}\] The first arrow comes from the adjunction map \(\text{id} \to Rg_* Lg^*\) and the last arrow from the adjunction map \(L(g')^*Rg'_* \to \text{id}\). We need the assumption on Tor independence to invert the arrow in the middle, see Lemma 0A9K. Alternatively, we can think of (0AA6) by adjointness of \(L(g')^*\) and \(R(g')_*\) as a natural transformation \[a \to a \circ Rg_* \circ Lg^* \leftarrow Rg'_* \circ a' \circ Lg^*\] were again the second arrow is invertible. If \(M \in D_\QCoh(\mathcal{O}_X)\) and \(K \in D_\QCoh(\mathcal{O}_Y)\) then on Yoneda functors this map is given by \[\begin{align*} \Hom_X(M, a(K)) & = \Hom_Y(Rf_*M, K) \\ & \to \Hom_Y(Rf_*M, Rg_* Lg^*K) \\ & = \Hom_{Y'}(Lg^*Rf_*M, Lg^*K) \\ & \leftarrow \Hom_{Y'}(Rf'_* L(g')^*M, Lg^*K) \\ & = \Hom_{X'}(L(g')^*M, a'(Lg^*K)) \\ & = \Hom_X(M, Rg'_*a'(Lg^*K)) \end{align*}\] (were the arrow pointing left is invertible by the base change theorem given in Derived Categories of Schemes, Lemma 08IB) which makes things a little bit more explicit.

In this section we first prove that the base change map satisfies some natural compatibilities with regards to stacking squares as in Cohomology, Remarks 0ATL and 0ATM for the usual base change map. We suggest the reader skip the rest of this section on a first reading.

Lemma

Consider a commutative diagram \[\xymatrix{ X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z' \ar[r]^m & Z }\] of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where \(f\) and \(l\) as well as \(g\) and \(m\) are Tor independent. Then the maps (0AA6) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).

Proof

It follows from the assumptions that \(g \circ f\) and \(m\) are Tor independent (details omitted), hence the statement makes sense. In this proof we write \(k^*\) in place of \(Lk^*\) and \(f_*\) instead of \(Rf_*\). Let \(a\), \(b\), and \(c\) be the right adjoints of Lemma 0A9E for \(f\), \(g\), and \(g \circ f\) and similarly for the primed versions. The arrow corresponding to the top square is the composition \[\gamma_{top} : k^* \circ a \to k^* \circ a \circ l_* \circ l^* \xleftarrow{\xi_{top}} k^* \circ k_* \circ a' \circ l^* \to a' \circ l^*\] where \(\xi_{top} : k_* \circ a' \to a \circ l_*\) is an isomorphism (hence can be inverted) and is the arrow “dual” to the base change map \(l^* \circ f_* \to f'_* \circ k^*\). The outer arrows come from the canonical maps \(1 \to l_* \circ l^*\) and \(k^* \circ k_* \to 1\). Similarly for the second square we have \[\gamma_{bot} : l^* \circ b \to l^* \circ b \circ m_* \circ m^* \xleftarrow{\xi_{bot}} l^* \circ l_* \circ b' \circ m^* \to b' \circ m^*\] For the outer rectangle we get \[\gamma_{rect} : k^* \circ c \to k^* \circ c \circ m_* \circ m^* \xleftarrow{\xi_{rect}} k^* \circ k_* \circ c' \circ m^* \to c' \circ m^*\] We have \((g \circ f)_* = g_* \circ f_*\) and hence \(c = a \circ b\) and similarly \(c' = a' \circ b'\). The statement of the lemma is that \(\gamma_{rect}\) is equal to the composition \[k^* \circ c = k^* \circ a \circ b \xrightarrow{\gamma_{top}} a' \circ l^* \circ b \xrightarrow{\gamma_{bot}} a' \circ b' \circ m^* = c' \circ m^*\] To see this we contemplate the following diagram: \[\xymatrix{ & & k^* \circ a \circ b \ar[d] \ar[lldd] \\ & & k^* \circ a \circ l_* \circ l^* \circ b \ar[ld] \\ k^* \circ a \circ b \circ m_* \circ m^* \ar[r] & k^* \circ a \circ l_* \circ l^* \circ b \circ m_* \circ m^* & k^* \circ k_* \circ a' \circ l^* \circ b \ar[u]_{\xi_{top}} \ar[d] \ar[ld] \\ & k^*\circ k_* \circ a' \circ l^* \circ b \circ m_* \circ m^* \ar[u]_{\xi_{top}} \ar[rd] & a' \circ l^* \circ b \ar[d] \\ k^* \circ k_* \circ a' \circ b' \circ m^* \ar[uu]_{\xi_{rect}} \ar[ddrr] & k^*\circ k_* \circ a' \circ l^* \circ l_* \circ b' \circ m^* \ar[u]_{\xi_{bot}} \ar[l] \ar[dr] & a' \circ l^* \circ b \circ m_* \circ m^* \\ & & a' \circ l^* \circ l_* \circ b' \circ m^* \ar[u]_{\xi_{bot}} \ar[d] \\ & & a' \circ b' \circ m^* }\] Going down the right hand side we have the composition and going down the left hand side we have \(\gamma_{rect}\). All the quadrilaterals on the right hand side of this diagram commute by Categories, Lemma 003F or more simply the discussion preceding Categories, Definition 003E. Hence we see that it suffices to show the diagram \[\xymatrix{ a \circ l_* \circ l^* \circ b \circ m_* & a \circ b \circ m_* \ar[l] \\ k_* \circ a' \circ l^* \circ b \circ m_* \ar[u]_{\xi_{top}} & \\ k_* \circ a' \circ l^* \circ l_* \circ b' \ar[u]_{\xi_{bot}} \ar[r] & k_* \circ a' \circ b' \ar[uu]_{\xi_{rect}} }\] becomes commutative if we invert the arrows \(\xi_{top}\), \(\xi_{bot}\), and \(\xi_{rect}\) (note that this is different from asking the diagram to be commutative). However, the diagram \[\xymatrix{ & a \circ l_* \circ l^* \circ b \circ m_* \\ a \circ l_* \circ l^* \circ l_* \circ b' \ar[ru]^{\xi_{bot}} & & k_* \circ a' \circ l^* \circ b \circ m_* \ar[ul]_{\xi_{top}} \\ & k_* \circ a' \circ l^* \circ l_* \circ b' \ar[ul]^{\xi_{top}} \ar[ur]_{\xi_{bot}} }\] commutes by Categories, Lemma 003F. Since the diagrams \[\vcenter{ \xymatrix{ a \circ l_* \circ l^* \circ b \circ m_* & a \circ b \circ m \ar[l] \\ a \circ l_* \circ l^* \circ l_* \circ b' \ar[u] & a \circ l_* \circ b' \ar[l] \ar[u] } } \quad\text{and}\quad \vcenter{ \xymatrix{ a \circ l_* \circ l^* \circ l_* \circ b' \ar[r] & a \circ l_* \circ b' \\ k_* \circ a' \circ l^* \circ l_* \circ b' \ar[u] \ar[r] & k_* \circ a' \circ b' \ar[u] } }\] commute (see references cited) and since the composition of \(l_* \to l_* \circ l^* \circ l_* \to l_*\) is the identity, we find that it suffices to prove that \[k \circ a' \circ b' \xrightarrow{\xi_{bot}} a \circ l_* \circ b \xrightarrow{\xi_{top}} a \circ b \circ m_*\] is equal to \(\xi_{rect}\) (via the identifications \(a \circ b = c\) and \(a' \circ b' = c'\)). This is the statement dual to Cohomology, Remark 0ATL and the proof is complete.

Lemma

Consider a commutative diagram \[\xymatrix{ X'' \ar[r]_{g'} \ar[d]_{f''} & X' \ar[r]_g \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{h'} & Y' \ar[r]^h & Y }\] of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where \(f\) and \(h\) as well as \(f'\) and \(h'\) are Tor independent. Then the maps (0AA6) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).

Proof

It follows from the assumptions that \(f\) and \(h \circ h'\) are Tor independent (details omitted), hence the statement makes sense. In this proof we write \(g^*\) in place of \(Lg^*\) and \(f_*\) instead of \(Rf_*\). Let \(a\), \(a'\), and \(a''\) be the right adjoints of Lemma 0A9E for \(f\), \(f'\), and \(f''\). The arrow corresponding to the right square is the composition \[\gamma_{right} : g^* \circ a \to g^* \circ a \circ h_* \circ h^* \xleftarrow{\xi_{right}} g^* \circ g_* \circ a' \circ h^* \to a' \circ h^*\] where \(\xi_{right} : g_* \circ a' \to a \circ h_*\) is an isomorphism (hence can be inverted) and is the arrow “dual” to the base change map \(h^* \circ f_* \to f'_* \circ g^*\). The outer arrows come from the canonical maps \(1 \to h_* \circ h^*\) and \(g^* \circ g_* \to 1\). Similarly for the left square we have \[\gamma_{left} : (g')^* \circ a' \to (g')^* \circ a' \circ (h')_* \circ (h')^* \xleftarrow{\xi_{left}} (g')^* \circ (g')_* \circ a'' \circ (h')^* \to a'' \circ (h')^*\] For the outer rectangle we get \[\gamma_{rect} : k^* \circ a \to k^* \circ a \circ m_* \circ m^* \xleftarrow{\xi_{rect}} k^* \circ k_* \circ a'' \circ m^* \to a'' \circ m^*\] where \(k = g \circ g'\) and \(m = h \circ h'\). We have \(k^* = (g')^* \circ g^*\) and \(m^* = (h')^* \circ h^*\). The statement of the lemma is that \(\gamma_{rect}\) is equal to the composition \[k^* \circ a = (g')^* \circ g^* \circ a \xrightarrow{\gamma_{right}} (g')^* \circ a' \circ h^* \xrightarrow{\gamma_{left}} a'' \circ (h')^* \circ h^* = a'' \circ m^*\] To see this we contemplate the following diagram \[\xymatrix{ & (g')^* \circ g^* \circ a \ar[d] \ar[ddl] \\ & (g')^* \circ g^* \circ a \circ h_* \circ h^* \ar[ld] \\ (g')^* \circ g^* \circ a \circ h_* \circ (h')_* \circ (h')^* \circ h^* & (g')^* \circ g^* \circ g_* \circ a' \circ h^* \ar[u]_{\xi_{right}} \ar[d] \ar[ld] \\ (g')^* \circ g^* \circ g_* \circ a' \circ (h')_* \circ (h')^* \circ h^* \ar[u]_{\xi_{right}} \ar[dr] & (g')^* \circ a' \circ h^* \ar[d] \\ (g')^* \circ g^* \circ g_* \circ (g')_* \circ a'' \circ (h')^* \circ h^* \ar[u]_{\xi_{left}} \ar[ddr] \ar[dr] & (g')^* \circ a' \circ (h')_* \circ (h')^* \circ h^* \\ & (g')^*\circ (g')_* \circ a'' \circ (h')^* \circ h^* \ar[u]_{\xi_{left}} \ar[d] \\ & a'' \circ (h')^* \circ h^* }\] Going down the right hand side we have the composition and going down the left hand side we have \(\gamma_{rect}\). All the quadrilaterals on the right hand side of this diagram commute by Categories, Lemma 003F or more simply the discussion preceding Categories, Definition 003E. Hence we see that it suffices to show that \[g_* \circ (g')_* \circ a'' \xrightarrow{\xi_{left}} g_* \circ a' \circ (h')_* \xrightarrow{\xi_{right}} a \circ h_* \circ (h')_*\] is equal to \(\xi_{rect}\). This is the statement dual to Cohomology, Remark 0ATM and the proof is complete.

Remark

Consider a commutative diagram \[\xymatrix{ X'' \ar[r]_{k'} \ar[d]_{f''} & X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{l'} \ar[d]_{g''} & Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z'' \ar[r]^{m'} & Z' \ar[r]^m & Z }\] of quasi-compact and quasi-separated schemes where all squares are cartesian and where \((f, l)\), \((g, m)\), \((f', l')\), \((g', m')\) are Tor independent pairs of maps. Let \(a\), \(a'\), \(a''\), \(b\), \(b'\), \(b''\) be the right adjoints of Lemma 0A9E for \(f\), \(f'\), \(f''\), \(g\), \(g'\), \(g''\). Let us label the squares of the diagram \(A\), \(B\), \(C\), \(D\) as follows \[\begin{matrix} A & B \\ C & D \end{matrix}\] Then the maps (0AA6) for the squares are (where we use \(k^* = Lk^*\), etc) \[\begin{matrix} \gamma_A : (k')^* \circ a' \to a'' \circ (l')^* & \gamma_B : k^* \circ a \to a' \circ l^* \\ \gamma_C : (l')^* \circ b' \to b'' \circ (m')^* & \gamma_D : l^* \circ b \to b' \circ m^* \end{matrix}\] For the \(2 \times 1\) and \(1 \times 2\) rectangles we have four further base change maps \[\begin{matrix} \gamma_{A + B} : (k \circ k')^* \circ a \to a'' \circ (l \circ l')^* \\ \gamma_{C + D} : (l \circ l')^* \circ b \to b'' \circ (m \circ m')^* \\ \gamma_{A + C} : (k')^* \circ (a' \circ b') \to (a'' \circ b'') \circ (m')^* \\ \gamma_{B + D} : k^* \circ (a \circ b) \to (a' \circ b') \circ m^* \end{matrix}\] By Lemma 0ATR we have \[\gamma_{A + B} = \gamma_A \circ \gamma_B, \quad \gamma_{C + D} = \gamma_C \circ \gamma_D\] and by Lemma 0ATQ we have \[\gamma_{A + C} = \gamma_C \circ \gamma_A, \quad \gamma_{B + D} = \gamma_D \circ \gamma_B\] Here it would be more correct to write \(\gamma_{A + B} = (\gamma_A \star \text{id}_{l^*}) \circ (\text{id}_{(k')^*} \star \gamma_B)\) with notation as in Categories, Section 003D and similarly for the others. However, we continue the abuse of notation used in the proofs of Lemmas 0ATQ and 0ATR of dropping \(\star\) products with identities as one can figure out which ones to add as long as the source and target of the transformation is known. Having said all of this we find (a priori) two transformations \[(k')^* \circ k^* \circ a \circ b \longrightarrow a'' \circ b'' \circ (m')^* \circ m^*\] namely \[\gamma_C \circ \gamma_A \circ \gamma_D \circ \gamma_B = \gamma_{A + C} \circ \gamma_{B + D}\] and \[\gamma_C \circ \gamma_D \circ \gamma_A \circ \gamma_B = \gamma_{C + D} \circ \gamma_{A + B}\] The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that \[\xymatrix{ (k')^* \circ a' \circ l^* \circ b \ar[r]_{\gamma_D} \ar[d]_{\gamma_A} & (k')^* \circ a' \circ b' \circ m^* \ar[d]^{\gamma_A} \\ a'' \circ (l')^* \circ l^* \circ b \ar[r]^{\gamma_D} & a'' \circ (l')^* \circ b' \circ m^* }\] commutes. This is true by Categories, Lemma 003F or more simply the discussion preceding Categories, Definition 003E.

Right adjoint of pushforward and base change, II

In this section we prove that the base change map of Section 0AA5 is an isomorphism in some cases. We first observe that it suffices to check over affine opens, provided formation of the right adjoint of pushforward commutes with restriction to opens.

Remark

Consider a cartesian diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] of quasi-compact and quasi-separated schemes with \((g, f)\) Tor independent. Let \(V \subset Y\) and \(V' \subset Y'\) be affine opens with \(g(V') \subset V\). Form the cartesian diagrams \[\vcenter{ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & Y } } \quad\text{and}\quad \vcenter{ \xymatrix{ U' \ar[r] \ar[d] & X' \ar[d] \\ V' \ar[r] & Y' } }\] Assume (0A9L) with respect to \(K\) and the first diagram and (0A9L) with respect to \(Lg^*K\) and the second diagram are isomorphisms. Then the restriction of the base change map (0AA6) \[L(g')^*a(K) \longrightarrow a'(Lg^*K)\] to \(U'\) is isomorphic to the base change map (0AA6) for \(K|_V\) and the cartesian diagram \[\xymatrix{ U' \ar[r] \ar[d] & U \ar[d] \\ V' \ar[r] & V }\] This follows from the fact that (0A9L) is a special case of the base change map (0AA6) and that the base change maps compose correctly if we stack squares horizontally, see Lemma 0ATR. Thus in order to check the base change map restricted to \(U'\) is an isomorphism it suffices to work with the last diagram.

Lemma

In diagram (0A9J) assume

1. \(g : Y' \to Y\) is a morphism of affine schemes,

2. \(f : X \to Y\) is proper, and

3. \(f\) and \(g\) are Tor independent.

Then the base change map (0AA6) induces an isomorphism \[L(g')^*a(K) \longrightarrow a'(Lg^*K)\] in the following cases

1. for all \(K \in D_\QCoh(\mathcal{O}_Y)\) if \(f\) is flat of finite presentation,

2. for all \(K \in D_\QCoh(\mathcal{O}_Y)\) if \(f\) is perfect and \(Y\) Noetherian,

3. for \(K \in D_\QCoh^+(\mathcal{O}_Y)\) if \(g\) has finite Tor dimension and \(Y\) Noetherian.

Proof

Write \(Y = \Spec(A)\) and \(Y' = \Spec(A')\). As a base change of an affine morphism, the morphism \(g'\) is affine. Let \(M\) be a perfect generator for \(D_\QCoh(\mathcal{O}_X)\), see Derived Categories of Schemes, Theorem 09IS. Then \(L(g')^*M\) is a generator for \(D_\QCoh(\mathcal{O}_{X'})\), see Derived Categories of Schemes, Remark 0BQT. Hence it suffices to show that (0AA6) induces an isomorphism [0E45]\[\begin{equation} R\Hom_{X'}(L(g')^*M, L(g')^*a(K)) \longrightarrow R\Hom_{X'}(L(g')^*M, a'(Lg^*K)) \end{equation}\] of global hom complexes, see Cohomology, Section 0B6A, as this will imply the cone of \(L(g')^*a(K) \to a'(Lg^*K)\) is zero. The structure of the proof is as follows: we will first show that these Hom complexes are isomorphic and in the last part of the proof we will show that the isomorphism is induced by (0E45).

The left hand side. Because \(M\) is perfect, the canonical map \[R\Hom_X(M, a(K)) \otimes^\mathbf{L}_A A' \longrightarrow R\Hom_{X'}(L(g')^*M, L(g')^*a(K))\] is an isomorphism by Derived Categories of Schemes, Lemma 0AA7. We can combine this with the isomorphism \(R\Hom_Y(Rf_*M, K) = R\Hom_X(M, a(K))\) of Lemma 0B6I to get that the left hand side equals \(R\Hom_Y(Rf_*M, K) \otimes^\mathbf{L}_A A'\).

The right hand side. Here we first use the isomorphism \[R\Hom_{X'}(L(g')^*M, a'(Lg^*K)) = R\Hom_{Y'}(Rf'_*L(g')^*M, Lg^*K)\] of Lemma 0B6I. Then we use the base change map \(Lg^*Rf_*M \to Rf'_*L(g')^*M\) is an isomorphism by Derived Categories of Schemes, Lemma 08IB. Hence we may rewrite this as \(R\Hom_{Y'}(Lg^*Rf_*M, Lg^*K)\). Since \(Y\), \(Y'\) are affine and \(K\), \(Rf_*M\) are in \(D_\QCoh(\mathcal{O}_Y)\) (Derived Categories of Schemes, Lemma 08D5) we have a canonical map \[\beta : R\Hom_Y(Rf_*M, K) \otimes^\mathbf{L}_A A' \longrightarrow R\Hom_{Y'}(Lg^*Rf_*M, Lg^*K)\] in \(D(A')\). This is the arrow More on Algebra, Equation (0E1X) where we have used Derived Categories of Schemes, Lemmas 06Z0 and 0A6H to translate back and forth into algebra.

1. If \(f\) is flat and of finite presentation, the complex \(Rf_*M\) is perfect on \(Y\) by Derived Categories of Schemes, Lemma 0B91 and \(\beta\) is an isomorphism by More on Algebra, Lemma 0A6A part (1).

2. If \(f\) is perfect and \(Y\) Noetherian, the complex \(Rf_*M\) is perfect on \(Y\) by More on Morphisms, Lemma 0B6G and \(\beta\) is an isomorphism as before.

3. If \(g\) has finite tor dimension and \(Y\) is Noetherian, the complex \(Rf_*M\) is pseudo-coherent on \(Y\) (Derived Categories of Schemes, Lemmas 08E2 and 08E8) and \(\beta\) is an isomorphism by More on Algebra, Lemma 0A6A part (4).

We conclude that we obtain the same answer as in the previous paragraph.

In the rest of the proof we show that the identifications of the left and right hand side of (0E45) given in the second and third paragraph are in fact given by (0E45). To make our formulas manageable we will use \((-, -)_X = R\Hom_X(-, -)\), use \(- \otimes A'\) in stead of \(- \otimes_A^\mathbf{L} A'\), and we will abbreviate \(g^* = Lg^*\) and \(f_* = Rf_*\). Consider the following commutative diagram \[\xymatrix{ ((g')^*M, (g')^*a(K))_{X'} \ar[d] & (M, a(K))_X \otimes A' \ar[l]^-\alpha \ar[d] & (f_*M, K)_Y \otimes A' \ar@{=}[l] \ar[d] \\ ((g')^*M, (g')^*a(g_*g^*K))_{X'} & (M, a(g_*g^*K))_X \otimes A' \ar[l]^-\alpha & (f_*M, g_*g^*K)_Y \otimes A' \ar@{=}[l] \ar@/_4pc/[dd]_{\mu'} \\ ((g')^*M, (g')^*g'_*a'(g^*K))_{X'} \ar[u] \ar[d] & (M, g'_*a'(g^*K))_X \otimes A' \ar[u] \ar[l]^-\alpha \ar[ld]^\mu & (f_*M, K) \otimes A' \ar[d]^\beta \\ ((g')^*M, a'(g^*K))_{X'} & (f'_*(g')^*M, g^*K)_{Y'} \ar@{=}[l] \ar[r] & (g^*f_*M, g^*K)_{Y'} }\] The arrows labeled \(\alpha\) are the maps from Derived Categories of Schemes, Lemma 0AA7 for the diagram with corners \(X', X, Y', Y\). The upper part of the diagram is commutative as the horizontal arrows are functorial in the entries. The middle vertical arrows come from the invertible transformation \(g'_* \circ a' \to a \circ g_*\) of Lemma 0A9K and therefore the middle square is commutative. Going down the left hand side is (0E45). The upper horizontal arrows provide the identifications used in the second paragraph of the proof. The lower horizontal arrows including \(\beta\) provide the identifications used in the third paragraph of the proof. Given \(E \in D(A)\), \(E' \in D(A')\), and \(c : E \to E'\) in \(D(A)\) we will denote \(\mu_c : E \otimes A' \to E'\) the map induced by \(c\) and the adjointness of restriction and base change; if \(c\) is clear we write \(\mu = \mu_c\), i.e., we drop \(c\) from the notation. The map \(\mu\) in the diagram is of this form with \(c\) given by the identification \((M, g'_*a(g^*K))_X = ((g')^*M, a'(g^*K))_{X'}\) ; the triangle involving \(\mu\) is commutative by Derived Categories of Schemes, Remark 0BZA.

Observe that \[\xymatrix{ (M, a(g_*g^*K))_X & (f_*M, g_* g^*K)_Y \ar@{=}[l] & (g^*f_*M, g^*K)_{Y'} \ar@{=}[l] \\ (M, g'_* a'(g^*K))_X \ar[u] & ((g')^*M, a'(g^*K))_{X'} \ar@{=}[l] & (f'_*(g')^*M, g^*K)_{Y'} \ar@{=}[l] \ar[u] }\] is commutative by the very definition of the transformation \(g'_* \circ a' \to a \circ g_*\). Letting \(\mu'\) be as above corresponding to the identification \((f_*M, g_*g^*K)_X = (g^*f_*M, g^*K)_{Y'}\), then the hexagon commutes as well. Thus it suffices to show that \(\beta\) is equal to the composition of \((f_*M, K)_Y \otimes A' \to (f_*M, g_*g^*K)_X \otimes A'\) and \(\mu'\). To do this, it suffices to prove the two induced maps \((f_*M, K)_Y \to (g^*f_*M, g^*K)_{Y'}\) are the same. In other words, it suffices to show the diagram \[\xymatrix{ R\Hom_A(E, K) \ar[rr]_{\text{induced by }\beta} \ar[rd] & & R\Hom_{A'}(E \otimes_A^\mathbf{L} A', K \otimes_A^\mathbf{L} A') \\ & R\Hom_A(E, K \otimes_A^\mathbf{L} A') \ar[ru] }\] commutes for all \(E, K \in D(A)\). Since this is how \(\beta\) is constructed in More on Algebra, Section 0E1V the proof is complete.

Right adjoint of pushforward and trace maps

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)\) be the right adjoint as in Lemma 0A9E. By Categories, Section 0036 we obtain a transformation of functors \[\text{Tr}_f : Rf_* \circ a \longrightarrow \text{id}\] The corresponding map \(\text{Tr}_{f, K} : Rf_*a(K) \longrightarrow K\) for \(K \in D_\QCoh(\mathcal{O}_Y)\) is sometimes called the trace map. This is the map which has the property that the bijection \[\Hom_X(L, a(K)) \longrightarrow \Hom_Y(Rf_*L, K)\] for \(L \in D_\QCoh(\mathcal{O}_X)\) which characterizes the right adjoint is given by \[\varphi \longmapsto \text{Tr}_{f, K} \circ Rf_*\varphi\] The map (0B6H) \[Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \longrightarrow R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)\] comes about by composition with \(\text{Tr}_{f, K}\). Every trace map we are going to consider in this section will be a special case of this trace map. Before we discuss some special cases we show that formation of the trace map commutes with base change.

Lemma

Suppose we have a diagram (0A9J) where \(f\) and \(g\) are tor independent. Then the maps \(1 \star \text{Tr}_f : Lg^* \circ Rf_* \circ a \to Lg^*\) and \(\text{Tr}_{f'} \star 1 : Rf'_* \circ a' \circ Lg^* \to Lg^*\) agree via the base change maps \(\beta : Lg^* \circ Rf_* \to Rf'_* \circ L(g')^*\) (Cohomology, Remark 08HY) and \(\alpha : L(g')^* \circ a \to a' \circ Lg^*\) (0AA6). More precisely, the diagram \[\xymatrix{ Lg^* \circ Rf_* \circ a \ar[d]_{\beta \star 1} \ar[r]_-{1 \star \text{Tr}_f} & Lg^* \\ Rf'_* \circ L(g')^* \circ a \ar[r]^{1 \star \alpha} & Rf'_* \circ a' \circ Lg^* \ar[u]_{\text{Tr}_{f'} \star 1} }\] of transformations of functors commutes.

Proof

In this proof we write \(f_*\) for \(Rf_*\) and \(g^*\) for \(Lg^*\) and we drop \(\star\) products with identities as one can figure out which ones to add as long as the source and target of the transformation is known. Recall that \(\beta : g^* \circ f_* \to f'_* \circ (g')^*\) is an isomorphism and that \(\alpha\) is defined using the isomorphism \(\beta^\vee : g'_* \circ a' \to a \circ g_*\) which is the adjoint of \(\beta\), see Lemma 0A9K and its proof. First we note that the top horizontal arrow of the diagram in the lemma is equal to the composition \[g^* \circ f_* \circ a \to g^* \circ f_* \circ a \circ g_* \circ g^* \to g^* \circ g_* \circ g^* \to g^*\] where the first arrow is the unit for \((g^*, g_*)\), the second arrow is \(\text{Tr}_f\), and the third arrow is the counit for \((g^*, g_*)\). This is a simple consequence of the fact that the composition \(g^* \to g^* \circ g_* \circ g^* \to g^*\) of unit and counit is the identity. Consider the diagram \[\xymatrix{ & g^* \circ f_* \circ a \ar[ld]_\beta \ar[d] \ar[r]_{\text{Tr}_f} & g^* \\ f'_* \circ (g')^* \circ a \ar[dr] & g^* \circ f_* \circ a \circ g_* \circ g^* \ar[d]_\beta \ar[ru] & g^* \circ f_* \circ g'_* \circ a' \circ g^* \ar[l]_{\beta^\vee} \ar[d]_\beta & f'_* \circ a' \circ g^* \ar[lu]_{\text{Tr}_{f'}} \\ & f'_* \circ (g')^* \circ a \circ g_* \circ g^* & f'_* \circ (g')^* \circ g'_* \circ a' \circ g^* \ar[ru] \ar[l]_{\beta^\vee} }\] In this diagram the two squares commute Categories, Lemma 003F or more simply the discussion preceding Categories, Definition 003E. The triangle commutes by the discussion above. By Categories, Lemma 0B65 the square \[\xymatrix{ g^* \circ f_* \circ g'_* \circ a' \ar[d]_{\beta^\vee} \ar[r]_-\beta & f'_* \circ (g')^* \circ g'_* \circ a' \ar[d] \\ g^* \circ f_* \circ a \circ g_* \ar[r] & \text{id} }\] commutes which implies the pentagon in the big diagram commutes. Since \(\beta\) and \(\beta^\vee\) are isomorphisms, and since going on the outside of the big diagram equals \(\text{Tr}_f \circ \alpha \circ \beta\) by definition this proves the lemma.

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)\) be the right adjoint of \(Rf_*\) as in Lemma 0A9E. By Categories, Section 0036 we obtain a transformation of functors \[\eta_f : \text{id} \to a \circ Rf_*\] which is called the unit of the adjunction.

Lemma

Suppose we have a diagram (0A9J) where \(f\) and \(g\) are tor independent. Then the maps \(1 \star \eta_f : L(g')^* \to L(g')^* \circ a \circ Rf_*\) and \(\eta_{f'} \star 1 : L(g')^* \to a' \circ Rf'_* \circ L(g')^*\) agree via the base change maps \(\beta : Lg^* \circ Rf_* \to Rf'_* \circ L(g')^*\) (Cohomology, Remark 08HY) and \(\alpha : L(g')^* \circ a \to a' \circ Lg^*\) (0AA6). More precisely, the diagram \[\xymatrix{ L(g')^* \ar[r]_-{1 \star \eta_f} \ar[d]_{\eta_{f'} \star 1} & L(g')^* \circ a \circ Rf_* \ar[d]^\alpha \\ a' \circ Rf'_* \circ L(g')^* & a' \circ Lg^* \circ Rf_* \ar[l]_-\beta }\] of transformations of functors commutes.

Proof

This proof is dual to the proof of Lemma 0B6J. In this proof we write \(f_*\) for \(Rf_*\) and \(g^*\) for \(Lg^*\) and we drop \(\star\) products with identities as one can figure out which ones to add as long as the source and target of the transformation is known. Recall that \(\beta : g^* \circ f_* \to f'_* \circ (g')^*\) is an isomorphism and that \(\alpha\) is defined using the isomorphism \(\beta^\vee : g'_* \circ a' \to a \circ g_*\) which is the adjoint of \(\beta\), see Lemma 0A9K and its proof. First we note that the left vertical arrow of the diagram in the lemma is equal to the composition \[(g')^* \to (g')^* \circ g'_* \circ (g')^* \to (g')^* \circ g'_* \circ a' \circ f'_* \circ (g')^* \to a' \circ f'_* \circ (g')^*\] where the first arrow is the unit for \(((g')^*, g'_*)\), the second arrow is \(\eta_{f'}\), and the third arrow is the counit for \(((g')^*, g'_*)\). This is a simple consequence of the fact that the composition \((g')^* \to (g')^* \circ (g')_* \circ (g')^* \to (g')^*\) of unit and counit is the identity. Consider the diagram \[\xymatrix{ & (g')^* \circ a \circ f_* \ar[r] & (g')^* \circ a \circ g_* \circ g^* \circ f_* \ar[ld]_\beta \\ (g')^* \ar[ru]^{\eta_f} \ar[dd]_{\eta_{f'}} \ar[rd] & (g')^* \circ a \circ g_* \circ f'_* \circ (g')^* & (g')^* \circ g'_* \circ a' \circ g^* \circ f_* \ar[u]_{\beta^\vee} \ar[ld]_\beta \ar[d] \\ & (g')^* \circ g'_* \circ a' \circ f'_* \circ (g')^* \ar[ld] \ar[u]_{\beta^\vee} & a' \circ g^* \circ f_* \ar[lld]^\beta \\ a' \circ f'_* \circ (g')^* }\] In this diagram the two squares commute Categories, Lemma 003F or more simply the discussion preceding Categories, Definition 003E. The triangle commutes by the discussion above. By the dual of Categories, Lemma 0B65 the square \[\xymatrix{ \text{id} \ar[r] \ar[d] & g'_* \circ a' \circ g^* \circ f_* \ar[d]^\beta \\ g'_* \circ a' \circ g^* \circ f_* \ar[r]^{\beta^\vee} & a \circ g_* \circ f'_* \circ (g')^* }\] commutes which implies the pentagon in the big diagram commutes. Since \(\beta\) and \(\beta^\vee\) are isomorphisms, and since going on the outside of the big diagram equals \(\beta \circ \alpha \circ \eta_f\) by definition this proves the lemma.

Example

Let \(A \to B\) be a ring map. Let \(Y = \Spec(A)\) and \(X = \Spec(B)\) and \(f : X \to Y\) the morphism corresponding to \(A \to B\). As seen in Example 0A9F the right adjoint of \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) sends an object \(K\) of \(D(A) = D_\QCoh(\mathcal{O}_Y)\) to \(R\Hom(B, K)\) in \(D(B) = D_\QCoh(\mathcal{O}_X)\). The trace map is the map \[\text{Tr}_{f, K} : R\Hom(B, K) \longrightarrow R\Hom(A, K) = K\] induced by the \(A\)-module map \(A \to B\).

Right adjoint of pushforward and pullback

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(a\) be the right adjoint of pushforward as in Lemma 0A9E. For \(K, L \in D_\QCoh(\mathcal{O}_Y)\) there is a canonical map \[Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(L) \longrightarrow a(K \otimes_{\mathcal{O}_Y}^\mathbf{L} L)\] Namely, this map is adjoint to a map \[Rf_*(Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(L)) = K \otimes^\mathbf{L}_{\mathcal{O}_Y} Rf_*(a(L)) \longrightarrow K \otimes^\mathbf{L}_{\mathcal{O}_Y} L\] (equality by Derived Categories of Schemes, Lemma 08EU) for which we use the trace map \(Rf_*a(L) \to L\). When \(L = \mathcal{O}_Y\) we obtain a map [0A9S]\[\begin{equation} Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(\mathcal{O}_Y) \longrightarrow a(K) \end{equation}\] functorial in \(K\) and compatible with distinguished triangles.

Lemma

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. The map \(Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(L) \to a(K \otimes_{\mathcal{O}_Y}^\mathbf{L} L)\) defined above for \(K, L \in D_\QCoh(\mathcal{O}_Y)\) is an isomorphism if \(K\) is perfect. In particular, (0A9S) is an isomorphism if \(K\) is perfect.

Proof

Let \(K^\vee\) be the “dual” to \(K\), see Cohomology, Lemma 08DQ. For \(M \in D_\QCoh(\mathcal{O}_X)\) we have \[\begin{align*} \Hom_{D(\mathcal{O}_Y)}(Rf_*M, K \otimes^\mathbf{L}_{\mathcal{O}_Y} L) & = \Hom_{D(\mathcal{O}_Y)}( Rf_*M \otimes^\mathbf{L}_{\mathcal{O}_Y} K^\vee, L) \\ & = \Hom_{D(\mathcal{O}_X)}( M \otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*K^\vee, a(L)) \\ & = \Hom_{D(\mathcal{O}_X)}(M, Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(L)) \end{align*}\] Second equality by the definition of \(a\) and the projection formula (Cohomology, Lemma 0B54) or the more general Derived Categories of Schemes, Lemma 08EU. Hence the result by the Yoneda lemma.

Lemma

Suppose we have a diagram (0A9J) where \(f\) and \(g\) are tor independent. Let \(K \in D_\QCoh(\mathcal{O}_Y)\). The diagram \[\xymatrix{ L(g')^*(Lf^*K \otimes^\mathbf{L}_{\mathcal{O}_X} a(\mathcal{O}_Y)) \ar[r] \ar[d] & L(g')^*a(K) \ar[d] \\ L(f')^*Lg^*K \otimes_{\mathcal{O}_{X'}}^\mathbf{L} a'(\mathcal{O}_{Y'}) \ar[r] & a'(Lg^*K) }\] commutes where the horizontal arrows are the maps (0A9S) for \(K\) and \(Lg^*K\) and the vertical maps are constructed using Cohomology, Remark 08HY and (0AA6).

Proof

In this proof we will write \(f_*\) for \(Rf_*\) and \(f^*\) for \(Lf^*\), etc, and we will write \(\otimes\) for \(\otimes^\mathbf{L}_{\mathcal{O}_X}\), etc. Let us write (0A9S) as the composition \[\begin{align*} f^*K \otimes a(\mathcal{O}_Y) & \to a(f_*(f^*K \otimes a(\mathcal{O}_Y))) \\ & \leftarrow a(K \otimes f_*a(\mathcal{O}_K)) \\ & \to a(K \otimes \mathcal{O}_Y) \\ & \to a(K) \end{align*}\] Here the first arrow is the unit \(\eta_f\), the second arrow is \(a\) applied to Cohomology, Equation (0B53) which is an isomorphism by Derived Categories of Schemes, Lemma 08EU, the third arrow is \(a\) applied to \(\text{id}_K \otimes \text{Tr}_f\), and the fourth arrow is \(a\) applied to the isomorphism \(K \otimes \mathcal{O}_Y = K\). The proof of the lemma consists in showing that each of these maps gives rise to a commutative square as in the statement of the lemma. For \(\eta_f\) and \(\text{Tr}_f\) this is Lemmas 0B6K and 0B6J. For the arrow using Cohomology, Equation (0B53) this is Cohomology, Remark 0B6B. For the multiplication map it is clear. This finishes the proof.

Lemma

Let \(f : X \to Y\) be a proper morphism of Noetherian schemes. Let \(V \subset Y\) be an open such that \(f^{-1}(V) \to V\) is an isomorphism. Then for \(K \in D_\QCoh^+(\mathcal{O}_Y)\) the map (0A9S) restricts to an isomorphism over \(f^{-1}(V)\).

Proof

By Lemma 0A9P the map (0A9L) is an isomorphism for objects of \(D_\QCoh^+(\mathcal{O}_Y)\). Hence Lemma 0B6P tells us the restriction of (0A9S) for \(K\) to \(f^{-1}(V)\) is the map (0A9S) for \(K|_V\) and \(f^{-1}(V) \to V\). Thus it suffices to show that the map is an isomorphism when \(f\) is the identity morphism. This is clear.

Lemma

Let \(f : X \to Y\) and \(g : Y \to Z\) be composable morphisms of quasi-compact and quasi-separated schemes and set \(h = g \circ f\). Let \(a, b, c\) be the adjoints of Lemma 0A9E for \(f, g, h\). For any \(K \in D_\QCoh(\mathcal{O}_Z)\) the diagram \[\xymatrix{ Lf^*(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} b(\mathcal{O}_Z)) \otimes_{\mathcal{O}_X}^\mathbf{L} a(\mathcal{O}_Y) \ar@{=}[d] \ar[r] & a(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} b(\mathcal{O}_Z)) \ar[r] & a(b(K)) \ar@{=}[d] \\ Lh^*K \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*b(\mathcal{O}_Z) \otimes_{\mathcal{O}_X}^\mathbf{L} a(\mathcal{O}_Y) \ar[r] & Lh^*K \otimes_{\mathcal{O}_X}^\mathbf{L} c(\mathcal{O}_Z) \ar[r] & c(K) }\] is commutative where the arrows are (0A9S) and we have used \(Lh^* = Lf^* \circ Lg^*\) and \(c = a \circ b\).

Proof

In this proof we will write \(f_*\) for \(Rf_*\) and \(f^*\) for \(Lf^*\), etc, and we will write \(\otimes\) for \(\otimes^\mathbf{L}_{\mathcal{O}_X}\), etc. The composition of the top arrows is adjoint to a map \[g_*f_*(f^*(g^*K \otimes b(\mathcal{O}_Z)) \otimes a(\mathcal{O}_Y)) \to K\] The left hand side is equal to \(K \otimes g_*f_*(f^*b(\mathcal{O}_Z) \otimes a(\mathcal{O}_Y))\) by Derived Categories of Schemes, Lemma 08EU and inspection of the definitions shows the map comes from the map \[g_*f_*(f^*b(\mathcal{O}_Z) \otimes a(\mathcal{O}_Y)) \xleftarrow{g_*\epsilon} g_*(b(\mathcal{O}_Z) \otimes f_*a(\mathcal{O}_Y)) \xrightarrow{g_*\alpha} g_*(b(\mathcal{O}_Z)) \xrightarrow{\beta} \mathcal{O}_Z\] tensored with \(\text{id}_K\). Here \(\epsilon\) is the isomorphism from Derived Categories of Schemes, Lemma 08EU and \(\beta\) comes from the counit map \(g_*b \to \text{id}\). Similarly, the composition of the lower horizontal arrows is adjoint to \(\text{id}_K\) tensored with the composition \[g_*f_*(f^*b(\mathcal{O}_Z) \otimes a(\mathcal{O}_Y)) \xrightarrow{g_*f_*\delta} g_*f_*(ab(\mathcal{O}_Z)) \xrightarrow{g_*\gamma} g_*(b(\mathcal{O}_Z)) \xrightarrow{\beta} \mathcal{O}_Z\] where \(\gamma\) comes from the counit map \(f_*a \to \text{id}\) and \(\delta\) is the map whose adjoint is the composition \[f_*(f^*b(\mathcal{O}_Z) \otimes a(\mathcal{O}_Y)) \xleftarrow{\epsilon} b(\mathcal{O}_Z) \otimes f_*a(\mathcal{O}_Y) \xrightarrow{\alpha} b(\mathcal{O}_Z)\] By general properties of adjoint functors, adjoint maps, and counits (see Categories, Section 0036) we have \(\gamma \circ f_*\delta = \alpha \circ \epsilon^{-1}\) as desired.

Right adjoint of pushforward for closed immersions

Let \(i : (Z, \mathcal{O}_Z) \to (X, \mathcal{O}_X)\) be a morphism of ringed spaces such that \(i\) is a homeomorphism onto a closed subset and such that \(i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z\) is surjective. (For example a closed immersion of schemes.) Let \(\mathcal{I} = \Ker(i^\sharp)\). For a sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) the sheaf \[\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Z, \mathcal{F})\] a sheaf of \(\mathcal{O}_X\)-modules annihilated by \(\mathcal{I}\). Hence by Modules, Lemma 08KS there is a sheaf of \(\mathcal{O}_Z\)-modules, which we will denote \(\SheafHom(\mathcal{O}_Z, \mathcal{F})\), such that \[i_*\SheafHom(\mathcal{O}_Z, \mathcal{F}) = \SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Z, \mathcal{F})\] as \(\mathcal{O}_X\)-modules. We spell out what this means.

Lemma

With notation as above. The functor \(\SheafHom(\mathcal{O}_Z, -)\) is a right adjoint to the functor \(i_* : \textit{Mod}(\mathcal{O}_Z) \to \textit{Mod}(\mathcal{O}_X)\). For \(V \subset Z\) open we have \[\Gamma(V, \SheafHom(\mathcal{O}_Z, \mathcal{F})) = \{s \in \Gamma(U, \mathcal{F}) \mid \mathcal{I}s = 0\}\] where \(U \subset X\) is an open whose intersection with \(Z\) is \(V\).

Proof

Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}_Z\)-modules. Then \[\Hom_{\mathcal{O}_X}(i_*\mathcal{G}, \mathcal{F}) = \Hom_{i_*\mathcal{O}_Z}(i_*\mathcal{G}, \SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Z, \mathcal{F})) = \Hom_{\mathcal{O}_Z}(\mathcal{G}, \SheafHom(\mathcal{O}_Z, \mathcal{F}))\] The first equality by Modules, Lemma 0A6F and the second by the fully faithfulness of \(i_*\), see Modules, Lemma 08KS. The description of sections is left to the reader.

The functor \[\textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_Z), \quad \mathcal{F} \longmapsto \SheafHom(\mathcal{O}_Z, \mathcal{F})\] is left exact and has a derived extension \[R\SheafHom(\mathcal{O}_Z, -) : D(\mathcal{O}_X) \to D(\mathcal{O}_Z).\]

Lemma

With notation as above. The functor \(R\SheafHom(\mathcal{O}_Z, -)\) is the right adjoint of the functor \(Ri_* : D(\mathcal{O}_Z) \to D(\mathcal{O}_X)\).

Proof

This is a consequence of the fact that \(i_*\) and \(\SheafHom(\mathcal{O}_Z, -)\) are adjoint functors by Lemma 0A75. See Derived Categories, Lemma 09T5.

Lemma

With notation as above. We have \[Ri_*R\SheafHom(\mathcal{O}_Z, K) = R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Z, K)\] in \(D(\mathcal{O}_X)\) for all \(K\) in \(D(\mathcal{O}_X)\).

Proof

This is immediate from the construction of the functor \(R\SheafHom(\mathcal{O}_Z, -)\).

Lemma

With notation as above. For \(M \in D(\mathcal{O}_Z)\) we have \[R\SheafHom_{\mathcal{O}_X}(Ri_*M, K) = Ri_*R\SheafHom_{\mathcal{O}_Z}(M, R\SheafHom(\mathcal{O}_Z, K))\] in \(D(\mathcal{O}_Z)\) for all \(K\) in \(D(\mathcal{O}_X)\).

Proof

This is immediate from the construction of the functor \(R\SheafHom(\mathcal{O}_Z, -)\) and the fact that if \(\mathcal{K}^\bullet\) is a K-injective complex of \(\mathcal{O}_X\)-modules, then \(\SheafHom(\mathcal{O}_Z, \mathcal{K}^\bullet)\) is a K-injective complex of \(\mathcal{O}_Z\)-modules, see Derived Categories, Lemma 08BJ.

Lemma

Let \(i : Z \to X\) be a pseudo-coherent closed immersion of schemes (any closed immersion if \(X\) is locally Noetherian). Then

1. \(R\SheafHom(\mathcal{O}_Z, -)\) maps \(D^+_\QCoh(\mathcal{O}_X)\) into \(D^+_\QCoh(\mathcal{O}_Z)\), and

2. if \(X = \Spec(A)\) and \(Z = \Spec(B)\), then the diagram \[\xymatrix{ D^+(B) \ar[r] & D_\QCoh^+(\mathcal{O}_Z) \\ D^+(A) \ar[r] \ar[u]^{R\Hom(B, -)} & D_\QCoh^+(\mathcal{O}_X) \ar[u]_{R\SheafHom(\mathcal{O}_Z, -)} }\] is commutative.

Proof

To explain the parenthetical remark, if \(X\) is locally Noetherian, then \(i\) is pseudo-coherent by More on Morphisms, Lemma 0684.

Let \(K\) be an object of \(D^+_\QCoh(\mathcal{O}_X)\). To prove (1), by Morphisms, Lemma 01QY it suffices to show that \(i_*\) applied to \(H^n(R\SheafHom(\mathcal{O}_Z, K))\) produces a quasi-coherent module on \(X\). By Lemma 0A77 this means we have to show that \(R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Z, K)\) is in \(D_\QCoh(\mathcal{O}_X)\). Since \(i\) is pseudo-coherent the sheaf \(\mathcal{O}_Z\) is a pseudo-coherent \(\mathcal{O}_X\)-module. Hence the result follows from Derived Categories of Schemes, Lemma 0A6H.

Assume \(X = \Spec(A)\) and \(Z = \Spec(B)\) as in (2). Let \(I^\bullet\) be a bounded below complex of injective \(A\)-modules representing an object \(K\) of \(D^+(A)\). Then we know that \(R\Hom(B, K) = \Hom_A(B, I^\bullet)\) viewed as a complex of \(B\)-modules. Choose a quasi-isomorphism \[\widetilde{I^\bullet} \longrightarrow \mathcal{I}^\bullet\] where \(\mathcal{I}^\bullet\) is a bounded below complex of injective \(\mathcal{O}_X\)-modules. It follows from the description of the functor \(\SheafHom(\mathcal{O}_Z, -)\) in Lemma 0A75 that there is a map \[\Hom_A(B, I^\bullet) \longrightarrow \Gamma(Z, \SheafHom(\mathcal{O}_Z, \mathcal{I}^\bullet))\] Observe that \(\SheafHom(\mathcal{O}_Z, \mathcal{I}^\bullet)\) represents \(R\SheafHom(\mathcal{O}_Z, \widetilde{K})\). Applying the universal property of the \(\widetilde{\ }\) functor we obtain a map \[\widetilde{\Hom_A(B, I^\bullet)} \longrightarrow R\SheafHom(\mathcal{O}_Z, \widetilde{K})\] in \(D(\mathcal{O}_Z)\). We may check that this map is an isomorphism in \(D(\mathcal{O}_Z)\) after applying \(i_*\). However, once we apply \(i_*\) we obtain the isomorphism of Derived Categories of Schemes, Lemma 0A6H via the identification of Lemma 0A77.

Lemma

Let \(i : Z \to X\) be a closed immersion of schemes. Assume \(X\) is a locally Noetherian. Then \(R\SheafHom(\mathcal{O}_Z, -)\) maps \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) into \(D^+_{\textit{Coh}}(\mathcal{O}_Z)\).

Proof

The question is local on \(X\), hence we may assume that \(X\) is affine. Say \(X = \Spec(A)\) and \(Z = \Spec(B)\) with \(A\) Noetherian and \(A \to B\) surjective. In this case, we can apply Lemma 0A78 to translate the question into algebra. The corresponding algebra result is a consequence of Dualizing Complexes, Lemma 0A72.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(i : Z \to X\) be a pseudo-coherent closed immersion (if \(X\) is Noetherian, then any closed immersion is pseudo-coherent). Let \(a : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Z)\) be the right adjoint to \(Ri_*\). Then there is a functorial isomorphism \[a(K) = R\SheafHom(\mathcal{O}_Z, K)\] for \(K \in D_\QCoh^+(\mathcal{O}_X)\).

Proof

(The parenthetical statement follows from More on Morphisms, Lemma 0684.) By Lemma 0A76 the functor \(R\SheafHom(\mathcal{O}_Z, -)\) is a right adjoint to \(Ri_* : D(\mathcal{O}_Z) \to D(\mathcal{O}_X)\). Moreover, by Lemma 0A78 and Lemma 0A9I both \(R\SheafHom(\mathcal{O}_Z, -)\) and \(a\) map \(D_\QCoh^+(\mathcal{O}_X)\) into \(D_\QCoh^+(\mathcal{O}_Z)\). Hence we obtain the isomorphism by uniqueness of adjoint functors.

Example

If \(i : Z \to X\) is closed immersion of Noetherian schemes, then the diagram \[\xymatrix{ i_*a(K) \ar[rr]_-{\text{Tr}_{i, K}} \ar@{=}[d] & & K \ar@{=}[d] \\ i_*R\SheafHom(\mathcal{O}_Z, K) \ar@{=}[r] & R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Z, K) \ar[r] & K }\] is commutative for \(K \in D_\QCoh^+(\mathcal{O}_X)\). Here the horizontal equality sign is Lemma 0A77 and the lower horizontal arrow is induced by the map \(\mathcal{O}_X \to i_*\mathcal{O}_Z\). The commutativity of the diagram is a consequence of Lemma 0A9X.

Right adjoint of pushforward for closed immersions and base change

Consider a cartesian diagram of schemes \[\xymatrix{ Z' \ar[r]_{i'} \ar[d]_g & X' \ar[d]^f \\ Z \ar[r]^i & X }\] where \(i\) is a closed immersion. If \(Z\) and \(X'\) are tor independent over \(X\), then there is a canonical base change map [0E2K]\[\begin{equation} Lg^*R\SheafHom(\mathcal{O}_Z, K) \longrightarrow R\SheafHom(\mathcal{O}_{Z'}, Lf^*K) \end{equation}\] in \(D(\mathcal{O}_{Z'})\) functorial for \(K\) in \(D(\mathcal{O}_X)\). Namely, by adjointness of Lemma 0A76 such an arrow is the same thing as a map \[Ri'_*Lg^*R\SheafHom(\mathcal{O}_Z, K) \longrightarrow Lf^*K\] in \(D(\mathcal{O}_{X'})\). By tor independence we have \(Ri'_* \circ Lg^* = Lf^* \circ Ri_*\) (see Derived Categories of Schemes, Lemma 0E23). Thus this is the same thing as a map \[Lf^*Ri_*R\SheafHom(\mathcal{O}_Z, K) \longrightarrow Lf^*K\] For this we can use \(Lf^*(can)\) where \(can : Ri_* R\SheafHom(\mathcal{O}_Z, K) \to K\) is the counit of the adjunction.

Lemma

In the situation above, the map (0E2K) is an isomorphism if and only if the base change map \[Lf^*R\SheafHom_{\mathcal{O}_X}(\mathcal{O}_Z, K) \longrightarrow R\SheafHom_{\mathcal{O}_{X'}}(\mathcal{O}_{Z'}, Lf^*K)\] of Cohomology, Remark 08I3 is an isomorphism.

Proof

The statement makes sense because \(\mathcal{O}_{Z'} = Lf^*\mathcal{O}_Z\) by the assumed tor independence. Since \(i'_*\) is exact and faithful we see that it suffices to show the map (0E2K) is an isomorphism after applying \(Ri'_*\). Since \(Ri'_* \circ Lg^* = Lf^* \circ Ri_*\) by the assumed tor independence and Derived Categories of Schemes, Lemma 0E23 we obtain a map \[Lf^*Ri_*R\SheafHom(\mathcal{O}_Z, K) \longrightarrow Ri'_*R\SheafHom(\mathcal{O}_{Z'}, Lf^*K)\] whose source and target are as in the statement of the lemma by Lemma 0A77. We omit the verification that this is the same map as the one constructed in Cohomology, Remark 08I3.

Lemma

In the situation above, assume \(f\) is flat and \(i\) pseudo-coherent. Then (0E2K) is an isomorphism for \(K\) in \(D^+_\QCoh(\mathcal{O}_X)\).

Proof

First proof. To prove this map is an isomorphism, we may work locally. Hence we may assume \(X\), \(X'\), \(Z\), \(Z'\) are affine, say corresponding to the rings \(A\), \(A'\), \(B\), \(B'\). Then \(B\) and \(A'\) are tor independent over \(A\). By Lemma 0E2L it suffices to check that \[R\Hom_A(B, K) \otimes_A^\mathbf{L} A' = R\Hom_{A'}(B', K \otimes_A^\mathbf{L} A')\] in \(D(A')\) for all \(K \in D^+(A)\). Here we use Derived Categories of Schemes, Lemma 0A6H and the fact that \(B\), resp. \(B'\) is pseudo-coherent as an \(A\)-module, resp. \(A'\)-module to compare derived hom on the level of rings and schemes. The displayed equality follows from More on Algebra, Lemma 0ATK part (3). See also the discussion in Dualizing Complexes, Section 0E28.

Second proof⁴. Let \(z' \in Z'\) with image \(z \in Z\). First show that (0E2K) on stalks at \(z'\) induces the map \[R\Hom(\mathcal{O}_{Z, z}, K_z) \otimes_{\mathcal{O}_{Z, x}}^\mathbf{L} \mathcal{O}_{Z', z'} \longrightarrow R\Hom(\mathcal{O}_{Z', z'}, K_z \otimes_{\mathcal{O}_{X, z}}^\mathbf{L} \mathcal{O}_{X', z'})\] from Dualizing Complexes, Equation (0E29). Namely, the constructions of these maps are identical. Then apply Dualizing Complexes, Lemma 0BZM.

Lemma

Let \(i : Z \to X\) be a pseudo-coherent closed immersion of schemes. Let \(M \in D_\QCoh(\mathcal{O}_X)\) locally have tor-amplitude in \([a, \infty)\). Let \(K \in D_\QCoh^+(\mathcal{O}_X)\). Then there is a canonical isomorphism \[R\SheafHom(\mathcal{O}_Z, K) \otimes_{\mathcal{O}_Z}^\mathbf{L} Li^*M = R\SheafHom(\mathcal{O}_Z, K \otimes_{\mathcal{O}_X}^\mathbf{L} M)\] in \(D(\mathcal{O}_Z)\).

Proof

A map from LHS to RHS is the same thing as a map \[Ri_*R\SheafHom(\mathcal{O}_Z, K) \otimes_{\mathcal{O}_X}^\mathbf{L} M \longrightarrow K \otimes_{\mathcal{O}_X}^\mathbf{L} M\] by Lemmas 0A76 and 0A77. For this map we take the counit \(Ri_*R\SheafHom(\mathcal{O}_Z, K) \to K\) tensored with \(\text{id}_M\). To see this map is an isomorphism under the hypotheses given, translate into algebra using Lemma 0A78 and then for example use More on Algebra, Lemma 0ATK part (3). Instead of using Lemma 0A78 you can look at stalks as in the second proof of Lemma 0E2M.

Right adjoint of pushforward for finite morphisms

If \(i : Z \to X\) is a closed immersion of schemes, then there is a right adjoint \(\SheafHom(\mathcal{O}_Z, -)\) to the functor \(i_* : \textit{Mod}(\mathcal{O}_Z) \to \textit{Mod}(\mathcal{O}_X)\) whose derived extension \(R\SheafHom(\mathcal{O}_Z, -)\) is the right adjoint to \(Ri_* : D(\mathcal{O}_Z) \to D(\mathcal{O}_X)\). See Section 0A74. In the case of a finite morphism \(f : Y \to X\) this strategy cannot work, as the functor \(f_* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)\) is not exact in general and hence does not have a right adjoint. A replacement is to consider the exact functor \(\textit{Mod}(f_*\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)\) and consider the corresponding right adjoint and its derived extension.

Let \(f : Y \to X\) be an affine morphism of schemes. For a sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) the sheaf \[\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, \mathcal{F})\] is a sheaf of \(f_*\mathcal{O}_Y\)-modules. We obtain a functor \(\textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(f_*\mathcal{O}_Y)\) which we will denote \(\SheafHom(f_*\mathcal{O}_Y, -)\).

Lemma

With notation as above. The functor \(\SheafHom(f_*\mathcal{O}_Y, -)\) is a right adjoint to the restriction functor \(\textit{Mod}(f_*\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)\). For an affine open \(U \subset X\) we have \[\Gamma(U, \SheafHom(f_*\mathcal{O}_Y, \mathcal{F})) = \Hom_A(B, \mathcal{F}(U))\] where \(A = \mathcal{O}_X(U)\) and \(B = \mathcal{O}_Y(f^{-1}(U))\).

Proof

Adjointness follows from Modules, Lemma 0A6F. As \(f\) is affine we see that \(f_*\mathcal{O}_Y\) is the quasi-coherent sheaf corresponding to \(B\) viewed as an \(A\)-module. Hence the description of sections over \(U\) follows from Schemes, Lemma 01I7.

The functor \(\SheafHom(f_*\mathcal{O}_Y, -)\) is left exact. Let \[R\SheafHom(f_*\mathcal{O}_Y, -) : D(\mathcal{O}_X) \longrightarrow D(f_*\mathcal{O}_Y)\] be its derived extension.

Lemma

With notation as above. The functor \(R\SheafHom(f_*\mathcal{O}_Y, -)\) is the right adjoint of the functor \(D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X)\).

Proof

Follows from Lemma 0BUZ and Derived Categories, Lemma 09T5.

Lemma

With notation as above. The composition \[D(\mathcal{O}_X) \xrightarrow{R\SheafHom(f_*\mathcal{O}_Y, -)} D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X)\] is the functor \(K \mapsto R\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K)\).

Proof

This is immediate from the construction.

Lemma

Let \(f : Y \to X\) be a finite pseudo-coherent morphism of schemes (a finite morphism of Noetherian schemes is pseudo-coherent). The functor \(R\SheafHom(f_*\mathcal{O}_Y, -)\) maps \(D_\QCoh^+(\mathcal{O}_X)\) into \(D_\QCoh^+(f_*\mathcal{O}_Y)\). If \(X\) is quasi-compact and quasi-separated, then the diagram \[\xymatrix{ D_\QCoh^+(\mathcal{O}_X) \ar[rr]_a \ar[rd]_{R\SheafHom(f_*\mathcal{O}_Y, -)} & & D_\QCoh^+(\mathcal{O}_Y) \ar[ld]^\Phi \\ & D_\QCoh^+(f_*\mathcal{O}_Y) }\] is commutative, where \(a\) is the right adjoint of Lemma 0A9E for \(f\) and \(\Phi\) is the equivalence of Derived Categories of Schemes, Lemma 0AVW.

Proof

(The parenthetical remark follows from More on Morphisms, Lemma 0684.) Since \(f\) is pseudo-coherent, the \(\mathcal{O}_X\)-module \(f_*\mathcal{O}_Y\) is pseudo-coherent, see More on Morphisms, Lemma 0AVX. Thus \(R\SheafHom(f_*\mathcal{O}_Y, -)\) maps \(D_\QCoh^+(\mathcal{O}_X)\) into \(D_\QCoh^+(f_*\mathcal{O}_Y)\), see Derived Categories of Schemes, Lemma 0A6H. Then \(\Phi \circ a\) and \(R\SheafHom(f_*\mathcal{O}_Y, -)\) agree on \(D_\QCoh^+(\mathcal{O}_X)\) because these functors are both right adjoint to the restriction functor \(D_\QCoh^+(f_*\mathcal{O}_Y) \to D_\QCoh^+(\mathcal{O}_X)\). To see this use Lemmas 0A9I and 0BV0.

Remark

If \(f : Y \to X\) is a finite morphism of Noetherian schemes, then the diagram \[\xymatrix{ Rf_*a(K) \ar[r]_-{\text{Tr}_{f, K}} \ar@{=}[d] & K \ar@{=}[d] \\ R\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K) \ar[r] & K }\] is commutative for \(K \in D_\QCoh^+(\mathcal{O}_X)\). This follows from Lemma 0AX2. The lower horizontal arrow is induced by the map \(\mathcal{O}_X \to f_*\mathcal{O}_Y\) and the upper horizontal arrow is the trace map discussed in Section 0AWG.

Right adjoint of pushforward for proper flat morphisms

For proper, flat, and finitely presented morphisms of quasi-compact and quasi-separated schemes the right adjoint of pushforward enjoys some remarkable properties.

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a morphism of schemes which is proper, flat, and of finite presentation. Let \(a\) be the right adjoint for \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) of Lemma 0A9E. Then \(a\) commutes with direct sums.

Proof

Let \(P\) be a perfect object of \(D(\mathcal{O}_X)\). By Derived Categories of Schemes, Lemma 0B91 the complex \(Rf_*P\) is perfect on \(Y\). Let \(K_i\) be a family of objects of \(D_\QCoh(\mathcal{O}_Y)\). Then \[\begin{align*} \Hom_{D(\mathcal{O}_X)}(P, a(\bigoplus K_i)) & = \Hom_{D(\mathcal{O}_Y)}(Rf_*P, \bigoplus K_i) \\ & = \bigoplus \Hom_{D(\mathcal{O}_Y)}(Rf_*P, K_i) \\ & = \bigoplus \Hom_{D(\mathcal{O}_X)}(P, a(K_i)) \end{align*}\] because a perfect object is compact (Derived Categories of Schemes, Proposition 09M1). Since \(D_\QCoh(\mathcal{O}_X)\) has a perfect generator (Derived Categories of Schemes, Theorem 09IS) we conclude that the map \(\bigoplus a(K_i) \to a(\bigoplus K_i)\) is an isomorphism, i.e., \(a\) commutes with direct sums.

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a morphism of schemes which is proper, flat, and of finite presentation. Let \(a\) be the right adjoint for \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) of Lemma 0A9E. Then

1. for every closed \(T \subset Y\) if \(Q \in D_\QCoh(Y)\) is supported on \(T\), then \(a(Q)\) is supported on \(f^{-1}(T)\),

2. for every quasi-compact open \(V \subset Y\) and any \(K \in D_\QCoh(\mathcal{O}_Y)\) the map (0A9L) is an isomorphism.

Proof

This follows from Lemmas 0A9N, 0A9P, and 0E4I. Strictly speaking part (1) follows only for those \(T\) whose complement is a quasi-compact open; however the general case then follows as any closed subset is an intersection of such.

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a morphism of schemes which is proper, flat, and of finite presentation. The map (0A9S) is an isomorphism for every object \(K\) of \(D_\QCoh(\mathcal{O}_Y)\).

Proof

By Lemma 0E4I we know that \(a\) commutes with direct sums. Hence the collection of objects of \(D_\QCoh(\mathcal{O}_Y)\) for which (0A9S) is an isomorphism is a strictly full, saturated, triangulated subcategory of \(D_\QCoh(\mathcal{O}_Y)\) which is moreover preserved under taking direct sums. Since \(D_\QCoh(\mathcal{O}_Y)\) is a module category (Derived Categories of Schemes, Theorem 09M5) generated by a single perfect object (Derived Categories of Schemes, Theorem 09IS) we can argue as in More on Algebra, Remark 09PB to see that it suffices to prove (0A9S) is an isomorphism for a single perfect object. However, the result holds for perfect objects, see Lemma 0A9T.

The following lemma shows that the base change map (0AA6) is an isomorphism for proper, flat morphisms of finite presentation. We will see in Example 0AAC that this does not remain true for perfect proper morphisms; in that case one has to make a tor independence condition.

Lemma

Let \(g : Y' \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(f : X \to Y\) be a proper, flat morphism of finite presentation. Then the base change map (0AA6) is an isomorphism for all \(K \in D_\QCoh(\mathcal{O}_Y)\).

Proof

By Lemma 0E4J formation of the functors \(a\) and \(a'\) commutes with restriction to opens of \(Y\) and \(Y'\). Hence we may assume \(Y' \to Y\) is a morphism of affine schemes, see Remark 0E9S. In this case the statement follows from Lemma 0AA8.

Remark

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a proper, flat morphism of finite presentation. Let \(a\) be the adjoint of Lemma 0A9E for \(f\). In this situation, \(\omega_{X/Y}^\bullet = a(\mathcal{O}_Y)\) is sometimes called the relative dualizing complex. By Lemma 0E4K there is a functorial isomorphism \(a(K) = Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet\) for \(K \in D_\QCoh(\mathcal{O}_Y)\). Moreover, the trace map \[\text{Tr}_{f, \mathcal{O}_Y} : Rf_*\omega_{X/Y}^\bullet \to \mathcal{O}_Y\] of Section 0AWG induces the trace map for all \(K\) in \(D_\QCoh(\mathcal{O}_Y)\). More precisely the diagram \[\xymatrix{ Rf_*a(K) \ar[rrr]_{\text{Tr}_{f, K}} \ar@{=}[d] & & & K \ar@{=}[d] \\ Rf_*(Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet) \ar@{=}[r] & K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\omega_{X/Y}^\bullet \ar[rr]^-{\text{id}_K \otimes \text{Tr}_{f, \mathcal{O}_Y}} & & K }\] where the equality on the lower right is Derived Categories of Schemes, Lemma 08EU. If \(g : Y' \to Y\) is a morphism of quasi-compact and quasi-separated schemes and \(X' = Y' \times_Y X\), then by Lemma 0AAB we have \(\omega_{X'/Y'}^\bullet = L(g')^*\omega_{X/Y}^\bullet\) where \(g' : X' \to X\) is the projection and by Lemma 0B6J the trace map \[\text{Tr}_{f', \mathcal{O}_{Y'}} : Rf'_*\omega_{X'/Y'}^\bullet \to \mathcal{O}_{Y'}\] for \(f' : X' \to Y'\) is the base change of \(\text{Tr}_{f, \mathcal{O}_Y}\) via the base change isomorphism.

Remark

Let \(f : X \to Y\), \(\omega^\bullet_{X/Y}\), and \(\text{Tr}_{f, \mathcal{O}_Y}\) be as in Remark 0B6S. Let \(K\) and \(M\) be in \(D_\QCoh(\mathcal{O}_X)\) with \(M\) pseudo-coherent (for example perfect). Suppose given a map \(K \otimes_{\mathcal{O}_X}^\mathbf{L} M \to \omega^\bullet_{X/Y}\) which corresponds to an isomorphism \(K \to R\SheafHom_{\mathcal{O}_X}(M, \omega^\bullet_{X/Y})\) via Cohomology, Equation (08DI). Then the relative cup product (Cohomology, Remark 0B68) \[Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*M \to Rf_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} M) \to Rf_*\omega^\bullet_{X/Y} \xrightarrow{\text{Tr}_{f, \mathcal{O}_Y}} \mathcal{O}_Y\] determines an isomorphism \(Rf_*K \to R\SheafHom_{\mathcal{O}_Y}(Rf_*M, \mathcal{O}_Y)\). Namely, since \(\omega^\bullet_{X/Y} = a(\mathcal{O}_Y)\) the canonical map (0B6H) \[Rf_*R\SheafHom_{\mathcal{O}_X}(M, \omega^\bullet_{X/Y}) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*M, \mathcal{O}_Y)\] is an isomorphism by Lemma 0A9Q and Remark 0GEV and the fact that \(M\) and \(Rf_*M\) are pseudo-coherent, see Derived Categories of Schemes, Lemma 0CSD. To see that the relative cup product induces this isomorphism use the commutativity of the diagram in Cohomology, Remark 0G7A.

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a morphism of schemes which is proper, flat, and of finite presentation with relative dualizing complex \(\omega_{X/Y}^\bullet\) (Remark 0B6S). Then

1. \(\omega_{X/Y}^\bullet\) is a \(Y\)-perfect object of \(D(\mathcal{O}_X)\),

2. \(Rf_*\omega_{X/Y}^\bullet\) has vanishing cohomology sheaves in positive degrees,

3. \(\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(\omega_{X/Y}^\bullet, \omega_{X/Y}^\bullet)\) is an isomorphism.

Proof

In view of the fact that formation of \(\omega_{X/Y}^\bullet\) commutes with base change (see Remark 0B6S), we may and do assume that \(Y\) is affine. For a perfect object \(E\) of \(D(\mathcal{O}_X)\) we have \[\begin{align*} Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet) & = Rf_*R\SheafHom_{\mathcal{O}_X}(E^\vee, \omega_{X/Y}^\bullet) \\ & = R\SheafHom_{\mathcal{O}_Y}(Rf_*E^\vee, \mathcal{O}_Y) \\ & = (Rf_*E^\vee)^\vee \end{align*}\] For the first equality, see Cohomology, Lemma 08DQ. For the second equality, see Lemma 0A9Q, Remark 0GEV, and Derived Categories of Schemes, Lemma 0B91. The third equality is the definition of the dual. In particular these references also show that the outcome is a perfect object of \(D(\mathcal{O}_Y)\). We conclude that \(\omega_{X/Y}^\bullet\) is \(Y\)-perfect by More on Morphisms, Lemma 0GET. This proves (1).

Let \(M\) be an object of \(D_\QCoh(\mathcal{O}_Y)\). Then \[\begin{align*} \Hom_Y(M, Rf_*\omega_{X/Y}^\bullet) & = \Hom_X(Lf^*M, \omega_{X/Y}^\bullet) \\ & = \Hom_Y(Rf_*Lf^*M, \mathcal{O}_Y) \\ & = \Hom_Y(M \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\mathcal{O}_X, \mathcal{O}_Y) \end{align*}\] The first equality holds by Cohomology, Lemma 079W. The second equality by construction of \(a\). The third equality by Derived Categories of Schemes, Lemma 08EU. Recall \(Rf_*\mathcal{O}_X\) is perfect of tor amplitude in \([0, N]\) for some \(N\), see Derived Categories of Schemes, Lemma 0B91. Thus we can represent \(Rf_*\mathcal{O}_X\) by a complex of finite projective modules sitting in degrees \([0, N]\) (using More on Algebra, Lemma 0658 and the fact that \(Y\) is affine). Hence if \(M = \mathcal{O}_Y[-i]\) for some \(i > 0\), then the last group is zero. Since \(Y\) is affine we conclude that \(H^i(Rf_*\omega_{X/Y}^\bullet) = 0\) for \(i > 0\). This proves (2).

Let \(E\) be a perfect object of \(D_\QCoh(\mathcal{O}_X)\). Then we have \[\begin{align*} \Hom_X(E, R\SheafHom_{\mathcal{O}_X}(\omega_{X/Y}^\bullet, \omega_{X/Y}^\bullet) & = \Hom_X(E \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet, \omega_{X/Y}^\bullet) \\ & = \Hom_Y(Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet), \mathcal{O}_Y) \\ & = \Hom_Y(Rf_*(R\SheafHom_{\mathcal{O}_X}(E^\vee, \omega_{X/Y}^\bullet)), \mathcal{O}_Y) \\ & = \Hom_Y(R\SheafHom_{\mathcal{O}_Y}(Rf_*E^\vee, \mathcal{O}_Y), \mathcal{O}_Y) \\ & = R\Gamma(Y, Rf_*E^\vee) \\ & = \Hom_X(E, \mathcal{O}_X) \end{align*}\] The first equality holds by Cohomology, Lemma 08DJ. The second equality is the definition of \(\omega_{X/Y}^\bullet\). The third equality comes from the construction of the dual perfect complex \(E^\vee\), see Cohomology, Lemma 08DQ. The fourth equality follows from the equality \(Rf_*R\SheafHom_{\mathcal{O}_X}(E^\vee, \omega_{X/Y}^\bullet) = R\SheafHom_{\mathcal{O}_Y}(Rf_*E^\vee, \mathcal{O}_Y)\) shown in the first paragraph of the proof. The fifth equality holds by double duality for perfect complexes (Cohomology, Lemma 08DQ) and the fact that \(Rf_*E\) is perfect by Derived Categories of Schemes, Lemma 0B91. The last equality is Leray for \(f\). This string of equalities essentially shows (3) holds by the Yoneda lemma. Namely, the object \(R\SheafHom(\omega_{X/Y}^\bullet, \omega_{X/Y}^\bullet)\) is in \(D_\QCoh(\mathcal{O}_X)\) by Derived Categories of Schemes, Lemma 0A6H. Taking \(E = \mathcal{O}_X\) in the above we get a map \(\alpha : \mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(\omega_{X/Y}^\bullet, \omega_{X/Y}^\bullet)\) corresponding to \(\text{id}_{\mathcal{O}_X} \in \Hom_X(\mathcal{O}_X, \mathcal{O}_X)\). Since all the isomorphisms above are functorial in \(E\) we see that the cone on \(\alpha\) is an object \(C\) of \(D_\QCoh(\mathcal{O}_X)\) such that \(\Hom(E, C) = 0\) for all perfect \(E\). Since the perfect objects generate (Derived Categories of Schemes, Theorem 09IS) we conclude that \(\alpha\) is an isomorphism.

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to Y\) be a proper, flat morphism of finite presentation with relative dualizing complex \(\omega_{X/Y}^\bullet\) (Remark 0B6S). There is a canonical isomorphism [0E2Q]\[\begin{equation} \mathcal{O}_X = c(L\text{pr}_1^*\omega_{X/Y}^\bullet) = c(L\text{pr}_2^*\omega_{X/Y}^\bullet) \end{equation}\] and a canonical isomorphism [0E2R]\[\begin{equation} \omega_{X/Y}^\bullet = c\left(L\text{pr}_1^*\omega_{X/Y}^\bullet \otimes_{\mathcal{O}_{X \times_Y X}}^\mathbf{L} L\text{pr}_2^*\omega_{X/Y}^\bullet\right) \end{equation}\] where \(c\) is the right adjoint of Lemma 0A9E for the diagonal \(\Delta : X \to X \times_Y X\).

Proof

Let \(a\) be the right adjoint to \(Rf_*\) as in Lemma 0A9E. Consider the cartesian square \[\xymatrix{ X \times_Y X \ar[r]_q \ar[d]_p & X \ar[d]_f \\ X \ar[r]^f & Y }\] Let \(b\) be the right adjoint for \(p\) as in Lemma 0A9E. Then \[\begin{align*} \omega_{X/Y}^\bullet & = c(b(\omega_{X/Y}^\bullet)) \\ & = c(Lp^*\omega_{X/Y}^\bullet \otimes_{\mathcal{O}_{X \times_Y X}}^\mathbf{L} b(\mathcal{O}_X)) \\ & = c(Lp^*\omega_{X/Y}^\bullet \otimes_{\mathcal{O}_{X \times_Y X}}^\mathbf{L} Lq^*a(\mathcal{O}_Y)) \\ & = c(Lp^*\omega_{X/Y}^\bullet \otimes_{\mathcal{O}_{X \times_Y X}}^\mathbf{L} Lq^*\omega_{X/Y}^\bullet) \end{align*}\] as in (0E2R). Explanation as follows:

1. The first equality holds as \(\text{id} = c \circ b\) because \(\text{id}_X = p \circ \Delta\).

2. The second equality holds by Lemma 0E4K.

3. The third holds by Lemma 0AAB and the fact that \(\mathcal{O}_X = Lf^*\mathcal{O}_Y\).

4. The fourth holds because \(\omega_{X/Y}^\bullet = a(\mathcal{O}_Y)\).

Equation (0E2Q) is proved in exactly the same way.

Remark

Lemma 0E2P means our relative dualizing complex is rigid in a sense analogous to the notion introduced in [vdB-rigid]. Namely, since the functor on the right of (0E2R) is “quadratic” in \(\omega_{X/Y}^\bullet\) and the functor on the left of (0E2R) is “linear” this “pins down” the complex \(\omega_{X/Y}^\bullet\) to some extent. There is an approach to duality theory using “rigid” (relative) dualizing complexes, see for example [Neeman-rigid], [Yekutieli-rigid], and [Yekutieli-Zhang]. We will return to this in Section 0E2S.

Right adjoint of pushforward for perfect proper morphisms

The correct generality for this section would be to consider perfect proper morphisms of quasi-compact and quasi-separated schemes, see [LN].

Lemma

Let \(f : X \to Y\) be a perfect proper morphism of Noetherian schemes. Let \(a\) be the right adjoint for \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) of Lemma 0A9E. Then \(a\) commutes with direct sums.

Proof

Let \(P\) be a perfect object of \(D(\mathcal{O}_X)\). By More on Morphisms, Lemma 0B6G the complex \(Rf_*P\) is perfect on \(Y\). Let \(K_i\) be a family of objects of \(D_\QCoh(\mathcal{O}_Y)\). Then \[\begin{align*} \Hom_{D(\mathcal{O}_X)}(P, a(\bigoplus K_i)) & = \Hom_{D(\mathcal{O}_Y)}(Rf_*P, \bigoplus K_i) \\ & = \bigoplus \Hom_{D(\mathcal{O}_Y)}(Rf_*P, K_i) \\ & = \bigoplus \Hom_{D(\mathcal{O}_X)}(P, a(K_i)) \end{align*}\] because a perfect object is compact (Derived Categories of Schemes, Proposition 09M1). Since \(D_\QCoh(\mathcal{O}_X)\) has a perfect generator (Derived Categories of Schemes, Theorem 09IS) we conclude that the map \(\bigoplus a(K_i) \to a(\bigoplus K_i)\) is an isomorphism, i.e., \(a\) commutes with direct sums.

Lemma

Let \(f : X \to Y\) be a perfect proper morphism of Noetherian schemes. Let \(a\) be the right adjoint for \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) of Lemma 0A9E. Then

1. for every closed \(T \subset Y\) if \(Q \in D_\QCoh(Y)\) is supported on \(T\), then \(a(Q)\) is supported on \(f^{-1}(T)\),

2. for every open \(V \subset Y\) and any \(K \in D_\QCoh(\mathcal{O}_Y)\) the map (0A9L) is an isomorphism, and

Proof

This follows from Lemmas 0A9N, 0A9P, and 0A9R.

Lemma

Let \(f : X \to Y\) be a perfect proper morphism of Noetherian schemes. The map (0A9S) is an isomorphism for every object \(K\) of \(D_\QCoh(\mathcal{O}_Y)\).

Proof

By Lemma 0A9R we know that \(a\) commutes with direct sums. Hence the collection of objects of \(D_\QCoh(\mathcal{O}_Y)\) for which (0A9S) is an isomorphism is a strictly full, saturated, triangulated subcategory of \(D_\QCoh(\mathcal{O}_Y)\) which is moreover preserved under taking direct sums. Since \(D_\QCoh(\mathcal{O}_Y)\) is a module category (Derived Categories of Schemes, Theorem 09M5) generated by a single perfect object (Derived Categories of Schemes, Theorem 09IS) we can argue as in More on Algebra, Remark 09PB to see that it suffices to prove (0A9S) is an isomorphism for a single perfect object. However, the result holds for perfect objects, see Lemma 0A9T.

Lemma

Let \(f : X \to Y\) be a perfect proper morphism of Noetherian schemes. Let \(g : Y' \to Y\) be a morphism with \(Y'\) Noetherian. If \(X\) and \(Y'\) are tor independent over \(Y\), then the base change map (0AA6) is an isomorphism for all \(K \in D_\QCoh(\mathcal{O}_Y)\).

Proof

By Lemma 0AAA formation of the functors \(a\) and \(a'\) commutes with restriction to opens of \(Y\) and \(Y'\). Hence we may assume \(Y' \to Y\) is a morphism of affine schemes, see Remark 0E9S. In this case the statement follows from Lemma 0AA8.

Right adjoint of pushforward for effective Cartier divisors

Let \(X\) be a scheme and let \(i : D \to X\) be the inclusion of an effective Cartier divisor. Denote \(\mathcal{N} = i^*\mathcal{O}_X(D)\) the normal sheaf of \(i\), see Morphisms, Section 01R1 and Divisors, Section 01WQ. Recall that \(R\SheafHom(\mathcal{O}_D, -)\) denotes the right adjoint to \(i_* : D(\mathcal{O}_D) \to D(\mathcal{O}_X)\) and has the property \(i_*R\SheafHom(\mathcal{O}_D, -) = R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_D, -)\), see Section 0A74.

Lemma

As above, let \(X\) be a scheme and let \(D \subset X\) be an effective Cartier divisor. There is a canonical isomorphism \(R\SheafHom(\mathcal{O}_D, \mathcal{O}_X) = \mathcal{N}[-1]\) in \(D(\mathcal{O}_D)\).

Proof

Equivalently, we are saying that \(R\SheafHom(\mathcal{O}_D, \mathcal{O}_X)\) has a unique nonzero cohomology sheaf in degree \(1\) and that this sheaf is isomorphic to \(\mathcal{N}\). Since \(i_*\) is exact and fully faithful, it suffices to prove that \(i_*R\SheafHom(\mathcal{O}_D, \mathcal{O}_X)\) is isomorphic to \(i_*\mathcal{N}[-1]\). We have \(i_*R\SheafHom(\mathcal{O}_D, \mathcal{O}_X) = R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_D, \mathcal{O}_X)\) by Lemma 0A77. We have a resolution \[0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_D \to 0\] where \(\mathcal{I}\) is the ideal sheaf of \(D\) which we can use to compute. Since \(R\SheafHom_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{O}_X) = \mathcal{O}_X\) and \(R\SheafHom_{\mathcal{O}_X}(\mathcal{I}, \mathcal{O}_X) = \mathcal{O}_X(D)\) by a local computation, we see that \[R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_D, \mathcal{O}_X) = (\mathcal{O}_X \to \mathcal{O}_X(D))\] where on the right hand side we have \(\mathcal{O}_X\) in degree \(0\) and \(\mathcal{O}_X(D)\) in degree \(1\). The result follows from the short exact sequence \[0 \to \mathcal{O}_X \to \mathcal{O}_X(D) \to i_*\mathcal{N} \to 0\] coming from the fact that \(D\) is the zero scheme of the canonical section of \(\mathcal{O}_X(D)\) and from the fact that \(\mathcal{N} = i^*\mathcal{O}_X(D)\).

For every object \(K\) of \(D(\mathcal{O}_X)\) there is a canonical map [0B4C]\[\begin{equation} Li^*K \otimes_{\mathcal{O}_D}^\mathbf{L} R\SheafHom(\mathcal{O}_D, \mathcal{O}_X) \longrightarrow R\SheafHom(\mathcal{O}_D, K) \end{equation}\] in \(D(\mathcal{O}_D)\) functorial in \(K\) and compatible with distinguished triangles. Namely, this map is adjoint to a map \[i_*(Li^*K \otimes^\mathbf{L}_{\mathcal{O}_D} R\SheafHom(\mathcal{O}_D, \mathcal{O}_X)) = K \otimes^\mathbf{L}_{\mathcal{O}_X} R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_D, \mathcal{O}_X) \longrightarrow K\] where the equality is Cohomology, Lemma 0B55 and the arrow comes from the canonical map \(R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_D, \mathcal{O}_X) \to \mathcal{O}_X\) induced by \(\mathcal{O}_X \to i_*\mathcal{O}_D\).

If \(K \in D_\QCoh(\mathcal{O}_X)\), then (0B4C) is equal to (0A9S) via the identification \(a(K) = R\SheafHom(\mathcal{O}_D, K)\) of Lemma 0A9X. If \(K \in D_\QCoh(\mathcal{O}_X)\) and \(X\) is Noetherian, then the following lemma is a special case of Lemma 0A9U.

Lemma

As above, let \(X\) be a scheme and let \(D \subset X\) be an effective Cartier divisor. Then (0B4C) combined with Lemma 0B4B defines an isomorphism \[Li^*K \otimes_{\mathcal{O}_D}^\mathbf{L} \mathcal{N}[-1] \longrightarrow R\SheafHom(\mathcal{O}_D, K)\] functorial in \(K\) in \(D(\mathcal{O}_X)\).

Proof

Since \(i_*\) is exact and fully faithful on modules, to prove the map is an isomorphism, it suffices to show that it is an isomorphism after applying \(i_*\). We will use the short exact sequences \(0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_D \to 0\) and \(0 \to \mathcal{O}_X \to \mathcal{O}_X(D) \to i_*\mathcal{N} \to 0\) used in the proof of Lemma 0B4B without further mention. By Cohomology, Lemma 0B55 which was used to define the map (0B4C) the left hand side becomes \[K \otimes_{\mathcal{O}_X}^\mathbf{L} i_*\mathcal{N}[-1] = K \otimes_{\mathcal{O}_X}^\mathbf{L} (\mathcal{O}_X \to \mathcal{O}_X(D))\] The right hand side becomes \[\begin{align*} R\SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_D, K) & = R\SheafHom_{\mathcal{O}_X}((\mathcal{I} \to \mathcal{O}_X), K) \\ & = R\SheafHom_{\mathcal{O}_X}((\mathcal{I} \to \mathcal{O}_X), \mathcal{O}_X) \otimes_{\mathcal{O}_X}^\mathbf{L} K \end{align*}\] the final equality by Cohomology, Lemma 08DQ. Since the map comes from the isomorphism \[R\SheafHom_{\mathcal{O}_X}((\mathcal{I} \to \mathcal{O}_X), \mathcal{O}_X) = (\mathcal{O}_X \to \mathcal{O}_X(D))\] the lemma is clear.

Right adjoint of pushforward in examples

In this section we compute the right adjoint to pushforward in some examples. The isomorphisms are canonical but only in the weakest possible sense, i.e., we do not prove or claim that these isomorphisms are compatible with various operations such as base change and compositions of morphisms. There is a huge literature on these types of issues; the reader can start with the material in [RD], [Conrad-GD] (these citations use a different starting point for duality but address the issue of constructing canonical representatives for relative dualizing complexes) and then continue looking at works by Joseph Lipman and collaborators.

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_Y\)-module of rank \(n + 1\) with determinant \(\mathcal{L} = \wedge^{n + 1}(\mathcal{E})\). Let \(f : X = \mathbf{P}(\mathcal{E}) \to Y\) be the projection. Let \(a\) be the right adjoint for \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) of Lemma 0A9E. Then there is an isomorphism \[c : f^*\mathcal{L}(-n - 1)[n] \longrightarrow a(\mathcal{O}_Y)\] In particular, if \(\mathcal{E} = \mathcal{O}_Y^{\oplus n + 1}\), then \(X = \mathbf{P}^n_Y\) and we obtain \(a(\mathcal{O}_Y) = \mathcal{O}_X(-n - 1)[n]\).

Proof

In (the proof of) Cohomology of Schemes, Lemma 01XX we constructed a canonical isomorphism \[R^nf_*(f^*\mathcal{L}(-n - 1)) \longrightarrow \mathcal{O}_Y\] Moreover, \(Rf_*(f^*\mathcal{L}(-n - 1))[n] = R^nf_*(f^*\mathcal{L}(-n - 1))\), i.e., the other higher direct images are zero. Thus we find an isomorphism \[Rf_*(f^*\mathcal{L}(-n - 1)[n]) \longrightarrow \mathcal{O}_Y\] This isomorphism determines \(c\) as in the statement of the lemma because \(a\) is the right adjoint of \(Rf_*\). By Lemma 0A9P construction of the \(a\) is local on the base. In particular, to check that \(c\) is an isomorphism, we may work locally on \(Y\). In other words, we may assume \(Y\) is affine and \(\mathcal{E} = \mathcal{O}_Y^{\oplus n + 1}\). In this case the sheaves \(\mathcal{O}_X, \mathcal{O}_X(-1), \ldots, \mathcal{O}_X(-n)\) generate \(D_\QCoh(X)\), see Derived Categories of Schemes, Lemma 0A9V. Hence it suffices to show that \(c : \mathcal{O}_X(-n - 1)[n] \to a(\mathcal{O}_Y)\) is transformed into an isomorphism under the functors \[F_{i, p}(-) = \Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(i), (-)[p])\] for \(i \in \{-n, \ldots, 0\}\) and \(p \in \mathbf{Z}\). For \(F_{0, p}\) this holds by construction of the arrow \(c\)! For \(i \in \{-n, \ldots, -1\}\) we have \[\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(i), \mathcal{O}_X(-n - 1)[n + p]) = H^p(X, \mathcal{O}_X(-n - 1 - i)) = 0\] by the computation of cohomology of projective space (Cohomology of Schemes, Lemma 01XT) and we have \[\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(i), a(\mathcal{O}_Y)[p]) = \Hom_{D(\mathcal{O}_Y)}(Rf_*\mathcal{O}_X(i), \mathcal{O}_Y[p]) = 0\] because \(Rf_*\mathcal{O}_X(i) = 0\) by the same lemma. Hence the source and the target of \(F_{i, p}(c)\) vanish and \(F_{i, p}(c)\) is necessarily an isomorphism. This finishes the proof.

Example

The base change map (0AA6) is not an isomorphism if \(f\) is perfect proper and \(g\) is perfect. Let \(k\) be a field. Let \(Y = \mathbf{A}^2_k\) and let \(f : X \to Y\) be the blowup of \(Y\) in the origin. Denote \(E \subset X\) the exceptional divisor. Then we can factor \(f\) as \[X \xrightarrow{i} \mathbf{P}^1_Y \xrightarrow{p} Y\] This gives a factorization \(a = c \circ b\) where \(a\), \(b\), and \(c\) are the right adjoints of Lemma 0A9E of \(Rf_*\), \(Rp_*\), and \(Ri_*\). Denote \(\mathcal{O}(n)\) the Serre twist of the structure sheaf on \(\mathbf{P}^1_Y\) and denote \(\mathcal{O}_X(n)\) its restriction to \(X\). Note that \(X \subset \mathbf{P}^1_Y\) is cut out by a degree one equation, hence \(\mathcal{O}(X) = \mathcal{O}(1)\). By Lemma 0A9W we have \(b(\mathcal{O}_Y) = \mathcal{O}(-2)[1]\). By Lemma 0A9X we have \[a(\mathcal{O}_Y) = c(b(\mathcal{O}_Y)) = c(\mathcal{O}(-2)[1]) = R\SheafHom(\mathcal{O}_X, \mathcal{O}(-2)[1]) = \mathcal{O}_X(-1)\] Last equality by Lemma 0AA4. Let \(Y' = \Spec(k)\) be the origin in \(Y\). The restriction of \(a(\mathcal{O}_Y)\) to \(X' = E = \mathbf{P}^1_k\) is an invertible sheaf of degree \(-1\) placed in cohomological degree \(0\). But on the other hand, \(a'(\mathcal{O}_{\Spec(k)}) = \mathcal{O}_E(-2)[1]\) which is an invertible sheaf of degree \(-2\) placed in cohomological degree \(-1\), so different. In this example the hypothesis of Tor independence in Lemma 0AA8 is violated.

Lemma

Let \(Y\) be a ringed space. Let \(\mathcal{I} \subset \mathcal{O}_Y\) be a sheaf of ideals. Set \(\mathcal{O}_X = \mathcal{O}_Y/\mathcal{I}\) and \(\mathcal{N} = \SheafHom_{\mathcal{O}_Y}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_X)\). There is a canonical isomorphism \(c : \mathcal{N} \to \SheafExt^1_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\).

Proof

Consider the canonical short exact sequence [0BQX]\[\begin{equation} 0 \to \mathcal{I}/\mathcal{I}^2 \to \mathcal{O}_Y/\mathcal{I}^2 \to \mathcal{O}_X \to 0 \end{equation}\] Let \(U \subset X\) be open and let \(s \in \mathcal{N}(U)\). Then we can pushout (0BQX) via \(s\) to get an extension \(E_s\) of \(\mathcal{O}_X|_U\) by \(\mathcal{O}_X|_U\). This in turn defines a section \(c(s)\) of \(\SheafExt^1_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\) over \(U\). See Cohomology, Lemma 08DK and Derived Categories, Lemma 06XV. Conversely, given an extension \[0 \to \mathcal{O}_X|_U \to \mathcal{E} \to \mathcal{O}_X|_U \to 0\] of \(\mathcal{O}_U\)-modules, we can find an open covering \(U = \bigcup U_i\) and sections \(e_i \in \mathcal{E}(U_i)\) mapping to \(1 \in \mathcal{O}_X(U_i)\). Then \(e_i\) defines a map \(\mathcal{O}_Y|_{U_i} \to \mathcal{E}|_{U_i}\) whose kernel contains \(\mathcal{I}^2\). In this way we see that \(\mathcal{E}|_{U_i}\) comes from a pushout as above. This shows that \(c\) is surjective. We omit the proof of injectivity.

Lemma

Let \(Y\) be a ringed space. Let \(\mathcal{I} \subset \mathcal{O}_Y\) be a sheaf of ideals. Set \(\mathcal{O}_X = \mathcal{O}_Y/\mathcal{I}\). If \(\mathcal{I}\) is Koszul-regular (Divisors, Definition 063D) then composition on \(R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\) defines isomorphisms \[\wedge^i(\SheafExt^1_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)) \longrightarrow \SheafExt^i_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\] for all \(i\).

Proof

By composition we mean the map \[R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \otimes_{\mathcal{O}_Y}^\mathbf{L} R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \longrightarrow R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\] of Cohomology, Lemma 0A8V. This induces multiplication maps \[\SheafExt^a_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \otimes_{\mathcal{O}_Y} \SheafExt^b_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \longrightarrow \SheafExt^{a + b}_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\] Please compare with More on Algebra, Equation (068H). The statement of the lemma means that the induced map \[\SheafExt^1_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \otimes \ldots \otimes \SheafExt^1_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \longrightarrow \SheafExt^i_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\] factors through the wedge product and then induces an isomorphism. To see this is true we may work locally on \(Y\). Hence we may assume that we have global sections \(f_1, \ldots, f_r\) of \(\mathcal{O}_Y\) which generate \(\mathcal{I}\) and which form a Koszul regular sequence. Denote \[\mathcal{A} = \mathcal{O}_Y\langle \xi_1, \ldots, \xi_r\rangle\] the sheaf of strictly commutative differential graded \(\mathcal{O}_Y\)-algebras which is a (divided power) polynomial algebra on \(\xi_1, \ldots, \xi_r\) in degree \(-1\) over \(\mathcal{O}_Y\) with differential \(\text{d}\) given by the rule \(\text{d}\xi_i = f_i\). Let us denote \(\mathcal{A}^\bullet\) the underlying complex of \(\mathcal{O}_Y\)-modules which is the Koszul complex mentioned above. Thus the canonical map \(\mathcal{A}^\bullet \to \mathcal{O}_X\) is a quasi-isomorphism. We obtain quasi-isomorphisms \[R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \to \SheafHom^\bullet(\mathcal{A}^\bullet, \mathcal{A}^\bullet) \to \SheafHom^\bullet(\mathcal{A}^\bullet, \mathcal{O}_X)\] by Cohomology, Lemma 08DM. The differentials of the latter complex are zero, and hence \[\SheafExt^i_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X) \cong \SheafHom_{\mathcal{O}_Y}(\mathcal{A}^{-i}, \mathcal{O}_X)\] For \(j \in \{1, \ldots, r\}\) let \(\delta_j : \mathcal{A} \to \mathcal{A}\) be the derivation of degree \(1\) with \(\delta_j(\xi_i) = \delta_{ij}\) (Kronecker delta). A computation shows that \(\delta_j \circ \text{d} = - \text{d} \circ \delta_j\) which shows that we get a morphism of complexes \[\delta_j : \mathcal{A}^\bullet \to \mathcal{A}^\bullet[1].\] Whence \(\delta_j\) defines a section of the corresponding \(\SheafExt\)-sheaf. Another computation shows that \(\delta_1, \ldots, \delta_r\) map to a basis for \(\SheafHom_{\mathcal{O}_Y}(\mathcal{A}^{-1}, \mathcal{O}_X)\) over \(\mathcal{O}_X\). Since it is clear that \(\delta_j \circ \delta_j = 0\) and \(\delta_j \circ \delta_{j'} = - \delta_{j'} \circ \delta_j\) as endomorphisms of \(\mathcal{A}\) and hence in the \(\SheafExt\)-sheaves we obtain the statement that our map above factors through the exterior power. To see we get the desired isomorphism the reader checks that the elements \[\delta_{j_1} \circ \ldots \circ \delta_{j_i}\] for \(j_1 < \ldots < j_i\) map to a basis of the sheaf \(\SheafHom_{\mathcal{O}_Y}(\mathcal{A}^{-i}, \mathcal{O}_X)\) over \(\mathcal{O}_X\).

Lemma

Let \(Y\) be a ringed space. Let \(\mathcal{I} \subset \mathcal{O}_Y\) be a sheaf of ideals. Set \(\mathcal{O}_X = \mathcal{O}_Y/\mathcal{I}\) and \(\mathcal{N} = \SheafHom_{\mathcal{O}_Y}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_X)\). If \(\mathcal{I}\) is Koszul-regular (Divisors, Definition 063D) then \[R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_Y) = \wedge^r \mathcal{N}[-r]\] where \(r : Y \to \{0, 1, 2, \ldots \}\) sends \(y\) to the minimal number of generators of \(\mathcal{I}\) needed in a neighbourhood of \(y\).

Proof

We can use Lemmas 0BQW and 0BQY to see that we have isomorphisms \(\wedge^i\mathcal{N} \to \SheafExt^i_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\) for \(i \geq 0\). Thus it suffices to show that the map \(\mathcal{O}_Y \to \mathcal{O}_X\) induces an isomorphism \[\SheafExt^r_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_Y) \longrightarrow \SheafExt^r_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)\] and that \(\SheafExt^i_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_Y)\) is zero for \(i \not = r\). These statements are local on \(Y\). Thus we may assume that we have global sections \(f_1, \ldots, f_r\) of \(\mathcal{O}_Y\) which generate \(\mathcal{I}\) and which form a Koszul regular sequence. Let \(\mathcal{A}^\bullet\) be the Koszul complex on \(f_1, \ldots, f_r\) as introduced in the proof of Lemma 0BQY. Then \[R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_Y) = \SheafHom^\bullet(\mathcal{A}^\bullet, \mathcal{O}_Y)\] by Cohomology, Lemma 08DM. Denote \(1 : \mathcal{A}^\bullet \to \mathcal{O}_Y\) the map of differential graded \(\mathcal{O}_Y\)-algebras given by the identity map of \(\mathcal{A}^0 = \mathcal{O}_Y \to \mathcal{O}_Y\) in degree \(0\). With \(\delta_j\) as in the proof of Lemma 0BQY we get an isomorphism of graded \(\mathcal{O}_Y\)-modules \[\mathcal{O}_Y\langle \delta_1, \ldots, \delta_r\rangle \longrightarrow \SheafHom^\bullet(\mathcal{A}^\bullet, \mathcal{O}_Y)\] by mapping \(\delta_{j_1} \ldots \delta_{j_i}\) to \(1 \circ \delta_{j_1} \circ \ldots \circ \delta_{j_i}\) in degree \(i\). Via this isomorphism the differential on the right hand side induces a differential \(\text{d}\) on the left hand side. By our sign rules we have \(\text{d}(1) = - \sum f_j \delta_j\). Since \(\delta_j : \mathcal{A}^\bullet \to \mathcal{A}^\bullet[1]\) is a morphism of complexes, it follows that \[\text{d}(\delta_{j_1} \ldots \delta_{j_i}) = (- \sum f_j \delta_j )\delta_{j_1} \ldots \delta_{j_i}\] Observe that we have \(\text{d} = \sum f_j \delta_j\) on the differential graded algebra \(\mathcal{A}\). Therefore the map defined by the rule \[1 \circ \delta_{j_1} \ldots \delta_{j_i} \longmapsto (\delta_{j_1} \circ \ldots \circ \delta_{j_i})(\xi_1 \ldots \xi_r)\] will define an isomorphism of complexes \[\SheafHom^\bullet(\mathcal{A}^\bullet, \mathcal{O}_Y) \longrightarrow \mathcal{A}^\bullet[-r]\] if \(r\) is odd and commuting with differentials up to sign if \(r\) is even. In any case these complexes have isomorphic cohomology, which shows the desired vanishing. The isomorphism on cohomology in degree \(r\) under the map \[\SheafHom^\bullet(\mathcal{A}^\bullet, \mathcal{O}_Y) \longrightarrow \SheafHom^\bullet(\mathcal{A}^\bullet, \mathcal{O}_X)\] also follows in a straightforward manner from this. (We observe that our choice of conventions regarding Koszul complexes does intervene in the definition of the isomorphism \(R\SheafHom_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_Y) = \wedge^r \mathcal{N}[-r]\).)

Lemma

Let \(Y\) be a quasi-compact and quasi-separated scheme. Let \(i : X \to Y\) be a Koszul-regular closed immersion. Let \(a\) be the right adjoint of \(Ri_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) of Lemma 0A9E. Then there is an isomorphism \[\wedge^r\mathcal{N}[-r] \longrightarrow a(\mathcal{O}_Y)\] where \(\mathcal{N} = \SheafHom_{\mathcal{O}_X}(\mathcal{C}_{X/Y}, \mathcal{O}_X)\) is the normal sheaf of \(i\) (Morphisms, Section 01R1) and \(r\) is its rank viewed as a locally constant function on \(X\).

Proof

Recall, from Lemmas 0A9X and 0A77, that \(a(\mathcal{O}_Y)\) is an object of \(D_\QCoh(\mathcal{O}_X)\) whose pushforward to \(Y\) is \(R\SheafHom_{\mathcal{O}_Y}(i_*\mathcal{O}_X, \mathcal{O}_Y)\). Thus the result follows from Lemma 0BQZ.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a smooth proper morphism of relative dimension \(d\). Let \(a\) be the right adjoint of \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)\) as in Lemma 0A9E. Then there is an isomorphism \[\wedge^d \Omega_{X/S}[d] \longrightarrow a(\mathcal{O}_S)\] in \(D(\mathcal{O}_X)\).

Proof

Set \(\omega_{X/S}^\bullet = a(\mathcal{O}_S)\) as in Remark 0B6S. Let \(c\) be the right adjoint of Lemma 0A9E for \(\Delta : X \to X \times_S X\). Because \(\Delta\) is the diagonal of a smooth morphism it is a Koszul-regular immersion, see Divisors, Lemma 067U. In particular, \(\Delta\) is a perfect proper morphism (More on Morphisms, Lemma 068C) and we obtain \[\begin{align*} \mathcal{O}_X & = c(L\text{pr}_1^*\omega_{X/S}^\bullet) \\ & = L\Delta^*(L\text{pr}_1^*\omega_{X/S}^\bullet) \otimes_{\mathcal{O}_X}^\mathbf{L} c(\mathcal{O}_{X \times_S X}) \\ & = \omega_{X/S}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} c(\mathcal{O}_{X \times_S X}) \\ & = \omega_{X/S}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} \wedge^d(\mathcal{N}_\Delta)[-d] \end{align*}\] The first equality is (0E2Q) because \(\omega_{X/S}^\bullet = a(\mathcal{O}_S)\). The second equality by Lemma 0A9U. The third equality because \(\text{pr}_1 \circ \Delta = \text{id}_X\). The fourth equality by Lemma 0BR0. Observe that \(\wedge^d(\mathcal{N}_\Delta)\) is an invertible \(\mathcal{O}_X\)-module. Hence \(\wedge^d(\mathcal{N}_\Delta)[-d]\) is an invertible object of \(D(\mathcal{O}_X)\) and we conclude that \(a(\mathcal{O}_S) = \omega_{X/S}^\bullet = \wedge^d(\mathcal{C}_\Delta)[d]\). Since the conormal sheaf \(\mathcal{C}_\Delta\) of \(\Delta\) is \(\Omega_{X/S}\) by Morphisms, Lemma 08S2 the proof is complete.

Upper shriek functors

In this section, we construct the functors \(f^!\) for morphisms between schemes which are of finite type and separated over a fixed Noetherian base using compactifications. As is customary in coherent duality, there are a number of diagrams that have to be shown to be commutative. We suggest the reader, after reading the construction, skips the verification of the lemmas and continues to the next section where we discuss properties of the upper shriek functors.

Situation

Here \(S\) is a Noetherian scheme and \(\textit{FTS}_S\) is the category whose

1. objects are schemes \(X\) over \(S\) such that the structure morphism \(X \to S\) is both separated and of finite type, and

2. morphisms \(f : X \to Y\) between objects are morphisms of schemes over \(S\).

In Situation 0F42 given a morphism \(f : X \to Y\) in \(\textit{FTS}_S\), we will define an exact functor \[f^! : D_\QCoh^+(\mathcal{O}_Y) \to D_\QCoh^+(\mathcal{O}_X)\] of triangulated categories. Namely, we choose a compactification \(X \to \overline{X}\) over \(Y\) which is possible by More on Flatness, Theorem 0F41 and Lemma 0A9Z. Denote \(\overline{f} : \overline{X} \to Y\) the structure morphism. Let \(\overline{a} : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_{\overline{X}})\) be the right adjoint of \(R\overline{f}_*\) constructed in Lemma 0A9E. Then we set \[f^!K = \overline{a}(K)|_X\] for \(K \in D_\QCoh^+(\mathcal{O}_Y)\). The result is an object of \(D_\QCoh^+(\mathcal{O}_X)\) by Lemma 0A9I.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). The functor \(f^!\) is, up to canonical isomorphism, independent of the choice of the compactification.

Proof

The category of compactifications of \(X\) over \(Y\) is defined in More on Flatness, Section 0ATT. By More on Flatness, Theorem 0F41 and Lemma 0A9Z it is nonempty. To every choice of a compactification \[j : X \to \overline{X},\quad \overline{f} : \overline{X} \to Y\] the construction above associates the functor \(j^* \circ \overline{a} : D_\QCoh^+(\mathcal{O}_Y) \to D_\QCoh^+(\mathcal{O}_X)\) where \(\overline{a}\) is the right adjoint of \(R\overline{f}_*\) constructed in Lemma 0A9E.

Suppose given a morphism \(g : \overline{X}_1 \to \overline{X}_2\) between compactifications \(j_i : X \to \overline{X}_i\) over \(Y\) such that \(g^{-1}(j_2(X)) = j_1(X)\)⁵. Let \(\overline{c}\) be the right adjoint of Lemma 0A9E for \(g\). Then \(\overline{c} \circ \overline{a}_2 = \overline{a}_1\) because these functors are adjoint to \(R\overline{f}_{2, *} \circ Rg_* = R(\overline{f}_2 \circ g)_*\). By (0A9L) we have a canonical transformation \[j_1^* \circ \overline{c} \longrightarrow j_2^*\] of functors \(D^+_\QCoh(\mathcal{O}_{\overline{X}_2}) \to D^+_\QCoh(\mathcal{O}_X)\) which is an isomorphism by Lemma 0A9P. The composition \[j_1^* \circ \overline{a}_1 \longrightarrow j_1^* \circ \overline{c} \circ \overline{a}_2 \longrightarrow j_2^* \circ \overline{a}_2\] is an isomorphism of functors which we will denote by \(\alpha_g\).

Consider two compactifications \(j_i : X \to \overline{X}_i\), \(i = 1, 2\) of \(X\) over \(Y\). By More on Flatness, Lemma 0ATU part (b) we can find a compactification \(j : X \to \overline{X}\) with dense image and morphisms \(g_i : \overline{X} \to \overline{X}_i\) of compactifications. By More on Flatness, Lemma 0ATU part (c) we have \(g_i^{-1}(j_i(X)) = j(X)\). Hence we get isomorpisms \[\alpha_{g_i} : j^* \circ \overline{a} \longrightarrow j_i^* \circ \overline{a}_i\] by the previous paragraph. We obtain an isomorphism \[\alpha_{g_2} \circ \alpha_{g_1}^{-1} : j_1^* \circ \overline{a}_1 \to j_2^* \circ \overline{a}_2\] To finish the proof we have to show that these isomorphisms are well defined. We claim it suffices to show the composition of isomorphisms constructed in the previous paragraph is another (for a precise statement see the next paragraph). We suggest the reader check this is true on a napkin, but we will also completely spell it out in the rest of this paragraph. Namely, consider a second choice of a compactification \(j' : X \to \overline{X}'\) with dense image and morphisms of compactifications \(g'_i : \overline{X}' \to \overline{X}_i\). By More on Flatness, Lemma 0ATU we can find a compactification \(j'' : X \to \overline{X}''\) with dense image and morphisms of compactifications \(h : \overline{X}'' \to \overline{X}\) and \(h' : \overline{X}'' \to \overline{X}'\). We may even assume \(g_1 \circ h = g'_1 \circ h'\) and \(g_2 \circ h = g'_2 \circ h'\). The result of the next paragraph gives \[\alpha_{g_i} \circ \alpha_h = \alpha_{g_i \circ h} = \alpha_{g'_i \circ h'} = \alpha_{g'_i} \circ \alpha_{h'}\] for \(i = 1, 2\). Since these are all isomorphisms of functors we conclude that \(\alpha_{g_2} \circ \alpha_{g_1}^{-1} = \alpha_{g'_2} \circ \alpha_{g'_1}^{-1}\) as desired.

Suppose given compactifications \(j_i : X \to \overline{X}_i\) for \(i = 1, 2, 3\). Suppose given morphisms \(g : \overline{X}_1 \to \overline{X}_2\) and \(h : \overline{X}_2 \to \overline{X}_3\) of compactifications such that \(g^{-1}(j_2(X)) = j_1(X)\) and \(h^{-1}(j_2(X)) = j_3(X)\). Let \(\overline{a}_i\) be as above. The claim above means that \[\alpha_g \circ \alpha_h = \alpha_{g \circ h} : j_1^* \circ \overline{a}_1 \to j_3^* \circ \overline{a}_3\] Let \(\overline{c}\), resp. \(\overline{d}\) be the right adjoint of Lemma 0A9E for \(g\), resp. \(h\). Then \(\overline{c} \circ \overline{a}_2 = \overline{a}_1\) and \(\overline{d} \circ \overline{a}_3 = \overline{a}_2\) and there are canonical transformations \[j_1^* \circ \overline{c} \longrightarrow j_2^* \quad\text{and}\quad j_2^* \circ \overline{d} \longrightarrow j_3^*\] of functors \(D^+_\QCoh(\mathcal{O}_{\overline{X}_2}) \to D^+_\QCoh(\mathcal{O}_X)\) and \(D^+_\QCoh(\mathcal{O}_{\overline{X}_3}) \to D^+_\QCoh(\mathcal{O}_X)\) for the same reasons as above. Denote \(\overline{e}\) the right adjoint of Lemma 0A9E for \(h \circ g\). There is a canonical transformation \[j_1^* \circ \overline{e} \longrightarrow j_3^*\] of functors \(D^+_\QCoh(\mathcal{O}_{\overline{X}_3}) \to D^+_\QCoh(\mathcal{O}_X)\) given by (0A9L). Spelling things out we have to show that the composition \[\alpha_h \circ \alpha_g : j_1^* \circ \overline{a}_1 \to j_1^* \circ \overline{c} \circ \overline{a}_2 \to j_2^* \circ \overline{a}_2 \to j_2^* \circ \overline{d} \circ \overline{a}_3 \to j_3^* \circ \overline{a}_3\] is the same as the composition \[\alpha_{h \circ g} : j_1^* \circ \overline{a}_1 \to j_1^* \circ \overline{e} \circ \overline{a}_3 \to j_3^* \circ \overline{a}_3\] We split this into two parts. The first is to show that the diagram \[\xymatrix{ \overline{a}_1 \ar[r] \ar[d] & \overline{c} \circ \overline{a}_2 \ar[d] \\ \overline{e} \circ \overline{a}_3 \ar[r] & \overline{c} \circ \overline{d} \circ \overline{a}_3 }\] commutes where the lower horizontal arrow comes from the identification \(\overline{e} = \overline{c} \circ \overline{d}\). This is true because the corresponding diagram of total direct image functors \[\xymatrix{ R\overline{f}_{1, *} \ar[r] \ar[d] & Rg_* \circ R\overline{f}_{2, *} \ar[d] \\ R(h \circ g)_* \circ R\overline{f}_{3, *} \ar[r] & Rg_* \circ Rh_* \circ R\overline{f}_{3, *} }\] is commutative (insert future reference here). The second part is to show that the composition \[j_1^* \circ \overline{c} \circ \overline{d} \to j_2^* \circ \overline{d} \to j_3^*\] is equal to the map \[j_1^* \circ \overline{e} \to j_3^*\] via the identification \(\overline{e} = \overline{c} \circ \overline{d}\). This was proven in Lemma 0ATQ (note that in the current case the morphisms \(f', g'\) of that lemma are equal to \(\text{id}_X\)).

Lemma

In Situation 0F42 let \(f : X \to Y\) and \(g : Y \to Z\) be composable morphisms of \(\textit{FTS}_S\). Then there is a canonical isomorphism \((g \circ f)^! \to f^! \circ g^!\).

Proof

Choose a compactification \(i : Y \to \overline{Y}\) of \(Y\) over \(Z\). Choose a compactification \(X \to \overline{X}\) of \(X\) over \(\overline{Y}\). This uses More on Flatness, Theorem 0F41 and Lemma 0A9Z twice. Let \(\overline{a}\) be the right adjoint of Lemma 0A9E for \(\overline{X} \to \overline{Y}\) and let \(\overline{b}\) be the right adjoint of Lemma 0A9E for \(\overline{Y} \to Z\). Then \(\overline{a} \circ \overline{b}\) is the right adjoint of Lemma 0A9E for the composition \(\overline{X} \to Z\). Hence \(g^! = i^* \circ \overline{b}\) and \((g \circ f)^! = (X \to \overline{X})^* \circ \overline{a} \circ \overline{b}\). Let \(U\) be the inverse image of \(Y\) in \(\overline{X}\) so that we get the commutative diagram \[\xymatrix{ X \ar[r]_j \ar[d] & U \ar[dl] \ar[r]_{j'} & \overline{X} \ar[dl] \\ Y \ar[r]_i \ar[d] & \overline{Y} \ar[dl] \\ Z }\] Let \(\overline{a}'\) be the right adjoint of Lemma 0A9E for \(U \to Y\). Then \(f^! = j^* \circ \overline{a}'\). We obtain \[\gamma : (j')^* \circ \overline{a} \to \overline{a}' \circ i^*\] by (0A9L) and we can use it to define \[(g \circ f)^! = (j' \circ j)^* \circ \overline{a} \circ \overline{b} = j^* \circ (j')^* \circ \overline{a} \circ \overline{b} \to j^* \circ \overline{a}' \circ i^* \circ \overline{b} = f^! \circ g^!\] which is an isomorphism on objects of \(D_\QCoh^+(\mathcal{O}_Z)\) by Lemma 0A9P. To finish the proof we show that this isomorphism is independent of choices made.

Suppose we have two diagrams \[\vcenter{ \xymatrix{ X \ar[r]_{j_1} \ar[d] & U_1 \ar[dl] \ar[r]_{j'_1} & \overline{X}_1 \ar[dl] \\ Y \ar[r]_{i_1} \ar[d] & \overline{Y}_1 \ar[dl] \\ Z } } \quad\text{and}\quad \vcenter{ \xymatrix{ X \ar[r]_{j_2} \ar[d] & U_2 \ar[dl] \ar[r]_{j'_2} & \overline{X}_2 \ar[dl] \\ Y \ar[r]_{i_2} \ar[d] & \overline{Y}_2 \ar[dl] \\ Z } }\] We can first choose a compactification \(i : Y \to \overline{Y}\) with dense image of \(Y\) over \(Z\) which dominates both \(\overline{Y}_1\) and \(\overline{Y}_2\), see More on Flatness, Lemma 0ATU. By More on Flatness, Lemma 0ATV and Categories, Lemmas 04VI and 04VJ we can choose a compactification \(X \to \overline{X}\) with dense image of \(X\) over \(\overline{Y}\) with morphisms \(\overline{X} \to \overline{X}_1\) and \(\overline{X} \to \overline{X}_2\) and such that the composition \(\overline{X} \to \overline{Y} \to \overline{Y}_1\) is equal to the composition \(\overline{X} \to \overline{X}_1 \to \overline{Y}_1\) and such that the composition \(\overline{X} \to \overline{Y} \to \overline{Y}_2\) is equal to the composition \(\overline{X} \to \overline{X}_2 \to \overline{Y}_2\). Thus we see that it suffices to compare the maps determined by our diagrams when we have a commutative diagram as follows \[\xymatrix{ X \ar[rr]_{j_1} \ar@{=}[d] & & U_1 \ar[d] \ar[ddll] \ar[rr]_{j'_1} & & \overline{X}_1 \ar[d] \ar[ddll] \\ X \ar'[r][rr]^-{j_2} \ar[d] & & U_2 \ar'[dl][ddll] \ar'[r][rr]^-{j'_2} & & \overline{X}_2 \ar[ddll] \\ Y \ar[rr]^{i_1} \ar@{=}[d] & & \overline{Y}_1 \ar[d] \\ Y \ar[rr]^{i_2} \ar[d] & & \overline{Y}_2 \ar[dll] \\ Z }\] and moreover the compactifications \(X \to \overline{X}_1\) and \(Y \to \overline{Y}_2\) have dense image. We use \(\overline{a}_i\), \(\overline{a}'_i\), \(\overline{c}\), and \(\overline{c}'\) for the right adjoint of Lemma 0A9E for \(\overline{X}_i \to \overline{Y}_i\), \(U_i \to Y\), \(\overline{X}_1 \to \overline{X}_2\), and \(U_1 \to U_2\). Each of the squares \[\xymatrix{ X \ar[r] \ar[d] \ar@{}[dr]|A & U_1 \ar[d] \\ X \ar[r] & U_2 } \quad \xymatrix{ U_2 \ar[r] \ar[d] \ar@{}[dr]|B & \overline{X}_2 \ar[d] \\ Y \ar[r] & \overline{Y}_2 } \quad \xymatrix{ U_1 \ar[r] \ar[d] \ar@{}[dr]|C & \overline{X}_1 \ar[d] \\ Y \ar[r] & \overline{Y}_1 } \quad \xymatrix{ Y \ar[r] \ar[d] \ar@{}[dr]|D & \overline{Y}_1 \ar[d] \\ Y \ar[r] & \overline{Y}_2 } \quad \xymatrix{ X \ar[r] \ar[d] \ar@{}[dr]|E & \overline{X}_1 \ar[d] \\ X \ar[r] & \overline{X}_2 }\] is cartesian (see More on Flatness, Lemma 0ATU part (c) for A, D, E and recall that \(U_i\) is the inverse image of \(Y\) by \(\overline{X}_i \to \overline{Y}_i\) for B, C) and hence gives rise to a base change map (0A9L) as follows \[\begin{matrix} \gamma_A : j_1^* \circ \overline{c}' \to j_2^* & \gamma_B : (j_2')^* \circ \overline{a}_2 \to \overline{a}'_2 \circ i_2^* & \gamma_C : (j_1')^* \circ \overline{a}_1 \to \overline{a}'_1 \circ i_1^* \\ \gamma_D : i_1^* \circ \overline{d} \to i_2^* & \gamma_E : (j'_1 \circ j_1)^* \circ \overline{c} \to (j'_2 \circ j_2)^* \end{matrix}\] Denote \(f_1^! = j_1^* \circ \overline{a}'_1\), \(f_2^! = j_2^* \circ \overline{a}'_2\), \(g_1^! = i_1^* \circ \overline{b}_1\), \(g_2^! = i_2^* \circ \overline{b}_2\), \((g \circ f)_1^! = (j_1' \circ j_1)^* \circ \overline{a}_1 \circ \overline{b}_1\), and \((g \circ f)^!_2 = (j_2' \circ j_2)^* \circ \overline{a}_2 \circ \overline{b}_2\). The construction given in the first paragraph of the proof and in Lemma 0AA0 uses

1. \(\gamma_C\) for the map \((g \circ f)^!_1 \to f_1^! \circ g_1^!\),

2. \(\gamma_B\) for the map \((g \circ f)^!_2 \to f_2^! \circ g_2^!\),

3. \(\gamma_A\) for the map \(f_1^! \to f_2^!\),

4. \(\gamma_D\) for the map \(g_1^! \to g_2^!\), and

5. \(\gamma_E\) for the map \((g \circ f)^!_1 \to (g \circ f)^!_2\).

We have to show that the diagram \[\xymatrix{ (g \circ f)^!_1 \ar[r]_{\gamma_E} \ar[d]_{\gamma_C} & (g \circ f)^!_2 \ar[d]_{\gamma_B} \\ f_1^! \circ g_1^! \ar[r]^{\gamma_A \circ \gamma_D} & f_2^! \circ g_2^! }\] is commutative. We will use Lemmas 0ATQ and 0ATR and with (abuse of) notation as in Remark 0ATS (in particular dropping \(\star\) products with identity transformations from the notation). We can write \(\gamma_E = \gamma_A \circ \gamma_F\) where \[\xymatrix{ U_1 \ar[r] \ar[d] \ar@{}[rd]|F & \overline{X}_1 \ar[d] \\ U_2 \ar[r] & \overline{X}_2 }\] Thus we see that \[\gamma_B \circ \gamma_E = \gamma_B \circ \gamma_A \circ \gamma_F = \gamma_A \circ \gamma_B \circ \gamma_F\] the last equality because the two squares \(A\) and \(B\) only intersect in one point (similar to the last argument in Remark 0ATS). Thus it suffices to prove that \(\gamma_D \circ \gamma_C = \gamma_B \circ \gamma_F\). Since both of these are equal to the map (0A9L) for the square \[\xymatrix{ U_1 \ar[r] \ar[d] & \overline{X}_1 \ar[d] \\ Y \ar[r] & \overline{Y}_2 }\] we conclude.

Lemma

In Situation 0F42 the constructions of Lemmas 0AA0 and 0ATX define a pseudo functor from the category \(\textit{FTS}_S\) into the \(2\)-category of categories (see Categories, Definition 003N).

Proof

To show this we have to prove given morphisms \(f : X \to Y\), \(g : Y \to Z\), \(h : Z \to T\) that \[\xymatrix{ (h \circ g \circ f)^! \ar[r]_{\gamma_{A + B}} \ar[d]_{\gamma_{B + C}} & f^! \circ (h \circ g)^! \ar[d]^{\gamma_C} \\ (g \circ f)^! \circ h^! \ar[r]^{\gamma_A} & f^! \circ g^! \circ h^! }\] is commutative (for the meaning of the \(\gamma\)’s, see below). To do this we choose a compactification \(\overline{Z}\) of \(Z\) over \(T\), then a compactification \(\overline{Y}\) of \(Y\) over \(\overline{Z}\), and then a compactification \(\overline{X}\) of \(X\) over \(\overline{Y}\). This uses More on Flatness, Theorem 0F41 and Lemma 0A9Z. Let \(W \subset \overline{Y}\) be the inverse image of \(Z\) under \(\overline{Y} \to \overline{Z}\) and let \(U \subset V \subset \overline{X}\) be the inverse images of \(Y \subset W\) under \(\overline{X} \to \overline{Y}\). This produces the following diagram \[\xymatrix{ X \ar[d]_f \ar[r] & U \ar[r] \ar[d] \ar@{}[dr]|A & V \ar[d] \ar[r] \ar@{}[rd]|B & \overline{X} \ar[d] \\ Y \ar[d]_g \ar[r] & Y \ar[r] \ar[d] & W \ar[r] \ar[d] \ar@{}[rd]|C & \overline{Y} \ar[d] \\ Z \ar[d]_h \ar[r] & Z \ar[d] \ar[r] & Z \ar[d] \ar[r] & \overline{Z} \ar[d] \\ T \ar[r] & T \ar[r] & T \ar[r] & T }\] Without introducing tons of notation but arguing exactly as in the proof of Lemma 0ATX we see that the maps in the first displayed diagram use the maps (0A9L) for the rectangles \(A + B\), \(B + C\), \(A\), and \(C\) as indicated. Since by Lemmas 0ATQ and 0ATR we have \(\gamma_{A + B} = \gamma_A \circ \gamma_B\) and \(\gamma_{B + C} = \gamma_C \circ \gamma_B\) we conclude that the desired equality holds provided \(\gamma_A \circ \gamma_C = \gamma_C \circ \gamma_A\). This is true because the two squares \(A\) and \(C\) only intersect in one point (similar to the last argument in Remark 0ATS).

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). There are canonical maps \[\mu_{f, K} : Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \longrightarrow f^!K\] functorial in \(K\) in \(D^+_\QCoh(\mathcal{O}_Y)\). If \(g : Y \to Z\) is another morphism of \(\textit{FTS}_S\), then the diagram \[\xymatrix{ Lf^*(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} g^!\mathcal{O}_Z) \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \ar@{=}[d] \ar[r]_-{\mu_f} & f^!(Lg^*K \otimes_{\mathcal{O}_Y}^\mathbf{L} g^!\mathcal{O}_Z) \ar[r]_-{f^!\mu_g} & f^!g^!K \ar@{=}[d] \\ Lf^*Lg^*K \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^* g^!\mathcal{O}_Z \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \ar[r]^-{\mu_f} & Lf^*Lg^*K \otimes_{\mathcal{O}_X}^\mathbf{L} f^!g^!\mathcal{O}_Z \ar[r]^-{\mu_{g \circ f}} & f^!g^!K }\] commutes for all \(K \in D^+_\QCoh(\mathcal{O}_Z)\).

Proof

If \(f\) is proper, then \(f^! = a\) and we can use (0A9S) and if \(g\) is also proper, then Lemma 0B6R proves the commutativity of the diagram (in greater generality).

Let us define the map \(\mu_{f, K}\). Choose a compactification \(j : X \to \overline{X}\) of \(X\) over \(Y\). Since \(f^!\) is defined as \(j^* \circ \overline{a}\) we obtain \(\mu_{f, K}\) as the restriction of the map (0A9S) \[L\overline{f}^*K \otimes_{\mathcal{O}_{\overline{X}}}^\mathbf{L} \overline{a}(\mathcal{O}_Y) \longrightarrow \overline{a}(K)\] to \(X\). To see this is independent of the choice of the compactification we argue as in the proof of Lemma 0AA0. We urge the reader to read the proof of that lemma first.

Assume given a morphism \(g : \overline{X}_1 \to \overline{X}_2\) between compactifications \(j_i : X \to \overline{X}_i\) over \(Y\) such that \(g^{-1}(j_2(X)) = j_1(X)\). Denote \(\overline{c}\) the right adjoint for pushforward of Lemma 0A9E for the morphism \(g\). The maps \[L\overline{f}_1^*K \otimes_{\mathcal{O}_{\overline{X}}}^\mathbf{L} \overline{a}_1(\mathcal{O}_Y) \longrightarrow \overline{a}_1(K) \quad\text{and}\quad L\overline{f}_2^*K \otimes_{\mathcal{O}_{\overline{X}}}^\mathbf{L} \overline{a}_2(\mathcal{O}_Y) \longrightarrow \overline{a}_2(K)\] fit into the commutative diagram \[\xymatrix{ Lg^*(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)) \otimes^\mathbf{L} \overline{c}(\mathcal{O}_{\overline{X}_2}) \ar@{=}[d] \ar[r]_-\sigma & \overline{c}(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)) \ar[r] & \overline{c}(\overline{a}_2(K)) \ar@{=}[d] \\ L\overline{f}_1^*K \otimes^\mathbf{L} Lg^*\overline{a}_2(\mathcal{O}_Y) \otimes^\mathbf{L} \overline{c}(\mathcal{O}_{\overline{X}_2}) \ar[r]^-{1 \otimes \tau} & L\overline{f}_1^*K \otimes^\mathbf{L} \overline{a}_1(\mathcal{O}_Y) \ar[r] & \overline{a}_1(K) }\] by Lemma 0B6R. By Lemma 0B6Q the maps \(\sigma\) and \(\tau\) restrict to an isomorphism over \(X\). In fact, we can say more. Recall that in the proof of Lemma 0AA0 we used the map (0A9L) \(\gamma : j_1^* \circ \overline{c} \to j_2^*\) to construct our isomorphism \(\alpha_g : j_1^* \circ \overline{a}_1 \to j_2^* \circ \overline{a}_2\). Pulling back to map \(\sigma\) by \(j_1\) we obtain the identity map on \(j_2^*\left(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)\right)\) if we identify \(j_1^*\overline{c}(\mathcal{O}_{\overline{X}_2})\) with \(\mathcal{O}_X\) via \(j_1^* \circ \overline{c} \to j_2^*\), see Lemma 0B6P. Similarly, the map \(\tau : Lg^*\overline{a}_2(\mathcal{O}_Y) \otimes^\mathbf{L} \overline{c}(\mathcal{O}_{\overline{X}_2}) \to \overline{a}_1(\mathcal{O}_Y) = \overline{c}(\overline{a}_2(\mathcal{O}_Y))\) pulls back to the identity map on \(j_2^*\overline{a}_2(\mathcal{O}_Y)\). We conclude that pulling back by \(j_1\) and applying \(\gamma\) wherever we can we obtain a commutative diagram \[\xymatrix{ j_2^*\left(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)\right) \ar[r] \ar[d] & j_2^*\overline{a}_2(K) \\ j_1^*L\overline{f}_1^*K \otimes^\mathbf{L} j_2^*\overline{a}_2(\mathcal{O}_Y) & j_1^*(L\overline{f}_1^*K \otimes^\mathbf{L} \overline{a}_1(\mathcal{O}_Y)) \ar[r] \ar[l]_{1 \otimes \alpha_g} & j_1^* \overline{a}_1(K) \ar[lu]_{\alpha_g} }\] The commutativity of this diagram exactly tells us that the map \(\mu_{f, K}\) constructed using the compactification \(\overline{X}_1\) is the same as the map \(\mu_{f, K}\) constructed using the compactification \(\overline{X}_2\) via the identification \(\alpha_g\) used in the proof of Lemma 0AA0. Some categorical arguments exactly as in the proof of Lemma 0AA0 now show that \(\mu_{f, K}\) is well defined (small detail omitted).

Having said this, the commutativity of the diagram in the statement of our lemma follows from the construction of the isomorphism \((g \circ f)^! \to f^! \circ g^!\) (first part of the proof of Lemma 0ATX using \(\overline{X} \to \overline{Y} \to Z\)) and the result of Lemma 0B6R for \(\overline{X} \to \overline{Y} \to Z\).

Properties of upper shriek functors

Here are some properties of the upper shriek functors.

Lemma

In Situation 0F42 let \(Y\) be an object of \(\textit{FTS}_S\) and let \(j : X \to Y\) be an open immersion. Then there is a canonical isomorphism \(j^! = j^*\) of functors.

For an étale morphism \(f : X \to Y\) of \(\textit{FTS}_S\) we also have \(f^* \cong f^!\), see Lemma 0FWI.

Proof

In this case we may choose \(\overline{X} = Y\) as our compactification. Then the right adjoint of Lemma 0A9E for \(\text{id} : Y \to Y\) is the identity functor and hence \(j^! = j^*\) by definition.

Lemma

In Situation 0F42 let \[\xymatrix{ U \ar[r]_j \ar[d]_g & X \ar[d]^f \\ V \ar[r]^{j'} & Y }\] be a commutative diagram of \(\textit{FTS}_S\) where \(j\) and \(j'\) are open immersions. Then \(j^* \circ f^! = g^! \circ (j')^*\) as functors \(D^+_\QCoh(\mathcal{O}_Y) \to D^+(\mathcal{O}_U)\).

Proof

Let \(h = f \circ j = j' \circ g\). By Lemma 0ATX we have \(h^! = j^! \circ f^! = g^! \circ (j')^!\). By Lemma 0AU0 we have \(j^! = j^*\) and \((j')^! = (j')^*\).

Lemma

In Situation 0F42 let \(Y\) be an object of \(\textit{FTS}_S\) and let \(f : X = \mathbf{A}^1_Y \to Y\) be the projection. Then there is a (noncanonical) isomorphism \(f^!(-) \cong Lf^*(-) [1]\) of functors.

Proof

Since \(X = \mathbf{A}^1_Y \subset \mathbf{P}^1_Y\) and since \(\mathcal{O}_{\mathbf{P}^1_Y}(-2)|_X \cong \mathcal{O}_X\) this follows from Lemmas 0A9W and 0A9U.

Lemma

In Situation 0F42 let \(Y\) be an object of \(\textit{FTS}_S\) and let \(i : X \to Y\) be a closed immersion. Then there is a canonical isomorphism \(i^!(-) = R\SheafHom(\mathcal{O}_X, -)\) of functors.

Proof

This is a restatement of Lemma 0A9X.

Remark

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Using the lemmas above we can compute \(f^!\) locally as follows. Suppose that we are given affine opens \[\xymatrix{ U \ar[r]_j \ar[d]_g & X \ar[d]^f \\ V \ar[r]^i & Y }\] Since \(j^! \circ f^! = g^! \circ i^!\) (Lemma 0ATX) and since \(j^!\) and \(i^!\) are given by restriction (Lemma 0AU0) we see that \[(f^!E)|_U = g^!(E|_V)\] for any \(E \in D^+_\QCoh(\mathcal{O}_X)\). Write \(U = \Spec(A)\) and \(V = \Spec(R)\) and let \(\varphi : R \to A\) be the finite type ring map corresponding to \(g\). Choose a presentation \(A = P/I\) where \(P = R[x_1, \ldots, x_n]\) is a polynomial algebra in \(n\) variables over \(R\). Choose an object \(K \in D^+(R)\) corresponding to \(E|_V\) (Derived Categories of Schemes, Lemma 06Z0). Then we claim that \(f^!E|_U\) corresponds to \[\varphi^!(K) = R\Hom(A, K \otimes_R^\mathbf{L} P)[n]\] where \(R\Hom(A, -) : D(P) \to D(A)\) is the functor of Dualizing Complexes, Section 0A6Z and where \(\varphi^! : D(R) \to D(A)\) is the functor of Dualizing Complexes, Section 0BZI. Namely, the choice of presentation gives a factorization \[U \rightarrow \mathbf{A}^n_V \to \mathbf{A}^{n - 1}_V \to \ldots \to \mathbf{A}^1_V \to V\] Applying Lemma 0AA1 exactly \(n\) times we see that \((\mathbf{A}^n_V \to V)^!(E|_V)\) corresponds to \(K \otimes_R^\mathbf{L} P[n]\). By Lemmas 0A78 and 0AA2 the last step corresponds to applying \(R\Hom(A, -)\).

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Then \(f^!\) maps \(D_{\textit{Coh}}^+(\mathcal{O}_Y)\) into \(D_{\textit{Coh}}^+(\mathcal{O}_X)\).

Proof

The question is local on \(X\) hence we may assume that \(X\) and \(Y\) are affine schemes. In this case we can factor \(f : X \to Y\) as \[X \xrightarrow{i} \mathbf{A}^n_Y \to \mathbf{A}^{n - 1}_Y \to \ldots \to \mathbf{A}^1_Y \to Y\] where \(i\) is a closed immersion. The lemma follows from By Lemmas 0AA1 and 0A79 and Dualizing Complexes, Lemma 0A7J and induction.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). If \(K\) is a dualizing complex for \(Y\), then \(f^!K\) is a dualizing complex for \(X\).

Proof

The question is local on \(X\) hence we may assume that \(X\) and \(Y\) are affine schemes. In this case we can factor \(f : X \to Y\) as \[X \xrightarrow{i} \mathbf{A}^n_Y \to \mathbf{A}^{n - 1}_Y \to \ldots \to \mathbf{A}^1_Y \to Y\] where \(i\) is a closed immersion. By Lemma 0AA1 and Dualizing Complexes, Lemma 0A7J and induction we see that the \(p^!K\) is a dualizing complex on \(\mathbf{A}^n_Y\) where \(p : \mathbf{A}^n_Y \to Y\) is the projection. Similarly, by Dualizing Complexes, Lemma 0A7I and Lemmas 0A78 and 0AA2 we see that \(i^!\) transforms dualizing complexes into dualizing complexes.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Let \(K\) be a dualizing complex on \(Y\). Set \(D_Y(M) = R\SheafHom_{\mathcal{O}_Y}(M, K)\) for \(M \in D_{\textit{Coh}}(\mathcal{O}_Y)\) and \(D_X(E) = R\SheafHom_{\mathcal{O}_X}(E, f^!K)\) for \(E \in D_{\textit{Coh}}(\mathcal{O}_X)\). Then there is a canonical isomorphism \[f^!M \longrightarrow D_X(Lf^*D_Y(M))\] for \(M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)\).

Proof

Choose compactification \(j : X \subset \overline{X}\) of \(X\) over \(Y\) (More on Flatness, Theorem 0F41 and Lemma 0A9Z). Let \(a\) be the right adjoint of Lemma 0A9E for \(\overline{X} \to Y\). Set \(D_{\overline{X}}(E) = R\SheafHom_{\mathcal{O}_{\overline{X}}}(E, a(K))\) for \(E \in D_{\textit{Coh}}(\mathcal{O}_{\overline{X}})\). Since formation of \(R\SheafHom\) commutes with restriction to opens and since \(f^! = j^* \circ a\) we see that it suffices to prove that there is a canonical isomorphism \[a(M) \longrightarrow D_{\overline{X}}(L\overline{f}^*D_Y(M))\] for \(M \in D_{\textit{Coh}}(\mathcal{O}_Y)\). For \(F \in D_\QCoh(\mathcal{O}_X)\) we have \[\begin{align*} \Hom_{\overline{X}}( F, D_{\overline{X}}(L\overline{f}^*D_Y(M))) & = \Hom_{\overline{X}}( F \otimes_{\mathcal{O}_X}^\mathbf{L} L\overline{f}^*D_Y(M), a(K)) \\ & = \Hom_Y( R\overline{f}_*(F \otimes_{\mathcal{O}_X}^\mathbf{L} L\overline{f}^*D_Y(M)), K) \\ & = \Hom_Y( R\overline{f}_*(F) \otimes_{\mathcal{O}_Y}^\mathbf{L} D_Y(M), K) \\ & = \Hom_Y( R\overline{f}_*(F), D_Y(D_Y(M))) \\ & = \Hom_Y(R\overline{f}_*(F), M) \\ & = \Hom_{\overline{X}}(F, a(M)) \end{align*}\] The first equality by Cohomology, Lemma 08DJ. The second by definition of \(a\). The third by Derived Categories of Schemes, Lemma 08EU. The fourth equality by Cohomology, Lemma 08DJ and the definition of \(D_Y\). The fifth equality by Lemma 0A89. The final equality by definition of \(a\). Hence we see that \(a(M) = D_{\overline{X}}(L\overline{f}^*D_Y(M))\) by Yoneda’s lemma.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Assume \(f\) is perfect (e.g., flat). Then

1. \(f^!\mathcal{O}_Y\) is in \(D_{\textit{Coh}}^b(\mathcal{O}_X)\),

2. \(f^!\mathcal{O}_Y\) has finite tor dimension in \(D(f^{-1}\mathcal{O}_Y)\),

3. \(\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(f^!\mathcal{O}_Y, f^!\mathcal{O}_Y)\) is an isomorphism,

4. \(f^!\) maps \(D_{\textit{Coh}}^b(\mathcal{O}_Y)\) into \(D_{\textit{Coh}}^b(\mathcal{O}_X)\),

5. the map \(\mu_{f, K} : Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y \to f^!K\) of Lemma 0B6T is an isomorphism for all \(K \in D_\QCoh^+(\mathcal{O}_Y)\).

Proof

(A flat morphism of finite presentation is perfect, see More on Morphisms, Lemma 068A.) Assertions (a), (b), and (c) are local on \(X\). Thus we may assume \(X\) and \(Y\) are affine. Then Remark 0BV2 turns (a), (b), and (c) into (1)(a), (1)(b), and (1)(c) of Dualizing Complexes, Lemma 0BZW. (Use Derived Categories of Schemes, Lemmas 0DHY, 08E5, 08E8 0A6H to match the assertions.)

To prove (d) and (e) we begin with a series of preliminary remarks.

1. We already know that \(f^!\) sends \(D_{\textit{Coh}}^+(\mathcal{O}_Y)\) into \(D_{\textit{Coh}}^+(\mathcal{O}_X)\), see Lemma 0AU1.

2. If \(f\) is an open immersion, then \(f^! = f^*\) and (d) and (e) are true because we can take \(\overline{X} = Y\) in the construction of \(f^!\) and \(\mu_f\), see Lemma 0AU0.

3. If \(f\) is a perfect proper morphism, then (e) is true by Lemma 0A9U.

4. If there exists an open covering \(X = \bigcup U_i\) and (d) is true for \(U_i \to Y\), then (d) is true for \(X \to Y\). Same for (e). This holds because the construction of \(f^!\) and \(\mu_f\) commutes with passing to open subschemes.

5. If \(g : Y \to Z\) is a second perfect morphism in \(\textit{FTS}_S\) and (e) holds for \(f\) and \(g\), then \(f^!g^!\mathcal{O}_Z = Lf^*g^!\mathcal{O}_Z \otimes_{\mathcal{O}_X}^\mathbf{L} f^!\mathcal{O}_Y\) and (e) holds for \(g \circ f\) by the commutative diagram of Lemma 0B6T.

6. If (d) and (e) hold for both \(f\) and \(g\), then (d) and (e) hold for \(g \circ f\). Namely, then \(f^!g^!\mathcal{O}_Z\) is bounded above (by the previous point) and \(L(g \circ f)^*\) has finite cohomological dimension and (d) follows from (e) which we saw above.

From these points we see it suffices to prove (d) and (e) in case \(X\) is affine. Choose an immersion \(X \to \mathbf{A}^n_Y\) (Morphisms, Lemma 04II) which we factor as \(X \to U \to \mathbf{A}^n_Y \to Y\) where \(X \to U\) is a closed immersion and \(U \subset \mathbf{A}^n_Y\) is open. Note that \(X \to U\) is a perfect closed immersion by More on Morphisms, Lemma 068D. Thus it suffices to prove the lemma for a perfect closed immersion and for the projection \(\mathbf{A}^n_Y \to Y\).

Let \(f : X \to Y\) be a perfect closed immersion. We already know (d) holds. Let \(K \in D^b_{\textit{Coh}}(\mathcal{O}_Y)\). Then \(f^!K = R\SheafHom(\mathcal{O}_X, K)\) (Lemma 0AA2) and \(f_*f^!K = R\SheafHom_{\mathcal{O}_Y}(f_*\mathcal{O}_X, K)\). Since \(f\) is perfect, the complex \(f_*\mathcal{O}_X\) is perfect and hence \(R\SheafHom_{\mathcal{O}_Y}(f_*\mathcal{O}_X, K)\) is bounded above. This proves that (d) holds. Some details omitted.

Let \(f : \mathbf{A}^n_Y \to Y\) be the projection. Then (d) holds by repeated application of Lemma 0AA1. Finally, (e) is true because it holds for \(\mathbf{P}^n_Y \to Y\) (flat and proper) and because \(\mathbf{A}^n_Y \subset \mathbf{P}^n_Y\) is an open.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). If \(f\) is flat, then \(f^!\mathcal{O}_Y\) is a \(Y\)-perfect object of \(D(\mathcal{O}_X)\) and \(\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(f^!\mathcal{O}_Y, f^!\mathcal{O}_Y)\) is an isomorphism.

Proof

Both assertions are local on \(X\). Thus we may assume \(X\) and \(Y\) are affine. Then Remark 0BV2 turns the lemma into an algebra lemma, namely Dualizing Complexes, Lemma 0BZW. (Use Derived Categories of Schemes, Lemma 0DI2 to match the languages.)

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Assume \(f : X \to Y\) is a local complete intersection morphism. Then

1. \(f^!\mathcal{O}_Y\) is an invertible object of \(D(\mathcal{O}_X)\), and

2. \(f^!\) maps perfect complexes to perfect complexes.

Proof

Recall that a local complete intersection morphism is perfect, see More on Morphisms, Lemma 069H. By Lemma 0B6U it suffices to show that \(f^!\mathcal{O}_Y\) is an invertible object in \(D(\mathcal{O}_X)\). This question is local on \(X\) and \(Y\). Hence we may assume that \(X \to Y\) factors as \(X \to \mathbf{A}^n_Y \to Y\) where the first arrow is a Koszul regular immersion. See More on Morphisms, Section 068E. The result holds for \(\mathbf{A}^n_Y \to Y\) by Lemma 0AA1. Thus it suffices to prove the lemma when \(f\) is a Koszul regular immersion. Working locally once again we reduce to the case \(X = \Spec(A)\) and \(Y = \Spec(B)\), where \(A = B/(f_1, \ldots, f_r)\) for some regular sequence \(f_1, \ldots, f_r \in B\) (use that for Noetherian local rings the notion of Koszul regular and regular are the same, see More on Algebra, Lemma 09CC). Thus \(X \to Y\) is a composition \[X = X_r \to X_{r - 1} \to \ldots \to X_1 \to X_0 = Y\] where each arrow is the inclusion of an effective Cartier divisor. In this way we reduce to the case of an inclusion of an effective Cartier divisor \(i : D \to X\). In this case \(i^!\mathcal{O}_X = \mathcal{N}[1]\) by Lemma 0B4B and the proof is complete.

Base change for upper shriek

In Situation 0F42 let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a cartesian diagram in \(\textit{FTS}_S\) such that \(X\) and \(Y'\) are Tor independent over \(Y\). Our setup is currently not sufficient to construct a base change map \(L(g')^* \circ f^! \to (f')^! \circ Lg^*\) in this generality. The reason is that in general it will not be possible to choose a compactification \(j : X \to \overline{X}\) over \(Y\) such that \(\overline{X}\) and \(Y'\) are tor independent over \(Y\) and hence our construction of the base change map in Section 0AA5 does not apply⁶.

A partial remedy will be found in Section 0E2S. Namely, if the morphism \(f\) is flat, then there is a good notion of a relative dualizing complex and using Lemmas 0E9W 0E2Y, and 0B6U we may construct a canonical base change isomorphism. If we ever need to use this, we will add precise statements and proofs later in this chapter.

Lemma

In Situation 0F42 let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of \(\textit{FTS}_S\) with \(g\) flat. Then there is an isomorphism \(L(g')^* \circ f^! \to (f')^! \circ Lg^*\) on \(D_\QCoh^+(\mathcal{O}_Y)\).

Proof

Namely, because \(g\) is flat, for every choice of compactification \(j : X \to \overline{X}\) of \(X\) over \(Y\) the scheme \(\overline{X}\) is Tor independent of \(Y'\). Denote \(j' : X' \to \overline{X}'\) the base change of \(j\) and \(\overline{g}' : \overline{X}' \to \overline{X}\) the projection. We define the base change map as the composition \[L(g')^* \circ f^! = L(g')^* \circ j^* \circ a = (j')^* \circ L(\overline{g}')^* \circ a \longrightarrow (j')^* \circ a' \circ Lg^* = (f')^! \circ Lg^*\] where the middle arrow is the base change map (0AA6) and \(a\) and \(a'\) are the right adjoints to pushforward of Lemma 0A9E for \(\overline{X} \to Y\) and \(\overline{X}' \to Y'\). This construction is independent of the choice of compactification (we will formulate a precise lemma and prove it, if we ever need this result).

To finish the proof it suffices to show that the base change map \(L(g')^* \circ a \to a' \circ Lg^*\) is an isomorphism on \(D_\QCoh^+(\mathcal{O}_Y)\). By Lemma 0A9P formation of \(a\) and \(a'\) commutes with restriction to affine opens of \(Y\) and \(Y'\). Thus by Remark 0E9S we may assume that \(Y\) and \(Y'\) are affine. Thus the result by Lemma 0AA8.

Lemma

In Situation 0F42 let \(f : X \to Y\) be an étale morphism of \(\textit{FTS}_S\). Then \(f^! \cong f^*\) as functors on \(D^+_\QCoh(\mathcal{O}_Y)\).

Proof

We are going to use that an étale morphism is flat, syntomic, and a local complete intersection morphism (Morphisms, Lemma 02GQ and 02GS and More on Morphisms, Lemma 069K). By Lemma 0B6U it suffices to show \(f^!\mathcal{O}_Y = \mathcal{O}_X\). By Lemma 0B6V we know that \(f^!\mathcal{O}_Y\) is an invertible module. Consider the commutative diagram \[\xymatrix{ X \times_Y X \ar[r]_{p_2} \ar[d]_{p_1} & X \ar[d]^f \\ X \ar[r]^f & Y }\] and the diagonal \(\Delta : X \to X \times_Y X\). Since \(\Delta\) is an open immersion (by Morphisms, Lemmas 02GE and 02GK), by Lemma 0AU0 we have \(\Delta^! = \Delta^*\). By Lemma 0ATX we have \(\Delta^! \circ p_1^! \circ f^! = f^!\). By Lemma 0E9U applied to the diagram we have \(p_1^!\mathcal{O}_X = p_2^*f^!\mathcal{O}_Y\). Hence we conclude \[f^!\mathcal{O}_Y = \Delta^!p_1^!f^!\mathcal{O}_Y = \Delta^*(p_1^*f^!\mathcal{O}_Y \otimes p_1^!\mathcal{O}_X) = \Delta^*(p_2^*f^!\mathcal{O}_Y \otimes p_1^*f^!\mathcal{O}_Y) = (f^!\mathcal{O}_Y)^{\otimes 2}\] where in the second step we have used Lemma 0B6U once more. Thus \(f^!\mathcal{O}_Y = \mathcal{O}_X\) as desired.

In the rest of this section, we formulate some easy to prove results which would be consequences of a good theory of the base change map.

Lemma

In Situation 0F42 let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of \(\textit{FTS}_S\). Let \(E \in D^+_\QCoh(\mathcal{O}_Y)\) be an object such that \(Lg^*E\) is in \(D^+(\mathcal{O}_Y)\). If \(f\) is flat, then \(L(g')^*f^!E\) and \((f')^!Lg^*E\) restrict to isomorphic objects of \(D(\mathcal{O}_{U'})\) for \(U' \subset X'\) affine open mapping into affine opens of \(Y\), \(Y'\), and \(X\).

Proof

By our assumptions we immediately reduce to the case where \(X\), \(Y\), \(Y'\), and \(X'\) are affine. Say \(Y = \Spec(R)\), \(Y' = \Spec(R')\), \(X = \Spec(A)\), and \(X' = \Spec(A')\). Then \(A' = A \otimes_R R'\). Let \(E\) correspond to \(K \in D^+(R)\). Denoting \(\varphi : R \to A\) and \(\varphi' : R' \to A'\) the given maps we see from Remark 0BV2 that \(L(g')^*f^!E\) and \((f')^!Lg^*E\) correspond to \(\varphi^!(K) \otimes_A^\mathbf{L} A'\) and \((\varphi')^!(K \otimes_R^\mathbf{L} R')\) where \(\varphi^!\) and \((\varphi')^!\) are the functors from Dualizing Complexes, Section 0BZI. The result follows from Dualizing Complexes, Lemma 0BZS.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Assume \(f\) is flat. Set \(\omega_{X/Y}^\bullet = f^!\mathcal{O}_Y\) in \(D^b_{\textit{Coh}}(X)\). Let \(y \in Y\) and \(h : X_y \to X\) the projection. Then \(Lh^*\omega_{X/Y}^\bullet\) is a dualizing complex on \(X_y\).

Proof

The complex \(\omega_{X/Y}^\bullet\) is in \(D^b_{\textit{Coh}}\) by Lemma 0B6U. Being a dualizing complex is a local property. Hence by Lemma 0BZY it suffices to show that \((X_y \to y)^!\mathcal{O}_y\) is a dualizing complex on \(X_y\). This follows from Lemma 0AA3.

A duality theory

In this section we spell out what kind of a duality theory our very general results above give for finite type separated schemes over a fixed Noetherian base scheme.

Recall that a dualizing complex on a Noetherian scheme \(X\), is an object of \(D(\mathcal{O}_X)\) which affine locally gives a dualizing complex for the corresponding rings, see Definition 0A87.

Given a Noetherian scheme \(S\) denote \(\textit{FTS}_S\) the category of schemes which are of finite type and separated over \(S\). Then:

1. the functors \(f^!\) turn \(D_\QCoh^+\) into a pseudo functor on \(\textit{FTS}_S\),

2. if \(f : X \to Y\) is a proper morphism in \(\textit{FTS}_S\), then \(f^!\) is the restriction of the right adjoint of \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) to \(D_\QCoh^+(\mathcal{O}_Y)\) and there is a canonical isomorphism \[Rf_*R\SheafHom_{\mathcal{O}_X}(K, f^!M) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*K, M)\] for all \(K \in D_{\textit{Coh}}^-(\mathcal{O}_X)\) and \(M \in D_\QCoh^+(\mathcal{O}_Y)\),

3. if an object \(X\) of \(\textit{FTS}_S\) has a dualizing complex \(\omega_X^\bullet\), then the functor \(D_X = R\SheafHom_{\mathcal{O}_X}(-, \omega_X^\bullet)\) defines an involution of \(D_{\textit{Coh}}(\mathcal{O}_X)\) switching \(D_{\textit{Coh}}^+(\mathcal{O}_X)\) and \(D_{\textit{Coh}}^-(\mathcal{O}_X)\) and fixing \(D_{\textit{Coh}}^b(\mathcal{O}_X)\),

4. if \(f : X \to Y\) is a morphism of \(\textit{FTS}_S\) and \(\omega_Y^\bullet\) is a dualizing complex on \(Y\), then

   1. \(\omega_X^\bullet = f^!\omega_Y^\bullet\) is a dualizing complex for \(X\),

   2. \(f^!M = D_X(Lf^*D_Y(M))\) canonically for \(M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)\), and

   3. if in addition \(f\) is proper then \[Rf_*R\SheafHom_{\mathcal{O}_X}(K, \omega_X^\bullet) = R\SheafHom_{\mathcal{O}_Y}(Rf_*K, \omega_Y^\bullet)\] for \(K\) in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\),

5. if \(f : X \to Y\) is a closed immersion in \(\textit{FTS}_S\), then \(f^!(-) = R\SheafHom(\mathcal{O}_X, -)\),

6. if \(f : Y \to X\) is a finite morphism in \(\textit{FTS}_S\), then \(f_*f^!(-) = R\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, -)\),

7. if \(f : X \to Y\) is the inclusion of an effective Cartier divisor into an object of \(\textit{FTS}_S\), then \(f^!(-) = Lf^*(-) \otimes_{\mathcal{O}_X} f^*\mathcal{O}_Y(X)[-1]\),

8. if \(f : X \to Y\) is a Koszul regular immersion of codimension \(c\) into an object of \(\textit{FTS}_S\), then \(f^!(-) \cong Lf^*(-) \otimes_{\mathcal{O}_X} \wedge^c\mathcal{N}[-c]\), and

9. if \(f : X \to Y\) is a smooth proper morphism of relative dimension \(d\) in \(\textit{FTS}_S\), then \(f^!(-) \cong Lf^*(-) \otimes_{\mathcal{O}_X} \Omega^d_{X/Y}[d]\).

This follows from Lemmas 0A89, 0A9Q, 0A9X, 0AX2, 0AA4, 0BR0, 0BRT, 0ATX, 0ATY, 0AA2, 0AA3, 0AU2, and 0B6U and Example 0GEW. We have obtained our functors by a very abstract procedure which finally rests on invoking an existence theorem (Derived Categories, Proposition 0A8G). This means we have, in general, no explicit description of the functors \(f^!\). This can sometimes be a problem. But in fact, it is often enough to know the existence of a dualizing complex and the duality isomorphism to pin down \(f^!\).

Glueing dualizing complexes

We will now use glueing of dualizing complexes to get a theory which works for all finite type schemes over \(S\) given a pair \((S, \omega_S^\bullet)\) as in Situation 0AU4. This is similar to [RD, Remark on page 310].

Situation

Here \(S\) is a Noetherian scheme and \(\omega_S^\bullet\) is a dualizing complex.

In Situation 0AU4 let \(X\) be a scheme of finite type over \(S\). Let \(\mathcal{U} : X = \bigcup_{i = 1, \ldots, n} U_i\) be a finite open covering of \(X\) by objects of \(\textit{FTS}_S\), see Situation 0F42. All this means is that the morphisms \(U_i \to S\) are separated (as they are already of finite type). Every affine scheme of finite type over \(S\) is an object of \(\textit{FTS}_S\) by Schemes, Lemma 01KV hence such open coverings certainly exist. Then for each \(i, j, k \in \{1, \ldots, n\}\) the morphisms \(p_i : U_i \to S\), \(p_{ij} : U_i \cap U_j \to S\), and \(p_{ijk} : U_i \cap U_j \cap U_k \to S\) are separated and each of these schemes is an object of \(\textit{FTS}_S\). From such an open covering we obtain

1. \(\omega_i^\bullet = p_i^!\omega_S^\bullet\) a dualizing complex on \(U_i\), see Section 0AU3,

2. for each \(i, j\) a canonical isomorphism \(\varphi_{ij} : \omega_i^\bullet|_{U_i \cap U_j} \to \omega_j^\bullet|_{U_i \cap U_j}\), and

3. for each \(i, j, k\) we have \[\varphi_{ik}|_{U_i \cap U_j \cap U_k} = \varphi_{jk}|_{U_i \cap U_j \cap U_k} \circ \varphi_{ij}|_{U_i \cap U_j \cap U_k}\] in \(D(\mathcal{O}_{U_i \cap U_j \cap U_k})\).

Here, in (2) we use that \((U_i \cap U_j \to U_i)^!\) is given by restriction (Lemma 0AU0) and that we have canonical isomorphisms \[(U_i \cap U_j \to U_i)^! \circ p_i^! = p_{ij}^! = (U_i \cap U_j \to U_j)^! \circ p_j^!\] by Lemma 0ATX and to get (3) we use that the upper shriek functors form a pseudo functor by Lemma 0ATY.

In the situation just described a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\) is a pair \((K, \alpha_i)\) where \(K \in D(\mathcal{O}_X)\) and \(\alpha_i : K|_{U_i} \to \omega_i^\bullet\) are isomorphisms such that \(\varphi_{ij}\) is given by \(\alpha_j|_{U_i \cap U_j} \circ \alpha_i^{-1}|_{U_i \cap U_j}\). Since being a dualizing complex on a scheme is a local property we see that dualizing complexes normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\) are indeed dualizing complexes.

Lemma

In Situation 0AU4 let \(X\) be a scheme of finite type over \(S\) and let \(\mathcal{U}\) be a finite open covering of \(X\) by schemes separated over \(S\). If there exists a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\), then it is unique up to unique isomorphism.

Proof

If \((K, \alpha_i)\) and \((K', \alpha_i')\) are two, then we consider \(L = R\SheafHom_{\mathcal{O}_X}(K, K')\). By Lemma 0ATP and its proof, this is an invertible object of \(D(\mathcal{O}_X)\). Using \(\alpha_i\) and \(\alpha'_i\) we obtain an isomorphism \[\alpha_i^t \otimes \alpha'_i : L|_{U_i} \longrightarrow R\SheafHom_{\mathcal{O}_X}(\omega_i^\bullet, \omega_i^\bullet) = \mathcal{O}_{U_i}[0]\] This already implies that \(L = H^0(L)[0]\) in \(D(\mathcal{O}_X)\). Moreover, \(H^0(L)\) is an invertible sheaf with given trivializations on the opens \(U_i\) of \(X\). Finally, the condition that \(\alpha_j|_{U_i \cap U_j} \circ \alpha_i^{-1}|_{U_i \cap U_j}\) and \(\alpha'_j|_{U_i \cap U_j} \circ (\alpha'_i)^{-1}|_{U_i \cap U_j}\) both give \(\varphi_{ij}\) implies that the transition maps are \(1\) and we get an isomorphism \(H^0(L) = \mathcal{O}_X\).

Lemma

In Situation 0AU4 let \(X\) be a scheme of finite type over \(S\) and let \(\mathcal{U}\), \(\mathcal{V}\) be two finite open coverings of \(X\) by schemes separated over \(S\). If there exists a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\), then there exists a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{V}\) and these complexes are canonically isomorphic.

Proof

It suffices to prove this when \(\mathcal{U}\) is given by the opens \(U_1, \ldots, U_n\) and \(\mathcal{V}\) by the opens \(U_1, \ldots, U_{n + m}\). In fact, we may and do even assume \(m = 1\). To go from a dualizing complex \((K, \alpha_i)\) normalized relative to \(\omega_S^\bullet\) and \(\mathcal{V}\) to a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\) is achieved by forgetting about \(\alpha_i\) for \(i = n + 1\). Conversely, let \((K, \alpha_i)\) be a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\). To finish the proof we need to construct a map \(\alpha_{n + 1} : K|_{U_{n + 1}} \to \omega_{n + 1}^\bullet\) satisfying the desired conditions. To do this we observe that \(U_{n + 1} = \bigcup U_i \cap U_{n + 1}\) is an open covering. It is clear that \((K|_{U_{n + 1}}, \alpha_i|_{U_i \cap U_{n + 1}})\) is a dualizing complex normalized relative to \(\omega_S^\bullet\) and the covering \(U_{n + 1} = \bigcup U_i \cap U_{n + 1}\). On the other hand, by condition (0AU6) the pair \((\omega_{n + 1}^\bullet|_{U_{n + 1}}, \varphi_{n + 1i})\) is another dualizing complex normalized relative to \(\omega_S^\bullet\) and the covering \(U_{n + 1} = \bigcup U_i \cap U_{n + 1}\). By Lemma 0AU7 we obtain a unique isomorphism \[\alpha_{n + 1} : K|_{U_{n + 1}} \longrightarrow \omega_{n + 1}^\bullet\] compatible with the given local isomorphisms. It is a pleasant exercise to show that this means it satisfies the required property.

Lemma

In Situation 0AU4 let \(X\) be a scheme of finite type over \(S\) and let \(\mathcal{U}\) be a finite open covering of \(X\) by schemes separated over \(S\). Then there exists a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\).

Proof

Say \(\mathcal{U} : X = \bigcup_{i = 1, \ldots, n} U_i\). We prove the lemma by induction on \(n\). The base case \(n = 1\) is immediate. Assume \(n > 1\). Set \(X' = U_1 \cup \ldots \cup U_{n - 1}\) and let \((K', \{\alpha'_i\}_{i = 1, \ldots, n - 1})\) be a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}' : X' = \bigcup_{i = 1, \ldots, n - 1} U_i\). It is clear that \((K'|_{X' \cap U_n}, \alpha'_i|_{U_i \cap U_n})\) is a dualizing complex normalized relative to \(\omega_S^\bullet\) and the covering \(X' \cap U_n = \bigcup_{i = 1, \ldots, n - 1} U_i \cap U_n\). On the other hand, by condition (0AU6) the pair \((\omega_n^\bullet|_{X' \cap U_n}, \varphi_{ni})\) is another dualizing complex normalized relative to \(\omega_S^\bullet\) and the covering \(X' \cap U_n = \bigcup_{i = 1, \ldots, n - 1} U_i \cap U_n\). By Lemma 0AU7 we obtain a unique isomorphism \[\epsilon : K'|_{X' \cap U_n} \longrightarrow \omega_i^\bullet|_{X' \cap U_n}\] compatible with the given local isomorphisms. By Cohomology, Lemma 08DG we obtain \(K \in D(\mathcal{O}_X)\) together with isomorphisms \(\beta : K|_{X'} \to K'\) and \(\gamma : K|_{U_n} \to \omega_n^\bullet\) such that \(\epsilon = \gamma|_{X'\cap U_n} \circ \beta|_{X' \cap U_n}^{-1}\). Then we define \[\alpha_i = \alpha'_i \circ \beta|_{U_i}, i = 1, \ldots, n - 1, \text{ and } \alpha_n = \gamma\] We still need to verify that \(\varphi_{ij}\) is given by \(\alpha_j|_{U_i \cap U_j} \circ \alpha_i^{-1}|_{U_i \cap U_j}\). For \(i, j \leq n - 1\) this follows from the corresponding condition for \(\alpha_i'\). For \(i = j = n\) it is clear as well. If \(i < j = n\), then we get \[\alpha_n|_{U_i \cap U_n} \circ \alpha_i^{-1}|_{U_i \cap U_n} = \gamma|_{U_i \cap U_n} \circ \beta^{-1}|_{U_i \cap U_n} \circ (\alpha'_i)^{-1}|_{U_i \cap U_n} = \epsilon|_{U_i \cap U_n} \circ (\alpha'_i)^{-1}|_{U_i \cap U_n}\] This is equal to \(\alpha_{in}\) exactly because \(\epsilon\) is the unique map compatible with the maps \(\alpha_i'\) and \(\alpha_{ni}\).

Let \((S, \omega_S^\bullet)\) be as in Situation 0AU4. The upshot of the lemmas above is that given any scheme \(X\) of finite type over \(S\), there is a pair \((K, \alpha_U)\) given up to unique isomorphism, consisting of an object \(K \in D(\mathcal{O}_X)\) and isomorphisms \(\alpha_U : K|_U \to \omega_U^\bullet\) for every open subscheme \(U \subset X\) which is separated over \(S\). Here \(\omega_U^\bullet = (U \to S)^!\omega_S^\bullet\) is a dualizing complex on \(U\), see Section 0AU3. Moreover, if \(\mathcal{U} : X = \bigcup U_i\) is a finite open covering by opens which are separated over \(S\), then \((K, \alpha_{U_i})\) is a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\). Namely, uniqueness up to unique isomorphism by Lemma 0AU7, existence for one open covering by Lemma 0AU9, and the fact that \(K\) then works for all open coverings is Lemma 0AU8.

Definition

Let \(S\) be a Noetherian scheme and let \(\omega_S^\bullet\) be a dualizing complex on \(S\). Let \(X\) be a scheme of finite type over \(S\). The complex \(K\) constructed above is called the dualizing complex normalized relative to \(\omega_S^\bullet\) and is denoted \(\omega_X^\bullet\).

As the terminology suggest, a dualizing complex normalized relative to \(\omega_S^\bullet\) is not just an object of the derived category of \(X\) but comes equipped with the local isomorphisms described above. This does not conflict with setting \(\omega_X^\bullet = p^!\omega_S^\bullet\) where \(p : X \to S\) is the structure morphism if \(X\) is separated over \(S\). More generally we have the following sanity check.

Lemma

Let \((S, \omega_S^\bullet)\) be as in Situation 0AU4. Let \(f : X \to Y\) be a morphism of finite type schemes over \(S\). Let \(\omega_X^\bullet\) and \(\omega_Y^\bullet\) be dualizing complexes normalized relative to \(\omega_S^\bullet\). Then \(\omega_X^\bullet\) is a dualizing complex normalized relative to \(\omega_Y^\bullet\).

Proof

This is just a matter of bookkeeping. Choose a finite affine open covering \(\mathcal{V} : Y = \bigcup V_j\). For each \(j\) choose a finite affine open covering \(f^{-1}(V_j) = U_{ji}\). Set \(\mathcal{U} : X = \bigcup U_{ji}\). The schemes \(V_j\) and \(U_{ji}\) are separated over \(S\), hence we have the upper shriek functors for \(q_j : V_j \to S\), \(p_{ji} : U_{ji} \to S\) and \(f_{ji} : U_{ji} \to V_j\) and \(f_{ji}' : U_{ji} \to Y\). Let \((L, \beta_j)\) be a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{V}\). Let \((K, \gamma_{ji})\) be a dualizing complex normalized relative to \(\omega_S^\bullet\) and \(\mathcal{U}\). (In other words, \(L = \omega_Y^\bullet\) and \(K = \omega_X^\bullet\).) We can define \[\alpha_{ji} : K|_{U_{ji}} \xrightarrow{\gamma_{ji}} p_{ji}^!\omega_S^\bullet = f_{ji}^!q_j^!\omega_S^\bullet \xrightarrow{f_{ji}^!\beta_j^{-1}} f_{ji}^!(L|_{V_j}) = (f_{ji}')^!(L)\] To finish the proof we have to show that \(\alpha_{ji}|_{U_{ji} \cap U_{j'i'}} \circ \alpha_{j'i'}^{-1}|_{U_{ji} \cap U_{j'i'}}\) is the canonical isomorphism \((f_{ji}')^!(L)|_{U_{ji} \cap U_{j'i'}} \to (f_{j'i'}')^!(L)|_{U_{ji} \cap U_{j'i'}}\). This is formal and we omit the details.

Lemma

Let \((S, \omega_S^\bullet)\) be as in Situation 0AU4. Let \(j : X \to Y\) be an open immersion of schemes of finite type over \(S\). Let \(\omega_X^\bullet\) and \(\omega_Y^\bullet\) be dualizing complexes normalized relative to \(\omega_S^\bullet\). Then there is a canonical isomorphism \(\omega_X^\bullet = \omega_Y^\bullet|_X\).

Proof

Immediate from the construction of normalized dualizing complexes given just above Definition 0AUA.

Lemma

Let \((S, \omega_S^\bullet)\) be as in Situation 0AU4. Let \(f : X \to Y\) be a proper morphism of schemes of finite type over \(S\). Let \(\omega_X^\bullet\) and \(\omega_Y^\bullet\) be dualizing complexes normalized relative to \(\omega_S^\bullet\). Let \(a\) be the right adjoint of Lemma 0A9E for \(f\). Then there is a canonical isomorphism \(a(\omega_Y^\bullet) = \omega_X^\bullet\).

Proof

Let \(p : X \to S\) and \(q : Y \to S\) be the structure morphisms. If \(X\) and \(Y\) are separated over \(S\), then this follows from the fact that \(\omega_X^\bullet = p^!\omega_S^\bullet\), \(\omega_Y^\bullet = q^!\omega_S^\bullet\), \(f^! = a\), and \(f^! \circ q^! = p^!\) (Lemma 0ATX). In the general case we first use Lemma 0AUB to reduce to the case \(Y = S\). In this case \(X\) and \(Y\) are separated over \(S\) and we’ve just seen the result.

Let \((S, \omega_S^\bullet)\) be as in Situation 0AU4. For a scheme \(X\) of finite type over \(S\) denote \(\omega_X^\bullet\) the dualizing complex for \(X\) normalized relative to \(\omega_S^\bullet\). Define \(D_X(-) = R\SheafHom_{\mathcal{O}_X}(-, \omega_X^\bullet)\) as in Lemma 0A89. Let \(f : X \to Y\) be a morphism of finite type schemes over \(S\). Define \[f_{new}^! = D_X \circ Lf^* \circ D_Y : D_{\textit{Coh}}^+(\mathcal{O}_Y) \to D_{\textit{Coh}}^+(\mathcal{O}_X)\] If \(f : X \to Y\) and \(g : Y \to Z\) are composable morphisms between schemes of finite type over \(S\), define \[\begin{align*} (g \circ f)^!_{new} & = D_X \circ L(g \circ f)^* \circ D_Z \\ & = D_X \circ Lf^* \circ Lg^* \circ D_Z \\ & \to D_X \circ Lf^* \circ D_Y \circ D_Y \circ Lg^* \circ D_Z \\ & = f^!_{new} \circ g^!_{new} \end{align*}\] where the arrow is defined in Lemma 0A89. We collect the results together in the following lemma.

Lemma

Let \((S, \omega_S^\bullet)\) be as in Situation 0AU4. With \(f^!_{new}\) and \(\omega_X^\bullet\) defined for all (morphisms of) schemes of finite type over \(S\) as above:

1. the functors \(f^!_{new}\) and the arrows \((g \circ f)^!_{new} \to f^!_{new} \circ g^!_{new}\) turn \(D_{\textit{Coh}}^+\) into a pseudo functor from the category of schemes of finite type over \(S\) into the \(2\)-category of categories,

2. \(\omega_X^\bullet = (X \to S)^!_{new} \omega_S^\bullet\),

3. the functor \(D_X\) defines an involution of \(D_{\textit{Coh}}(\mathcal{O}_X)\) switching \(D_{\textit{Coh}}^+(\mathcal{O}_X)\) and \(D_{\textit{Coh}}^-(\mathcal{O}_X)\) and fixing \(D_{\textit{Coh}}^b(\mathcal{O}_X)\),

4. \(\omega_X^\bullet = f^!_{new}\omega_Y^\bullet\) for \(f : X \to Y\) a morphism of finite type schemes over \(S\),

5. \(f^!_{new}M = D_X(Lf^*D_Y(M))\) for \(M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)\), and

6. if in addition \(f\) is proper, then \(f^!_{new}\) is isomorphic to the restriction of the right adjoint of \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) to \(D_{\textit{Coh}}^+(\mathcal{O}_Y)\) and there is a canonical isomorphism \[Rf_*R\SheafHom_{\mathcal{O}_X}(K, f^!_{new}M) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*K, M)\] for \(K \in D^-_{\textit{Coh}}(\mathcal{O}_X)\) and \(M \in D_{\textit{Coh}}^+(\mathcal{O}_Y)\), and \[Rf_*R\SheafHom_{\mathcal{O}_X}(K, \omega_X^\bullet) = R\SheafHom_{\mathcal{O}_Y}(Rf_*K, \omega_Y^\bullet)\] for \(K \in D^-_{\textit{Coh}}(\mathcal{O}_X)\) and

If \(X\) is separated over \(S\), then \(\omega_X^\bullet\) is canonically isomorphic to \((X \to S)^!\omega_S^\bullet\) and if \(f\) is a morphism between schemes separated over \(S\), then there is a canonical isomorphism⁷ \(f_{new}^!K = f^!K\) for \(K\) in \(D_{\textit{Coh}}^+\).

Proof

Let \(f : X \to Y\), \(g : Y \to Z\), \(h : Z \to T\) be morphisms of schemes of finite type over \(S\). We have to show that \[\xymatrix{ (h \circ g \circ f)^!_{new} \ar[r] \ar[d] & f^!_{new} \circ (h \circ g)^!_{new} \ar[d] \\ (g \circ f)^!_{new} \circ h^!_{new} \ar[r] & f^!_{new} \circ g^!_{new} \circ h^!_{new} }\] is commutative. Let \(\eta_Y : \text{id} \to D_Y^2\) and \(\eta_Z : \text{id} \to D_Z^2\) be the canonical isomorphisms of Lemma 0A89. Then, using Categories, Lemma 003F, a computation (omitted) shows that both arrows \((h \circ g \circ f)^!_{new} \to f^!_{new} \circ g^!_{new} \circ h^!_{new}\) are given by \[1 \star \eta_Y \star 1 \star \eta_Z \star 1 : D_X \circ Lf^* \circ Lg^* \circ Lh^* \circ D_T \longrightarrow D_X \circ Lf^* \circ D_Y^2 \circ Lg^* \circ D_Z^2 \circ Lh^* \circ D_T\] This proves (1). Part (2) is immediate from the definition of \((X \to S)^!_{new}\) and the fact that \(D_S(\omega_S^\bullet) = \mathcal{O}_S\). Part (3) is Lemma 0A89. Part (4) follows by the same argument as part (2). Part (5) is the definition of \(f^!_{new}\).

Proof of (6). Let \(a\) be the right adjoint of Lemma 0A9E for the proper morphism \(f : X \to Y\) of schemes of finite type over \(S\). The issue is that we do not know \(X\) or \(Y\) is separated over \(S\) (and in general this won’t be true) hence we cannot immediately apply Lemma 0AU2 to \(f\) over \(S\). To get around this we use the canonical identification \(\omega_X^\bullet = a(\omega_Y^\bullet)\) of Lemma 0AUD. Hence \(f^!_{new}\) is the restriction of \(a\) to \(D_{\textit{Coh}}^+(\mathcal{O}_Y)\) by Lemma 0AU2 applied to \(f : X \to Y\) over the base scheme \(Y\)! The displayed equalities hold by Example 0GEW.

The final assertions follow from the construction of normalized dualizing complexes and the already used Lemma 0AU2.

Remark

Let \(S\) be a Noetherian scheme which has a dualizing complex. Let \(f : X \to Y\) be a morphism of schemes of finite type over \(S\). Then the functor \[f_{new}^! : D^+_{Coh}(\mathcal{O}_Y) \to D^+_{Coh}(\mathcal{O}_X)\] is independent of the choice of the dualizing complex \(\omega_S^\bullet\) up to canonical isomorphism. We sketch the proof. Any second dualizing complex is of the form \(\omega_S^\bullet \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{L}\) where \(\mathcal{L}\) is an invertible object of \(D(\mathcal{O}_S)\), see Lemma 0ATP. For any separated morphism \(p : U \to S\) of finite type we have \(p^!(\omega_S^\bullet \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{L}) = p^!(\omega_S^\bullet) \otimes^\mathbf{L}_{\mathcal{O}_U} Lp^*\mathcal{L}\) by Lemma 0A9T. Hence, if \(\omega_X^\bullet\) and \(\omega_Y^\bullet\) are the dualizing complexes normalized relative to \(\omega_S^\bullet\) we see that \(\omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}\) and \(\omega_Y^\bullet \otimes_{\mathcal{O}_Y}^\mathbf{L} Lb^*\mathcal{L}\) are the dualizing complexes normalized relative to \(\omega_S^\bullet \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{L}\) (where \(a : X \to S\) and \(b : Y \to S\) are the structure morphisms). Then the result follows as \[\begin{align*} & R\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet \otimes_{\mathcal{O}_Y}^\mathbf{L} Lb^*\mathcal{L}), \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}) \\ & = R\SheafHom_{\mathcal{O}_X}(Lf^*R(\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet) \otimes_{\mathcal{O}_Y}^\mathbf{L} Lb^*\mathcal{L}), \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}) \\ & = R\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet) \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}) \\ & = R\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet), \omega_X^\bullet) \end{align*}\] for \(K \in D^+_{Coh}(\mathcal{O}_Y)\). The last equality because \(La^*\mathcal{L}\) is invertible in \(D(\mathcal{O}_X)\).

Example

Let \(S\) be a Noetherian scheme and let \(\omega_S^\bullet\) be a dualizing complex. Let \(f : X \to Y\) be a proper morphism of finite type schemes over \(S\). Let \(\omega_X^\bullet\) and \(\omega_Y^\bullet\) be dualizing complexes normalized relative to \(\omega_S^\bullet\). In this situation we have \(a(\omega_Y^\bullet) = \omega_X^\bullet\) (Lemma 0AUD) and hence the trace map (Section 0AWG) is a canonical arrow \[\text{Tr}_f : Rf_*\omega_X^\bullet \longrightarrow \omega_Y^\bullet\] which produces the isomorphisms (Lemma 0AUE) \[\Hom_X(L, \omega_X^\bullet) = \Hom_Y(Rf_*L, \omega_Y^\bullet)\] and \[Rf_*R\SheafHom_{\mathcal{O}_X}(L, \omega_X^\bullet) = R\SheafHom_{\mathcal{O}_Y}(Rf_*L, \omega_Y^\bullet)\] for \(L\) in \(D_\QCoh(\mathcal{O}_X)\).

Remark

Let \(S\) be a Noetherian scheme and let \(\omega_S^\bullet\) be a dualizing complex. Let \(f : X \to Y\) be a finite morphism between schemes of finite type over \(S\). Let \(\omega_X^\bullet\) and \(\omega_Y^\bullet\) be dualizing complexes normalized relative to \(\omega_S^\bullet\). Then we have \[f_*\omega_X^\bullet = R\SheafHom(f_*\mathcal{O}_X, \omega_Y^\bullet)\] in \(D_\QCoh^+(f_*\mathcal{O}_X)\) by Lemmas 0AX2 and 0AUD and the trace map of Example 0B6X is the map \[\text{Tr}_f : Rf_*\omega_X^\bullet = f_*\omega_X^\bullet = R\SheafHom(f_*\mathcal{O}_X, \omega_Y^\bullet) \longrightarrow \omega_Y^\bullet\] which often goes under the name “evaluation at \(1\)”.

Remark

Let \(f : X \to Y\) be a flat proper morphism of finite type schemes over a pair \((S, \omega_S^\bullet)\) as in Situation 0AU4. The relative dualizing complex (Remark 0B6S) is \(\omega_{X/Y}^\bullet = a(\mathcal{O}_Y)\). By Lemma 0AUD we have the first canonical isomorphism in \[\omega_X^\bullet = a(\omega_Y^\bullet) = Lf^*\omega_Y^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet\] in \(D(\mathcal{O}_X)\). The second canonical isomorphism follows from the discussion in Remark 0B6S.

Dimension functions

We need a bit more information about how the dimension functions change when passing to a scheme of finite type over another.

Lemma

Let \(S\) be a Noetherian scheme and let \(\omega_S^\bullet\) be a dualizing complex. Let \(X\) be a scheme of finite type over \(S\) and let \(\omega_X^\bullet\) be the dualizing complex normalized relative to \(\omega_S^\bullet\). If \(x \in X\) is a closed point lying over a closed point \(s\) of \(S\), then \(\omega_{X, x}^\bullet\) is a normalized dualizing complex over \(\mathcal{O}_{X, x}\) provided that \(\omega_{S, s}^\bullet\) is a normalized dualizing complex over \(\mathcal{O}_{S, s}\).

Proof

We may replace \(X\) by an affine neighbourhood of \(x\), hence we may and do assume that \(f : X \to S\) is separated. Then \(\omega_X^\bullet = f^!\omega_S^\bullet\). We have to show that \(R\Hom_{\mathcal{O}_{X, x}}(\kappa(x), \omega_{X, x}^\bullet)\) is sitting in degree \(0\). Let \(i_x : x \to X\) denote the inclusion morphism which is a closed immersion as \(x\) is a closed point. Hence \(R\Hom_{\mathcal{O}_{X, x}}(\kappa(x), \omega_{X, x}^\bullet)\) represents \(i_x^!\omega_X^\bullet\) by Lemma 0AA2. Consider the commutative diagram \[\xymatrix{ x \ar[r]_{i_x} \ar[d]_\pi & X \ar[d]^f \\ s \ar[r]^{i_s} & S }\] By Morphisms, Lemma 01TF the extension \(\kappa(x)/\kappa(s)\) is finite and hence \(\pi\) is a finite morphism. We conclude that \[i_x^!\omega_X^\bullet = i_x^! f^! \omega_S^\bullet = \pi^! i_s^! \omega_S^\bullet\] Thus if \(\omega_{S, s}^\bullet\) is a normalized dualizing complex over \(\mathcal{O}_{S, s}\), then \(i_s^!\omega_S^\bullet = \kappa(s)[0]\) by the same reasoning as above. We have \[R\pi_*(\pi^!(\kappa(s)[0])) = R\SheafHom_{\mathcal{O}_s}(R\pi_*(\kappa(x)[0]), \kappa(s)[0]) = \widetilde{\Hom_{\kappa(s)}(\kappa(x), \kappa(s))}\] The first equality by Example 0GEW applied with \(L = \kappa(x)[0]\). The second equality holds because \(\pi_*\) is exact. Thus \(\pi^!(\kappa(s)[0])\) is supported in degree \(0\) and we win.

Lemma

Let \(S\) be a Noetherian scheme and let \(\omega_S^\bullet\) be a dualizing complex. Let \(f : X \to S\) be of finite type and let \(\omega_X^\bullet\) be the dualizing complex normalized relative to \(\omega_S^\bullet\). For all \(x \in X\) we have \[\delta_X(x) - \delta_S(f(x)) = \text{trdeg}_{\kappa(f(x))}(\kappa(x))\] where \(\delta_S\), resp. \(\delta_X\) is the dimension function of \(\omega_S^\bullet\), resp. \(\omega_X^\bullet\), see Lemma 0AWF.

Proof

We may replace \(X\) by an affine neighbourhood of \(x\). Hence we may and do assume there is a compactification \(X \subset \overline{X}\) over \(S\). Then we may replace \(X\) by \(\overline{X}\) and assume that \(X\) is proper over \(S\). We may also assume \(X\) is connected by replacing \(X\) by the connected component of \(X\) containing \(x\). Next, recall that both \(\delta_X\) and the function \(x \mapsto \delta_S(f(x)) + \text{trdeg}_{\kappa(f(x))}(\kappa(x))\) are dimension functions on \(X\), see Morphisms, Lemma 02JW (and the fact that \(S\) is universally catenary by Lemma 0AWF). By Topology, Lemma 02IB we see that the difference is locally constant, hence constant as \(X\) is connected. Thus it suffices to prove equality in any point of \(X\). By Properties, Lemma 02IL the scheme \(X\) has a closed point \(x\). Since \(X \to S\) is proper the image \(s\) of \(x\) is closed in \(S\). Thus we may apply Lemma 0AWL to conclude.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Let \(x \in X\) with image \(y \in Y\). Then \[H^i(f^!\mathcal{O}_Y)_x \not = 0 \Rightarrow - \dim_x(X_y) \leq i.\]

Proof

Since the statement is local on \(X\) we may assume \(X\) and \(Y\) are affine schemes. Write \(X = \Spec(A)\) and \(Y = \Spec(R)\). Then \(f^!\mathcal{O}_Y\) corresponds to the relative dualizing complex \(\omega_{A/R}^\bullet\) of Dualizing Complexes, Section 0E9M by Remark 0BV2. Thus the lemma follows from Dualizing Complexes, Lemma 0E9P.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Let \(x \in X\) with image \(y \in Y\). If \(f\) is flat, then \[H^i(f^!\mathcal{O}_Y)_x \not = 0 \Rightarrow - \dim_x(X_y) \leq i \leq 0.\] In fact, if all fibres of \(f\) have dimension \(\leq d\), then \(f^!\mathcal{O}_Y\) has tor-amplitude in \([-d, 0]\) as an object of \(D(X, f^{-1}\mathcal{O}_Y)\).

Proof

Arguing exactly as in the proof of Lemma 0BV5 this follows from Dualizing Complexes, Lemma 0E9Q.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Let \(x \in X\) with image \(y \in Y\). Assume

1. \(\mathcal{O}_{Y, y}\) is Cohen-Macaulay, and

2. \(\text{trdeg}_{\kappa(f(\xi))}(\kappa(\xi)) \leq r\) for any generic point \(\xi\) of an irreducible component of \(X\) containing \(x\).

Then \[H^i(f^!\mathcal{O}_Y)_x \not = 0 \Rightarrow - r \leq i\] and the stalk \(H^{-r}(f^!\mathcal{O}_Y)_x\) is \((S_2)\) as an \(\mathcal{O}_{X, x}\)-module.

Proof

After replacing \(X\) by an open neighbourhood of \(x\), we may assume every irreducible component of \(X\) passes through \(x\). Then arguing exactly as in the proof of Lemma 0BV5 this follows from Dualizing Complexes, Lemma 0E9R.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). If \(f\) is flat and quasi-finite, then \[f^!\mathcal{O}_Y = \omega_{X/Y}[0]\] for some coherent \(\mathcal{O}_X\)-module \(\omega_{X/Y}\) flat over \(Y\).

Proof

Consequence of Lemma 0BV6 and the fact that the cohomology sheaves of \(f^!\mathcal{O}_Y\) are coherent by Lemma 0AU1.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). If \(f\) is Cohen-Macaulay (More on Morphisms, Definition 045R), then \[f^!\mathcal{O}_Y = \omega_{X/Y}[d]\] for some coherent \(\mathcal{O}_X\)-module \(\omega_{X/Y}\) flat over \(Y\) where \(d\) is the locally constant function on \(X\) which gives the relative dimension of \(X\) over \(Y\).

Proof

The relative dimension \(d\) is well defined and locally constant by Morphisms, Lemma 02NM. The cohomology sheaves of \(f^!\mathcal{O}_Y\) are coherent by Lemma 0AU1. We will get flatness of \(\omega_{X/Y}\) from Lemma 0BV6 if we can show the other cohomology sheaves of \(f^!\mathcal{O}_Y\) are zero.

The question is local on \(X\), hence we may assume \(X\) and \(Y\) are affine and the morphism has relative dimension \(d\). If \(d = 0\), then the result follows directly from Lemma 0BV7. If \(d > 0\), then we may assume there is a factorization \[X \xrightarrow{g} \mathbf{A}^d_Y \xrightarrow{p} Y\] with \(g\) quasi-finite and flat, see More on Morphisms, Lemma 0BUU. Then \(f^! = g^! \circ p^!\). By Lemma 0AA1 we see that \(p^!\mathcal{O}_Y \cong \mathcal{O}_{\mathbf{A}^d_Y}[-d]\). We conclude by the case \(d = 0\).

Remark

Let \(S\) be a Noetherian scheme endowed with a dualizing complex \(\omega_S^\bullet\). In this case Lemmas 0BV5, 0BV6, 0BV7, and 0BV8 are true for any morphism \(f : X \to Y\) of finite type schemes over \(S\) but with \(f^!\) replaced by \(f_{new}^!\). This is clear because in each case the proof reduces immediately to the affine case and then \(f^! = f_{new}^!\) by Lemma 0AUE.

Dualizing modules

This section is a continuation of Dualizing Complexes, Section 0DW3.

Let \(X\) be a Noetherian scheme and let \(\omega_X^\bullet\) be a dualizing complex. Let \(n \in \mathbf{Z}\) be the smallest integer such that \(H^n(\omega_X^\bullet)\) is nonzero. In other words, \(-n\) is the maximal value of the dimension function associated to \(\omega_X^\bullet\) (Lemma 0AWF). Sometimes \(H^n(\omega_X^\bullet)\) is called a dualizing module or dualizing sheaf for \(X\) and then it is often denoted by \(\omega_X\). We will say “let \(\omega_X\) be a dualizing module” to indicate the above.

Care has to be taken when using dualizing modules \(\omega_X\) on Noetherian schemes \(X\):

1. the integer \(n\) may change when passing from \(X\) to an open \(U\) of \(X\) and then it won’t be true that \(\omega_X|_U = \omega_U\),

2. the dualizing complex isn’t unique; the dualizing module is only unique up to tensoring by an invertible module.

The second problem will often be irrelevant because we will work with \(X\) of finite type over a base change \(S\) which is endowed with a fixed dualizing complex \(\omega_S^\bullet\) and \(\omega_X^\bullet\) will be the dualizing complex normalized relative to \(\omega_S^\bullet\). The first problem will not occur if \(X\) is equidimensional, more precisely, if the dimension function associated to \(\omega_X^\bullet\) (Lemma 0AWF) maps every generic point of \(X\) to the same integer.

Example

Say \(S = \Spec(A)\) with \((A, \mathfrak m, \kappa)\) a local Noetherian ring, and \(\omega_S^\bullet\) corresponds to a normalized dualizing complex \(\omega_A^\bullet\). Then if \(f : X \to S\) is proper over \(S\) and \(\omega_X^\bullet = f^!\omega_S^\bullet\) the coherent sheaf \[\omega_X = H^{-\dim(X)}(\omega_X^\bullet)\] is a dualizing module and is often called the dualizing module of \(X\) (with \(S\) and \(\omega_S^\bullet\) being understood). We will see that this has good properties.

Example

Say \(X\) is an equidimensional scheme of finite type over a field \(k\). Then it is customary to take \(\omega_X^\bullet\) the dualizing complex normalized relative to \(k[0]\) and to refer to \[\omega_X = H^{-\dim(X)}(\omega_X^\bullet)\] as the dualizing module of \(X\). If \(X\) is separated over \(k\), then \(\omega_X^\bullet = f^!\mathcal{O}_{\Spec(k)}\) where \(f : X \to \Spec(k)\) is the structure morphism by Lemma 0AUE. If \(X\) is proper over \(k\), then this is a special case of Example 0AWI.

Lemma

Let \(X\) be a connected Noetherian scheme and let \(\omega_X\) be a dualizing module on \(X\). The support of \(\omega_X\) is the union of the irreducible components of maximal dimension with respect to any dimension function and \(\omega_X\) is a coherent \(\mathcal{O}_X\)-module having property \((S_2)\).

Proof

By our conventions discussed above there exists a dualizing complex \(\omega_X^\bullet\) such that \(\omega_X\) is the leftmost nonvanishing cohomology sheaf. Since \(X\) is connected, any two dimension functions differ by a constant (Topology, Lemma 02IB). Hence we may use the dimension function associated to \(\omega_X^\bullet\) (Lemma 0AWF). With these remarks in place, the lemma now follows from Dualizing Complexes, Lemma 0AWE and the definitions (in particular Cohomology of Schemes, Definition 0341).

Lemma

Let \(X/A\) with \(\omega_X^\bullet\) and \(\omega_X\) be as in Example 0AWI. Then

1. \(H^i(\omega_X^\bullet) \not = 0 \Rightarrow i \in \{-\dim(X), \ldots, 0\}\),

2. the dimension of the support of \(H^i(\omega_X^\bullet)\) is at most \(-i\),

3. \(\text{Supp}(\omega_X)\) is the union of the components of dimension \(\dim(X)\), and

4. \(\omega_X\) has property \((S_2)\).

Proof

Let \(\delta_X\) and \(\delta_S\) be the dimension functions associated to \(\omega_X^\bullet\) and \(\omega_S^\bullet\) as in Lemma 0AWM. As \(X\) is proper over \(A\), every closed subscheme of \(X\) contains a closed point \(x\) which maps to the closed point \(s \in S\) and \(\delta_X(x) = \delta_S(s) = 0\). Hence \(\delta_X(\xi) = \dim(\overline{\{\xi\}})\) for any point \(\xi \in X\). Hence we can check each of the statements of the lemma by looking at what happens over \(\Spec(\mathcal{O}_{X, x})\) in which case the result follows from Dualizing Complexes, Lemmas 0A7U and 0AWE. Some details omitted. The last two statements can also be deduced from Lemma 0AWK.

Lemma

Let \(X/A\) with dualizing module \(\omega_X\) be as in Example 0AWI. Let \(d = \dim(X_s)\) be the dimension of the closed fibre. If \(\dim(X) = d + \dim(A)\), then the dualizing module \(\omega_X\) represents the functor \[\mathcal{F} \longmapsto \Hom_A(H^d(X, \mathcal{F}), \omega_A)\] on the category of coherent \(\mathcal{O}_X\)-modules.

Proof

We have \[\begin{align*} \Hom_X(\mathcal{F}, \omega_X) & = \Ext^{-\dim(X)}_X(\mathcal{F}, \omega_X^\bullet) \\ & = \Hom_X(\mathcal{F}[\dim(X)], \omega_X^\bullet) \\ & = \Hom_X(\mathcal{F}[\dim(X)], f^!(\omega_A^\bullet)) \\ & = \Hom_S(Rf_*\mathcal{F}[\dim(X)], \omega_A^\bullet) \\ & = \Hom_A(H^d(X, \mathcal{F}), \omega_A) \end{align*}\] The first equality because \(H^i(\omega_X^\bullet) = 0\) for \(i < -\dim(X)\), see Lemma 0AWN and Derived Categories, Lemma 06XS. The second equality is follows from the definition of Ext groups. The third equality is our choice of \(\omega_X^\bullet\). The fourth equality holds because \(f^!\) is the right adjoint of Lemma 0A9E for \(f\), see Section 0AU3. The final equality holds because \(R^if_*\mathcal{F}\) is zero for \(i > d\) (Cohomology of Schemes, Lemma 02V7) and \(H^j(\omega_A^\bullet)\) is zero for \(j < -\dim(A)\).

Cohen-Macaulay schemes

This section is the continuation of Dualizing Complexes, Section 0DW4. Duality takes a particularly simple form for Cohen-Macaulay schemes.

Lemma

Let \(X\) be a locally Noetherian scheme with dualizing complex \(\omega_X^\bullet\).

1. \(X\) is Cohen-Macaulay \(\Leftrightarrow\) \(\omega_X^\bullet\) locally has a unique nonzero cohomology sheaf,

2. \(\mathcal{O}_{X, x}\) is Cohen-Macaulay \(\Leftrightarrow\) \(\omega_{X, x}^\bullet\) has a unique nonzero cohomology,

3. \(U = \{x \in X \mid \mathcal{O}_{X, x}\text{ is Cohen-Macaulay}\}\) is open and Cohen-Macaulay.

If \(X\) is connected and Cohen-Macaulay, then there is an integer \(n\) and a coherent Cohen-Macaulay \(\mathcal{O}_X\)-module \(\omega_X\) such that \(\omega_X^\bullet = \omega_X[-n]\).

Proof

By definition and Dualizing Complexes, Lemma 0A7G for every \(x \in X\) the complex \(\omega_{X, x}^\bullet\) is a dualizing complex over \(\mathcal{O}_{X, x}\). By Dualizing Complexes, Lemma 0AWS we see that (2) holds.

To see (3) assume that \(\mathcal{O}_{X, x}\) is Cohen-Macaulay. Let \(n_x\) be the unique integer such that \(H^{n_{x}}(\omega_{X, x}^\bullet)\) is nonzero. For an affine neighbourhood \(V \subset X\) of \(x\) we have \(\omega_X^\bullet|_V\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_V)\) hence there are finitely many nonzero coherent modules \(H^i(\omega_X^\bullet)|_V\). Thus after shrinking \(V\) we may assume only \(H^{n_x}\) is nonzero, see Modules, Lemma 01B9. In this way we see that \(\mathcal{O}_{X, v}\) is Cohen-Macaulay for every \(v \in V\). This proves that \(U\) is open as well as a Cohen-Macaulay scheme.

Proof of (1). The implication \(\Leftarrow\) follows from (2). The implication \(\Rightarrow\) follows from the discussion in the previous paragraph, where we showed that if \(\mathcal{O}_{X, x}\) is Cohen-Macaulay, then in a neighbourhood of \(x\) the complex \(\omega_X^\bullet\) has only one nonzero cohomology sheaf.

Assume \(X\) is connected and Cohen-Macaulay. The above shows that the map \(x \mapsto n_x\) is locally constant. Since \(X\) is connected it is constant, say equal to \(n\). Setting \(\omega_X = H^n(\omega_X^\bullet)\) we see that the lemma holds because \(\omega_X\) is Cohen-Macaulay by Dualizing Complexes, Lemma 0AWS (and Cohomology of Schemes, Definition 0343).

Lemma

Let \(X\) be a locally Noetherian scheme. If there exists a coherent sheaf \(\omega_X\) such that \(\omega_X[0]\) is a dualizing complex on \(X\), then \(X\) is a Cohen-Macaulay scheme.

Proof

This follows immediately from Dualizing Complexes, Lemma 0DW5 and our definitions.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Let \(x \in X\). If \(f\) is flat, then the following are equivalent

1. \(f\) is Cohen-Macaulay at \(x\),

2. \(f^!\mathcal{O}_Y\) has a unique nonzero cohomology sheaf in a neighbourhood of \(x\).

Proof

One direction of the lemma follows from Lemma 0BV8. To prove the converse, we may assume \(f^!\mathcal{O}_Y\) has a unique nonzero cohomology sheaf. Let \(y = f(x)\). Let \(\xi_1, \ldots, \xi_n \in X_y\) be the generic points of the fibre \(X_y\) specializing to \(x\). Let \(d_1, \ldots, d_n\) be the dimensions of the corresponding irreducible components of \(X_y\). The morphism \(f : X \to Y\) is Cohen-Macaulay at \(\eta_i\) by More on Morphisms, Lemma 045U. Hence by Lemma 0BV8 we see that \(d_1 = \ldots = d_n\). If \(d\) denotes the common value, then \(d = \dim_x(X_y)\). After shrinking \(X\) we may assume all fibres have dimension at most \(d\) (Morphisms, Lemma 02FZ). Then the only nonzero cohomology sheaf \(\omega = H^{-d}(f^!\mathcal{O}_Y)\) is flat over \(Y\) by Lemma 0BV6. Hence, if \(h : X_y \to X\) denotes the canonical morphism, then \(Lh^*(f^!\mathcal{O}_Y) = Lh^*(\omega[d]) = (h^*\omega)[d]\) by Derived Categories of Schemes, Lemma 0C0V. Thus \(h^*\omega[d]\) is the dualizing complex of \(X_y\) by Lemma 0BZZ. Hence \(X_y\) is Cohen-Macaulay by Lemma 0AWT. This proves \(f\) is Cohen-Macaulay at \(x\) as desired.

Remark

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Assume \(f\) is a Cohen-Macaulay morphism of relative dimension \(d\). Let \(\omega_{X/Y} = H^{-d}(f^!\mathcal{O}_Y)\) be the unique nonzero cohomology sheaf of \(f^!\mathcal{O}_Y\), see Lemma 0BV8. Then there is a canonical isomorphism \[f^!K = Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}[d]\] for \(K \in D^+_\QCoh(\mathcal{O}_Y)\), see Lemma 0B6U. In particular, if \(S\) has a dualizing complex \(\omega_S^\bullet\), \(\omega_Y^\bullet = (Y \to S)^!\omega_S^\bullet\), and \(\omega_X^\bullet = (X \to S)^!\omega_S^\bullet\) then we have \[\omega_X^\bullet = Lf^*\omega_Y^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}[d]\] Thus if further \(X\) and \(Y\) are connected and Cohen-Macaulay and if \(\omega_Y\) and \(\omega_X\) denote the unique nonzero cohomology sheaves of \(\omega_Y^\bullet\) and \(\omega_X^\bullet\), then we have \[\omega_X = f^*\omega_Y \otimes_{\mathcal{O}_X} \omega_{X/Y}.\] Similar results hold for \(X\) and \(Y\) arbitrary finite type schemes over \(S\) (i.e., not necessarily separated over \(S\)) with dualizing complexes normalized with respect to \(\omega_S^\bullet\) as in Section 0AU5.

Gorenstein schemes

This section is the continuation of Dualizing Complexes, Section 0DW6.

Definition

Let \(X\) be a scheme. We say \(X\) is Gorenstein if \(X\) is locally Noetherian and \(\mathcal{O}_{X, x}\) is Gorenstein for all \(x \in X\).

This definition makes sense because a Noetherian ring is said to be Gorenstein if and only if all of its local rings are Gorenstein, see Dualizing Complexes, Definition 0DW7.

Lemma

A Gorenstein scheme is Cohen-Macaulay.

Proof

Looking affine locally this follows from the corresponding result in algebra, namely Dualizing Complexes, Lemma 0DW8.

Lemma

A regular scheme is Gorenstein.

Proof

Looking affine locally this follows from the corresponding result in algebra, namely Dualizing Complexes, Lemma 0AWX.

Lemma

Let \(X\) be a locally Noetherian scheme.

1. If \(X\) has a dualizing complex \(\omega_X^\bullet\), then

   1. \(X\) is Gorenstein \(\Leftrightarrow\) \(\omega_X^\bullet\) is an invertible object of \(D(\mathcal{O}_X)\),

   2. \(\mathcal{O}_{X, x}\) is Gorenstein \(\Leftrightarrow\) \(\omega_{X, x}^\bullet\) is an invertible object of \(D(\mathcal{O}_{X, x})\),

   3. \(U = \{x \in X \mid \mathcal{O}_{X, x}\text{ is Gorenstein}\}\) is an open Gorenstein subscheme.

2. If \(X\) is Gorenstein, then \(X\) has a dualizing complex if and only if \(\mathcal{O}_X[0]\) is a dualizing complex.

Proof

Looking affine locally this follows from the corresponding result in algebra, namely Dualizing Complexes, Lemma 0DW9.

Lemma

If \(f : Y \to X\) is a local complete intersection morphism with \(X\) a Gorenstein scheme, then \(Y\) is Gorenstein.

Proof

By More on Morphisms, Lemma 07DB it suffices to prove the corresponding statement about ring maps. This is Dualizing Complexes, Lemma 0DWA.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is Gorenstein” is local in the syntomic topology.

Proof

Let \(\{S_i \to S\}\) be a syntomic covering. The scheme \(S\) is locally Noetherian if and only if each \(S_i\) is Noetherian, see Descent, Lemma 034C. Thus we may now assume \(S\) and \(S_i\) are locally Noetherian. If \(S\) is Gorenstein, then each \(S_i\) is Gorenstein by Lemma 0BVA. Conversely, if each \(S_i\) is Gorenstein, then for each point \(s \in S\) we can pick \(i\) and \(t \in S_i\) mapping to \(s\). Then \(\mathcal{O}_{S, s} \to \mathcal{O}_{S_i, t}\) is a flat local ring homomorphism with \(\mathcal{O}_{S_i, t}\) Gorenstein. Hence \(\mathcal{O}_{S, s}\) is Gorenstein by Dualizing Complexes, Lemma 0BJL.

Gorenstein morphisms

This section is one in a series. The corresponding sections for normal morphisms, regular morphisms, and Cohen-Macaulay morphisms can be found in More on Morphisms, Sections 038Z, 07R6, and 045Q.

The following lemma says that it does not make sense to define geometrically Gorenstein schemes, since these would be the same as Gorenstein schemes.

Lemma

Let \(X\) be a locally Noetherian scheme over the field \(k\). Let \(k'/k\) be a finitely generated field extension. Let \(x \in X\) be a point, and let \(x' \in X_{k'}\) be a point lying over \(x\). Then we have \[\mathcal{O}_{X, x}\text{ is Gorenstein} \Leftrightarrow \mathcal{O}_{X_{k'}, x'}\text{ is Gorenstein}\] If \(X\) is locally of finite type over \(k\), the same holds for any field extension \(k'/k\).

Proof

In both cases the ring map \(\mathcal{O}_{X, x} \to \mathcal{O}_{X_{k'}, x'}\) is a faithfully flat local homomorphism of Noetherian local rings. Thus if \(\mathcal{O}_{X_{k'}, x'}\) is Gorenstein, then so is \(\mathcal{O}_{X, x}\) by Dualizing Complexes, Lemma 0BJL. To go up, we use Dualizing Complexes, Lemma 0BJL as well. Thus we have to show that \[\mathcal{O}_{X_{k'}, x'}/\mathfrak m_x \mathcal{O}_{X_{k'}, x'} = \kappa(x) \otimes_k k'\] is Gorenstein. Note that in the first case \(k \to k'\) is finitely generated and in the second case \(k \to \kappa(x)\) is finitely generated. Hence this follows as property (A) holds for Gorenstein, see Dualizing Complexes, Lemma 0BJN.

The lemma above guarantees that the following is the correct definition of Gorenstein morphisms.

Definition

Let \(f : X \to Y\) be a morphism of schemes. Assume that all the fibres \(X_y\) are locally Noetherian schemes.

1. Let \(x \in X\), and \(y = f(x)\). We say that \(f\) is Gorenstein at \(x\) if \(f\) is flat at \(x\), and the local ring of the scheme \(X_y\) at \(x\) is Gorenstein.

2. We say \(f\) is a Gorenstein morphism if \(f\) is Gorenstein at every point of \(X\).

Here is a translation.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume all fibres of \(f\) are locally Noetherian. The following are equivalent

1. \(f\) is Gorenstein, and

2. \(f\) is flat and its fibres are Gorenstein schemes.

Proof

This follows directly from the definitions.

Lemma

A Gorenstein morphism is Cohen-Macaulay.

Proof

Follows from Lemma 0C00 and the definitions.

Lemma

A syntomic morphism is Gorenstein. Equivalently a flat local complete intersection morphism is Gorenstein.

Proof

Recall that a syntomic morphism is flat and its fibres are local complete intersections over fields, see Morphisms, Lemma 01UF. Since a local complete intersection over a field is a Gorenstein scheme by Lemma 0BVA we conclude. The properties “syntomic” and “flat and local complete intersection morphism” are equivalent by More on Morphisms, Lemma 069K.

Lemma

Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms. Assume that the fibres \(X_y\), \(Y_z\) and \(X_z\) of \(f\), \(g\), and \(g \circ f\) are locally Noetherian.

1. If \(f\) is Gorenstein at \(x\) and \(g\) is Gorenstein at \(f(x)\), then \(g \circ f\) is Gorenstein at \(x\).

2. If \(f\) and \(g\) are Gorenstein, then \(g \circ f\) is Gorenstein.

3. If \(g \circ f\) is Gorenstein at \(x\) and \(f\) is flat at \(x\), then \(f\) is Gorenstein at \(x\) and \(g\) is Gorenstein at \(f(x)\).

4. If \(g \circ f\) is Gorenstein and \(f\) is flat, then \(f\) is Gorenstein and \(g\) is Gorenstein at every point in the image of \(f\).

Proof

After translating into algebra this follows from Dualizing Complexes, Lemma 0BJL.

Lemma

Let \(f : X \to Y\) be a flat morphism of locally Noetherian schemes. If \(X\) is Gorenstein, then \(f\) is Gorenstein and \(\mathcal{O}_{Y, f(x)}\) is Gorenstein for all \(x \in X\).

Proof

After translating into algebra this follows from Dualizing Complexes, Lemma 0BJL.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume that all the fibres \(X_y\) are locally Noetherian schemes. Let \(Y' \to Y\) be locally of finite type. Let \(f' : X' \to Y'\) be the base change of \(f\). Let \(x' \in X'\) be a point with image \(x \in X\).

1. If \(f\) is Gorenstein at \(x\), then \(f' : X' \to Y'\) is Gorenstein at \(x'\).

2. If \(f\) is flat at \(x\) and \(f'\) is Gorenstein at \(x'\), then \(f\) is Gorenstein at \(x\).

3. If \(Y' \to Y\) is flat at \(f'(x')\) and \(f'\) is Gorenstein at \(x'\), then \(f\) is Gorenstein at \(x\).

Proof

Note that the assumption on \(Y' \to Y\) implies that for \(y' \in Y'\) mapping to \(y \in Y\) the field extension \(\kappa(y')/\kappa(y)\) is finitely generated. Hence also all the fibres \(X'_{y'} = (X_y)_{\kappa(y')}\) are locally Noetherian, see Varieties, Lemma 038R. Thus the lemma makes sense. Set \(y' = f'(x')\) and \(y = f(x)\). Hence we get the following commutative diagram of local rings \[\xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{Y', y'} \ar[u] & \mathcal{O}_{Y, y} \ar[l] \ar[u] }\] where the upper left corner is a localization of the tensor product of the upper right and lower left corners over the lower right corner.

Assume \(f\) is Gorenstein at \(x\). The flatness of \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) implies the flatness of \(\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}\), see Algebra, Lemma 00MQ. The fact that \(\mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) is Gorenstein implies that \(\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}\) is Gorenstein, see Lemma 0C03. Hence we see that \(f'\) is Gorenstein at \(x'\).

Assume \(f\) is flat at \(x\) and \(f'\) is Gorenstein at \(x'\). The fact that \(\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}\) is Gorenstein implies that \(\mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) is Gorenstein, see Lemma 0C03. Hence we see that \(f\) is Gorenstein at \(x\).

Assume \(Y' \to Y\) is flat at \(y'\) and \(f'\) is Gorenstein at \(x'\). The flatness of \(\mathcal{O}_{Y', y'} \to \mathcal{O}_{X', x'}\) and \(\mathcal{O}_{Y, y} \to \mathcal{O}_{Y', y'}\) implies the flatness of \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\), see Algebra, Lemma 00MQ. The fact that \(\mathcal{O}_{X', x'}/\mathfrak m_{y'}\mathcal{O}_{X', x'}\) is Gorenstein implies that \(\mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) is Gorenstein, see Lemma 0C03. Hence we see that \(f\) is Gorenstein at \(x\).

Lemma

Let \(f : X \to Y\) be a morphism of schemes which is flat and locally of finite type. Then formation of the set \(\{x \in X \mid f\text{ is Gorenstein at }x\}\) commutes with arbitrary base change.

Proof

The assumption implies any fibre of \(f\) is locally of finite type over a field and hence locally Noetherian and the same is true for any base change. Thus the statement makes sense. Looking at fibres we reduce to the following problem: let \(X\) be a scheme locally of finite type over a field \(k\), let \(K/k\) be a field extension, and let \(x_K \in X_K\) be a point with image \(x \in X\). Problem: show that \(\mathcal{O}_{X_K, x_K}\) is Gorenstein if and only if \(\mathcal{O}_{X, x}\) is Gorenstein.

The problem can be solved using a bit of algebra as follows. Choose an affine open \(\Spec(A) \subset X\) containing \(x\). Say \(x\) corresponds to \(\mathfrak p \subset A\). With \(A_K = A \otimes_k K\) we see that \(\Spec(A_K) \subset X_K\) contains \(x_K\). Say \(x_K\) corresponds to \(\mathfrak p_K \subset A_K\). Let \(\omega_A^\bullet\) be a dualizing complex for \(A\). By Dualizing Complexes, Lemma 0E0P \(\omega_A^\bullet \otimes_A A_K\) is a dualizing complex for \(A_K\). Now we are done because \(A_\mathfrak p \to (A_K)_{\mathfrak p_K}\) is a flat local homomorphism of Noetherian rings and hence \((\omega_A^\bullet)_\mathfrak p\) is an invertible object of \(D(A_\mathfrak p)\) if and only if \((\omega_A^\bullet)_\mathfrak p \otimes_{A_\mathfrak p} (A_K)_{\mathfrak p_K}\) is an invertible object of \(D((A_K)_{\mathfrak p_K})\). Some details omitted; hint: look at cohomology modules.

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Let \(x \in X\). If \(f\) is flat, then the following are equivalent

1. \(f\) is Gorenstein at \(x\),

2. \(f^!\mathcal{O}_Y\) is isomorphic to an invertible object in a neighbourhood of \(x\).

In particular, the set of points where \(f\) is Gorenstein is open in \(X\).

Proof

Set \(\omega^\bullet = f^!\mathcal{O}_Y\). By Lemma 0BZZ this is a bounded complex with coherent cohomology sheaves whose derived restriction \(Lh^*\omega^\bullet\) to the fibre \(X_y\) is a dualizing complex on \(X_y\). Denote \(i : x \to X_y\) the inclusion of a point. Then the following are equivalent

1. \(f\) is Gorenstein at \(x\),

2. \(\mathcal{O}_{X_y, x}\) is Gorenstein,

3. \(Lh^*\omega^\bullet\) is invertible in a neighbourhood of \(x\),

4. \(Li^* Lh^* \omega^\bullet\) has exactly one nonzero cohomology of dimension \(1\) over \(\kappa(x)\),

5. \(L(h \circ i)^* \omega^\bullet\) has exactly one nonzero cohomology of dimension \(1\) over \(\kappa(x)\),

6. \(\omega^\bullet\) is invertible in a neighbourhood of \(x\).

The equivalence of (1) and (2) is by definition (as \(f\) is flat). The equivalence of (2) and (3) follows from Lemma 0BFQ. The equivalence of (3) and (4) follows from More on Algebra, Lemma 0BYP. The equivalence of (4) and (5) holds because \(Li^* Lh^* = L(h \circ i)^*\). The equivalence of (5) and (6) holds by More on Algebra, Lemma 0BYP. Thus the lemma is clear.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Let \(x \in X\) with image \(s \in S\). Set \(d = \dim_x(X_s)\). The following are equivalent

1. \(f\) is Gorenstein at \(x\),

2. there exists an open neighbourhood \(U \subset X\) of \(x\) and a locally quasi-finite morphism \(U \to \mathbf{A}^d_S\) over \(S\) which is Gorenstein at \(x\),

3. there exists an open neighbourhood \(U \subset X\) of \(x\) and a locally quasi-finite Gorenstein morphism \(U \to \mathbf{A}^d_S\) over \(S\),

4. for any \(S\)-morphism \(g : U \to \mathbf{A}^d_S\) of an open neighbourhood \(U \subset X\) of \(x\) we have: \(g\) is quasi-finite at \(x\) \(\Rightarrow\) \(g\) is Gorenstein at \(x\).

In particular, the set of points where \(f\) is Gorenstein is open in \(X\).

Proof

Choose affine open \(U = \Spec(A) \subset X\) with \(x \in U\) and \(V = \Spec(R) \subset S\) with \(f(U) \subset V\). Then \(R \to A\) is a flat ring map of finite presentation. Let \(\mathfrak p \subset A\) be the prime ideal corresponding to \(x\). After replacing \(A\) by a principal localization we may assume there exists a quasi-finite map \(R[x_1, \ldots, x_d] \to A\), see Algebra, Lemma 00QE. Thus there exists at least one pair \((U, g)\) consisting of an open neighbourhood \(U \subset X\) of \(x\) and a locally⁸ quasi-finite morphism \(g : U \to \mathbf{A}^d_S\).

Having said this, the lemma translates into the following algebra problem (translation omitted). Given \(R \to A\) flat and of finite presentation, a prime \(\mathfrak p \subset A\) and \(\varphi : R[x_1, \ldots, x_d] \to A\) quasi-finite at \(\mathfrak p\) the following are equivalent

1. \(\Spec(\varphi)\) is Gorenstein at \(\mathfrak p\), and

2. \(\Spec(A) \to \Spec(R)\) is Gorenstein at \(\mathfrak p\).

3. \(\Spec(A) \to \Spec(R)\) is Gorenstein in an open neighbourhood of \(\mathfrak p\).

In each case \(R[x_1, \ldots, x_n] \to A\) is flat at \(\mathfrak p\) hence by openness of flatness (Algebra, Theorem 00RC), we may assume \(R[x_1, \ldots, x_n] \to A\) is flat (replace \(A\) by a suitable principal localization). By Algebra, Lemma 02JO there exists \(R_0 \subset R\) and \(R_0[x_1, \ldots, x_n] \to A_0\) such that \(R_0\) is of finite type over \(\mathbf{Z}\) and \(R_0 \to A_0\) is of finite type and \(R_0[x_1, \ldots, x_n] \to A_0\) is flat. Note that the set of points where a flat finite type morphism is Gorenstein commutes with base change by Lemma 0C07. In this way we reduce to the case where \(R\) is Noetherian.

Thus we may assume \(X\) and \(S\) affine and that we have a factorization of \(f\) of the form \[X \xrightarrow{g} \mathbf{A}^n_S \xrightarrow{p} S\] with \(g\) flat and quasi-finite and \(S\) Noetherian. Then \(X\) and \(\mathbf{A}^n_S\) are separated over \(S\) and we have \[f^!\mathcal{O}_S = g^!p^!\mathcal{O}_S = g^!\mathcal{O}_{\mathbf{A}^n_S}[n]\] by know properties of upper shriek functors (Lemmas 0ATX and 0AA1). Hence the equivalence of (a), (b), and (c) by Lemma 0C08.

Lemma

The property \(\mathcal{P}(f)=\)“the fibres of \(f\) are locally Noetherian and \(f\) is Gorenstein” is local in the fppf topology on the target and local in the syntomic topology on the source.

Proof

We have \(\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f)\) where \(\mathcal{P}_1(f)=\)“\(f\) is flat”, and \(\mathcal{P}_2(f)=\)“the fibres of \(f\) are locally Noetherian and Gorenstein”. We know that \(\mathcal{P}_1\) is local in the fppf topology on the source and the target, see Descent, Lemmas 02L2 and 036K. Thus we have to deal with \(\mathcal{P}_2\).

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{\varphi_i : Y_i \to Y\}_{i \in I}\) be an fppf covering of \(Y\). Denote \(f_i : X_i \to Y_i\) the base change of \(f\) by \(\varphi_i\). Let \(i \in I\) and let \(y_i \in Y_i\) be a point. Set \(y = \varphi_i(y_i)\). Note that \[X_{i, y_i} = \Spec(\kappa(y_i)) \times_{\Spec(\kappa(y))} X_y.\] and that \(\kappa(y_i)/\kappa(y)\) is a finitely generated field extension. Hence if \(X_y\) is locally Noetherian, then \(X_{i, y_i}\) is locally Noetherian, see Varieties, Lemma 038R. And if in addition \(X_y\) is Gorenstein, then \(X_{i, y_i}\) is Gorenstein, see Lemma 0C03. Thus \(\mathcal{P}_2\) is fppf local on the target.

Let \(\{X_i \to X\}\) be a syntomic covering of \(X\). Let \(y \in Y\). In this case \(\{X_{i, y} \to X_y\}\) is a syntomic covering of the fibre. Hence the locality of \(\mathcal{P}_2\) for the syntomic topology on the source follows from Lemma 0C01.

More on dualizing complexes

Some lemmas which don’t fit anywhere else very well.

Lemma

Let \(f : X \to Y\) be a morphism of locally Noetherian schemes. Assume

1. \(f\) is syntomic and surjective, or

2. \(f\) is a surjective flat local complete intersection morphism, or

3. \(f\) is a surjective Gorenstein morphism of finite type.

Then \(K \in D_\QCoh(\mathcal{O}_Y)\) is a dualizing complex on \(Y\) if and only if \(Lf^*K\) is a dualizing complex on \(X\).

Proof

Taking affine opens and using Derived Categories of Schemes, Lemma 06Z0 this translates into Dualizing Complexes, Lemma 0E4D.

Duality for proper schemes over fields

In this section we work out the consequences of the very general material above on dualizing complexes and duality for proper schemes over fields.

Lemma

Let \(X\) be a proper scheme over a field \(k\). There exists a dualizing complex \(\omega_X^\bullet\) with the following properties

1. \(H^i(\omega_X^\bullet)\) is nonzero only for \(i \in [-\dim(X), 0]\),

2. \(\omega_X = H^{-\dim(X)}(\omega_X^\bullet)\) is a coherent \((S_2)\)-module whose support is the irreducible components of dimension \(\dim(X)\),

3. the dimension of the support of \(H^i(\omega_X^\bullet)\) is at most \(-i\),

4. for \(x \in X\) closed the module \(H^i(\omega_{X, x}^\bullet) \oplus \ldots \oplus H^0(\omega_{X, x}^\bullet)\) is nonzero if and only if \(\text{depth}(\mathcal{O}_{X, x}) \leq -i\),

5. for \(K \in D_\QCoh(\mathcal{O}_X)\) there are functorial isomorphisms⁹ \[\Ext^i_X(K, \omega_X^\bullet) = \Hom_k(H^{-i}(X, K), k)\] compatible with shifts and distinguished triangles,

6. there are functorial isomorphisms \(\Hom(\mathcal{F}, \omega_X) = \Hom_k(H^{\dim(X)}(X, \mathcal{F}), k)\) for \(\mathcal{F}\) quasi-coherent on \(X\), and

7. if \(X \to \Spec(k)\) is smooth of relative dimension \(d\), then \(\omega_X^\bullet \cong \wedge^d\Omega_{X/k}[d]\) and \(\omega_X \cong \wedge^d\Omega_{X/k}\).

Proof

Denote \(f : X \to \Spec(k)\) the structure morphism. Let \(a\) be the right adjoint of pushforward of this morphism, see Lemma 0A9E. Consider the relative dualizing complex \[\omega_X^\bullet = a(\mathcal{O}_{\Spec(k)})\] Compare with Remark 0B6S. Since \(f\) is proper we have \(f^!(\mathcal{O}_{\Spec(k)}) = a(\mathcal{O}_{\Spec(k)})\) by definition, see Section 0A9Y. Applying Lemma 0AA3 we find that \(\omega_X^\bullet\) is a dualizing complex. Moreover, we see that \(\omega_X^\bullet\) and \(\omega_X\) are as in Example 0AWI and as in Example 0AWJ.

Parts (1), (2), and (3) follow from Lemma 0AWN.

For a closed point \(x \in X\) we see that \(\omega_{X, x}^\bullet\) is a normalized dualizing complex over \(\mathcal{O}_{X, x}\), see Lemma 0AWL. Part (4) then follows from Dualizing Complexes, Lemma 0AWR.

Part (5) holds by construction as \(a\) is the right adjoint to \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_{\Spec(k)}) = D(k)\) which we can identify with \(K \mapsto R\Gamma(X, K)\). We also use that the derived category \(D(k)\) of \(k\)-modules is the same as the category of graded \(k\)-vector spaces.

Part (6) follows from Lemma 0AWP for coherent \(\mathcal{F}\) and in general by unwinding (5) for \(K = \mathcal{F}[0]\) and \(i = -\dim(X)\).

Part (7) follows from Lemma 0BRT.

Remark

Let \(k\), \(X\), and \(\omega_X^\bullet\) be as in Lemma 0FVV. The identity on the complex \(\omega_X^\bullet\) corresponds, via the functorial isomorphism in part (5), to a map \[t : H^0(X, \omega_X^\bullet) \longrightarrow k\] For an arbitrary \(K\) in \(D_\QCoh(\mathcal{O}_X)\) the identification \(\Hom(K, \omega_X^\bullet)\) with \(H^0(X, K)^\vee\) in part (5) corresponds to the pairing \[\Hom_X(K, \omega_X^\bullet) \times H^0(X, K) \longrightarrow k,\quad (\alpha, \beta) \longmapsto t(\alpha(\beta))\] This follows from the functoriality of the isomorphisms in (5). Similarly for any \(i \in \mathbf{Z}\) we get the pairing \[\Ext^i_X(K, \omega_X^\bullet) \times H^{-i}(X, K) \longrightarrow k,\quad (\alpha, \beta) \longmapsto t(\alpha(\beta))\] Here we think of \(\alpha\) as a morphism \(K[-i] \to \omega_X^\bullet\) and \(\beta\) as an element of \(H^0(X, K[-i])\) in order to define \(\alpha(\beta)\). Observe that if \(K\) is general, then we only know that this pairing is nondegenerate on one side: the pairing induces an isomorphism of \(\Hom_X(K, \omega_X^\bullet)\), resp. \(\Ext^i_X(K, \omega_X^\bullet)\) with the \(k\)-linear dual of \(H^0(X, K)\), resp. \(H^{-i}(X, K)\) but in general not vice versa. If \(K\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\), then \(\Hom_X(K, \omega_X^\bullet)\), \(\Ext_X(K, \omega_X^\bullet)\), \(H^0(X, K)\), and \(H^i(X, K)\) are finite dimensional \(k\)-vector spaces (by Derived Categories of Schemes, Lemmas 0D0C and 0D0B) and the pairings are perfect in the usual sense.

Remark

We continue the discussion in Remark 0FVW and we use the same notation \(k\), \(X\), \(\omega_X^\bullet\), and \(t\). If \(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module we obtain perfect pairings \[\langle -, - \rangle : \Ext^i_X(\mathcal{F}, \omega_X^\bullet) \times H^{-i}(X,\mathcal{F}) \longrightarrow k,\quad (\alpha, \beta) \longmapsto t(\alpha(\beta))\] of finite dimensional \(k\)-vector spaces. These pairings satisfy the following (obvious) functoriality: if \(\varphi : \mathcal{F} \to \mathcal{G}\) is a homomorphism of coherent \(\mathcal{O}_X\)-modules, then we have \[\langle \alpha \circ \varphi, \beta \rangle = \langle \alpha, \varphi(\beta) \rangle\] for \(\alpha \in \Ext^i_X(\mathcal{G}, \omega_X^\bullet)\) and \(\beta \in H^{-i}(X, \mathcal{F})\). In other words, the \(k\)-linear map \(\Ext^i_X(\mathcal{G}, \omega_X^\bullet) \to \Ext^i_X(\mathcal{F}, \omega_X^\bullet)\) induced by \(\varphi\) is, via the pairings, the \(k\)-linear dual of the \(k\)-linear map \(H^{-i}(X, \mathcal{F}) \to H^{-i}(X, \mathcal{G})\) induced by \(\varphi\). Formulated in this manner, this still works if \(\varphi\) is a homomorphism of quasi-coherent \(\mathcal{O}_X\)-modules.

Lemma

Let \(k\), \(X\), and \(\omega_X^\bullet\) be as in Lemma 0FVV. Let \(t : H^0(X, \omega_X^\bullet) \to k\) be as in Remark 0FVW. Let \(E \in D(\mathcal{O}_X)\) be perfect. Then the pairings \[H^i(X, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} E^\vee) \times H^{-i}(X, E) \longrightarrow k, \quad (\xi, \eta) \longmapsto t((1_{\omega_X^\bullet} \otimes \epsilon)(\xi \cup \eta))\] are perfect for all \(i\). Here \(\cup\) denotes the cup product of Cohomology, Section 0FKU and \(\epsilon : E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E \to \mathcal{O}_X\) is as in Cohomology, Example 0FPC.

Proof

By replacing \(E\) with \(E[-i]\) this reduces to the case \(i = 0\). By Cohomology, Lemma 0FVB we see that the pairing is the same as the one discussed in Remark 0FVW whence the result by the discussion in that remark.

Lemma

Let \(X\) be a nonempty projective scheme over a field \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. Let \(\omega_X^\bullet\) and \(\omega_X\) be as in Lemma 0FVV, and set \(d = \dim(X)\). For a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) and \(p \in \mathbf{Z}\), consider the map \[\begin{aligned} a_{\mathcal{F}, p} : \Ext_X^{d - p}(\mathcal{F}, \omega_X) & = \Ext_X^{-p}(\mathcal{F}, \omega_X[d]) \\ & \longrightarrow \Ext_X^{-p}(\mathcal{F}, \omega_X^\bullet) \\ & = \Hom_k(H^p(X, \mathcal{F}), k). \end{aligned}\] Here the arrow is induced by the canonical map \(\omega_X[d] \to \omega_X^\bullet\). For an integer \(c \geq 0\), the following are equivalent.

1. The map \(a_{\mathcal{F}, p}\) is an isomorphism for every coherent \(\mathcal{F}\) and \(d - c \leq p \leq d\).

2. We have \(H^p(X, \mathcal{L}^{\otimes -m}) = 0\) for \(m \gg 0\) and \(d - c \leq p < d\).

3. For every coherent \(\mathcal{F}\) and \(d - c \leq p < d\), there is a surjection \(\mathcal{E} \to \mathcal{F}\) of coherent modules such that \[H^p(X, \mathcal{E}) \longrightarrow H^p(X, \mathcal{F})\] is zero.

4. We have \(H^{-d + i}(\omega_X^\bullet) = 0\) for \(0 < i \leq c\).

Proof

The canonical map in the statement exists because \(\omega_X^\bullet\) has no cohomology in degrees less than \(-d\). The final equality in the definition of \(a_{\mathcal{F},p}\) is Lemma 0FVV part (5).

Assume (1). For \(d - c \leq p < d\), apply (1) to \(\mathcal{F} = \mathcal{L}^{\otimes -m}\). Since this module is invertible, the source of \(a_{\mathcal{F},p}\) is \[H^{d - p}(X, \omega_X \otimes \mathcal{L}^{\otimes m}),\] which is zero for \(m \gg 0\) by Cohomology of Schemes, Lemma 0B5T. Thus (2) holds.

Assume (2). Given \(\mathcal{F}\) and \(p\) as in (3), choose \(m\) sufficiently large that \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated and \(H^p(X, \mathcal{L}^{\otimes -m}) = 0\). Such an \(m\) exists by Properties, Proposition 01Q3 and (2). We obtain a surjection \[(\mathcal{L}^{\otimes -m})^{\oplus r} \longrightarrow \mathcal{F}\] for some \(r\), and this surjection has the property required in (3).

Assume (3). For \(q \geq 0\), set \[T^q(\mathcal{F}) = \Ext_X^q(\mathcal{F}, \omega_X) \quad\text{and}\quad U^q(\mathcal{F}) = \Hom_k(H^{d - q}(X, \mathcal{F}), k).\] These are contravariant delta functors on coherent modules, and the maps \(a_{\mathcal{F},d-q}\) give a morphism between them. The map \(T^0(\mathcal{F}) \to U^0(\mathcal{F})\) is an isomorphism by Lemma 0FVV part (6).

For \(q > 0\), the functor \(T^q\) is co-effaceable with respect to surjections. Indeed, choose \(m \gg 0\) and a surjection \[(\mathcal{L}^{\otimes -m})^{\oplus r} \longrightarrow \mathcal{F}.\] By Cohomology of Schemes, Lemma 0B5T, we may choose \(m\) such that \[\Ext_X^q(\mathcal{L}^{\otimes -m}, \omega_X) = H^q(X, \omega_X \otimes \mathcal{L}^{\otimes m}) = 0.\] Condition (3), after taking \(k\)-linear duals, says that \(U^q\) is co-effaceable for \(1 \leq q \leq c\).

We finish by dimension shifting. Suppose by induction that \(T^{q - 1} \to U^{q - 1}\) is an isomorphism on every coherent module. Given \(\mathcal{F}\), take a surjection \(\mathcal{E}_1 \to \mathcal{F}\) which kills the map from \(U^q(\mathcal{F})\), and then a surjection \(\mathcal{E} \to \mathcal{E}_1\) with \(T^q(\mathcal{E}) = 0\). The composite \(\mathcal{E} \to \mathcal{F}\) kills the maps from both \(T^q(\mathcal{F})\) and \(U^q(\mathcal{F})\). If \(\mathcal{K}\) is its kernel, the two long exact sequences identify \(T^q(\mathcal{F})\) and \(U^q(\mathcal{F})\) with the respective cokernels of \[T^{q - 1}(\mathcal{E}) \longrightarrow T^{q - 1}(\mathcal{K}) \quad\text{and}\quad U^{q - 1}(\mathcal{E}) \longrightarrow U^{q - 1}(\mathcal{K}).\] The induction hypothesis identifies these cokernels. This proves (1).

It remains to compare (2) and (4). By duality and invertibility of \(\mathcal{L}\), we have \[\Hom_k(H^p(X, \mathcal{L}^{\otimes -m}), k) = H^{-p}(X, \omega_X^\bullet \otimes \mathcal{L}^{\otimes m}).\] The spectral sequence of Cohomology, Example 0BKM has terms \[H^a(X, H^b(\omega_X^\bullet) \otimes \mathcal{L}^{\otimes m})\] and converges to the group on the right. There are only finitely many nonzero coherent modules \(H^b(\omega_X^\bullet)\). Hence, for \(m \gg 0\), Serre vanishing makes the spectral sequence degenerate and gives \[\Hom_k(H^p(X, \mathcal{L}^{\otimes -m}), k) = H^0(X, H^{-p}(\omega_X^\bullet) \otimes \mathcal{L}^{\otimes m}).\] A coherent module \(\mathcal{G}\) is zero if and only if \(H^0(X, \mathcal{G} \otimes \mathcal{L}^{\otimes m})\) is zero for all sufficiently large \(m\): use global generation for the nontrivial direction. Taking \(p = d - i\) proves the equivalence of (2) and (4).

This is the intrinsic dualizing-complex form of [FGA, Exposé 149, Théorème 3 bis and Proposition 6, pp. 186–187].

Lemma

Let \(X\) be a proper scheme over a field \(k\) which is Cohen-Macaulay and equidimensional of dimension \(d\). The module \(\omega_X\) of Lemma 0FVV has the following properties

1. \(\omega_X\) is a dualizing module on \(X\) (Section 0AWH),

2. \(\omega_X\) is a coherent Cohen-Macaulay module whose support is \(X\),

3. there are functorial isomorphisms \(\Ext^i_X(K, \omega_X[d]) = \Hom_k(H^{-i}(X, K), k)\) compatible with shifts and distinguished triangles for \(K \in D_\QCoh(X)\),

4. there are functorial isomorphisms \(\Ext^{d - i}(\mathcal{F}, \omega_X) = \Hom_k(H^i(X, \mathcal{F}), k)\) for \(\mathcal{F}\) quasi-coherent on \(X\).

Proof

It is clear from Lemma 0FVV that \(\omega_X\) is a dualizing module (as it is the left most nonvanishing cohomology sheaf of a dualizing complex). We have \(\omega_X^\bullet = \omega_X[d]\) and \(\omega_X\) is Cohen-Macaulay as \(X\) is Cohen-Macualay, see Lemma 0AWT. The other statements follow from this combined with the corresponding statements of Lemma 0FVV.

Remark

Let \(X\) be a proper Cohen-Macaulay scheme over a field \(k\) which is equidimensional of dimension \(d\). Let \(\omega_X^\bullet\) and \(\omega_X\) be as in Lemma 0FVV. By Lemma 0FVZ we have \(\omega_X^\bullet = \omega_X[d]\). Let \(t : H^d(X, \omega_X) \to k\) be the map of Remark 0FVW. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module with dual \(\mathcal{E}^\vee\). Then we have perfect pairings \[H^i(X, \omega_X \otimes_{\mathcal{O}_X} \mathcal{E}^\vee) \times H^{d - i}(X, \mathcal{E}) \longrightarrow k,\quad (\xi, \eta) \longmapsto t(1 \otimes \epsilon)(\xi \cup \eta))\] where \(\cup\) is the cup-product and \(\epsilon : \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \mathcal{E} \to \mathcal{O}_X\) is the evaluation map. This is a special case of Lemma 0FVY.

Here is a sanity check for the dualizing complex.

Lemma

Let \(X\) be a proper scheme over a field \(k\). Let \(\omega_X^\bullet\) and \(\omega_X\) be as in Lemma 0FVV.

1. If \(X \to \Spec(k)\) factors as \(X \to \Spec(k') \to \Spec(k)\) for some field \(k'\), then \(\omega_X^\bullet\) and \(\omega_X\) are as in Lemma 0FVV for the morphism \(X \to \Spec(k')\).

2. If \(K/k\) is a field extension, then the pullback of \(\omega_X^\bullet\) and \(\omega_X\) to the base change \(X_K\) are as in Lemma 0FVV for the morphism \(X_K \to \Spec(K)\).

Proof

Denote \(f : X \to \Spec(k)\) the structure morphism and denote \(f' : X \to \Spec(k')\) the given factorization. In the proof of Lemma 0FVV we took \(\omega_X^\bullet = a(\mathcal{O}_{\Spec(k)})\) where \(a\) be is the right adjoint of Lemma 0A9E for \(f\). Thus we have to show \(a(\mathcal{O}_{\Spec(k)}) \cong a'(\mathcal{O}_{\Spec(k)})\) where \(a'\) be is the right adjoint of Lemma 0A9E for \(f'\). Since \(k' \subset H^0(X, \mathcal{O}_X)\) we see that \(k'/k\) is a finite extension (Cohomology of Schemes, Lemma 02O6). By uniqueness of adjoints we have \(a = a' \circ b\) where \(b\) is the right adjoint of Lemma 0A9E for \(g : \Spec(k') \to \Spec(k)\). Another way to say this: we have \(f^! = (f')^! \circ g^!\). Thus it suffices to show that \(\Hom_k(k', k) \cong k'\) as \(k'\)-modules, see Example 0A9F. This holds because these are \(k'\)-vector spaces of the same dimension (namely dimension \(1\)).

Proof of (2). This holds because we have base change for \(a\) by Lemma 0AA8. See discussion in Remark 0B6S.

Relative dualizing complexes

For a proper, flat morphism of finite presentation we have a rigid relative dualizing complex, see Remark 0B6S and Lemma 0E2P. For a separated and finite type morphism \(f : X \to Y\) of Noetherian schemes, we can consider \(f^!\mathcal{O}_Y\). In this section we define relative dualizing complexes for morphisms which are flat and locally of finite presentation (but not necessarily quasi-separated or quasi-compact) between schemes (not necessarily locally Noetherian). We show such complexes exist, are unique up to unique isomorphism, and agree with the cases mentioned above. Before reading this section, please read Dualizing Complexes, Section 0E2B.

Definition

Let \(X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Let \(W \subset X \times_S X\) be any open such that the diagonal \(\Delta_{X/S} : X \to X \times_S X\) factors through a closed immersion \(\Delta : X \to W\). A relative dualizing complex is a pair \((K, \xi)\) consisting of an object \(K \in D(\mathcal{O}_X)\) and a map \[\xi : \Delta_*\mathcal{O}_X \longrightarrow L\text{pr}_1^*K|_W\] in \(D(\mathcal{O}_W)\) such that

1. \(K\) is \(S\)-perfect (Derived Categories of Schemes, Definition 0DI0), and

2. \(\xi\) defines an isomorphism of \(\Delta_*\mathcal{O}_X\) with \(R\SheafHom_{\mathcal{O}_W}( \Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)\).

By Lemma 0A77 condition (2) is equivalent to the existence of an isomorphism \[\mathcal{O}_X \longrightarrow R\SheafHom(\mathcal{O}_X, L\text{pr}_1^*K|_W)\] in \(D(\mathcal{O}_X)\) whose pushforward via \(\Delta\) is equal to \(\xi\). Since \(R\SheafHom(\mathcal{O}_X, L\text{pr}_1^*K|_W)\) is independent of the choice of the open \(W\), so is the category of pairs \((K, \xi)\). If \(X \to S\) is separated, then we can choose \(W = X \times_S X\). We will reduce many of the arguments to the case of rings using the following lemma.

Lemma

Let \(X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Let \((K, \xi)\) be a relative dualizing complex. Then for any commutative diagram \[\xymatrix{ \Spec(A) \ar[d] \ar[r] & X \ar[d] \\ \Spec(R) \ar[r] & S }\] whose horizontal arrows are open immersions, the restriction of \(K\) to \(\Spec(A)\) corresponds via Derived Categories of Schemes, Lemma 06Z0 to a relative dualizing complex for \(R \to A\) in the sense of Dualizing Complexes, Definition 0E2C.

Proof

Since formation of \(R\SheafHom\) commutes with restrictions to opens we may as well assume \(X = \Spec(A)\) and \(S = \Spec(R)\). Observe that relatively perfect objects of \(D(\mathcal{O}_X)\) are pseudo-coherent and hence are in \(D_\QCoh(\mathcal{O}_X)\) (Derived Categories of Schemes, Lemma 08E5). Thus the statement makes sense. Observe that taking \(\Delta_*\), \(L\text{pr}_1^*\), and \(R\SheafHom\) is compatible with what happens on the algebraic side by Derived Categories of Schemes, Lemmas 0DJK, 08DW, 0A6H. For the last one we observe that \(L\text{pr}_1^*K\) is \(S\)-perfect (hence bounded below) and that \(\Delta_*\mathcal{O}_X\) is a pseudo-coherent object of \(D(\mathcal{O}_W)\); translated into algebra this means that \(A\) is pseudo-coherent as an \(A \otimes_R A\)-module which follows from More on Algebra, Lemma 0E1T applied to \(R \to A \otimes_R A \to A\). Thus we recover exactly the conditions in Dualizing Complexes, Definition 0E2C.

Lemma

Let \(X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Let \((K, \xi)\) be a relative dualizing complex. Then \(\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(K, K)\) is an isomorphism.

Proof

Looking affine locally this reduces using Lemma 0E2U to the algebraic case which is Dualizing Complexes, Lemma 0E2G.

Lemma

Let \(X \to S\) be a morphism of schemes which is flat and locally of finite presentation. If \((K, \xi)\) and \((L, \eta)\) are two relative dualizing complexes on \(X/S\), then there is a unique isomorphism \(K \to L\) sending \(\xi\) to \(\eta\).

Proof

Let \(U \subset X\) be an affine open mapping into an affine open of \(S\). Then there is an isomorphism \(K|_U \to L|_U\) by Lemma 0E2U and Dualizing Complexes, Lemma 0E2D. The reader can reuse the argument of that lemma in the schemes case to obtain a proof in this case. We will instead use a glueing argument.

Suppose we have an isomorphism \(\alpha : K \to L\). Then \(\alpha(\xi) = u \eta\) for some invertible section \(u \in H^0(W, \Delta_*\mathcal{O}_X) = H^0(X, \mathcal{O}_X)\). (Because both \(\eta\) and \(\alpha(\xi)\) are generators of an invertible \(\Delta_*\mathcal{O}_X\)-module by assumption.) Hence after replacing \(\alpha\) by \(u^{-1}\alpha\) we see that \(\alpha(\xi) = \eta\). Since the automorphism group of \(K\) is \(H^0(X, \mathcal{O}_X^*)\) by Lemma 0E2V there is at most one such \(\alpha\).

Let \(\mathcal{B}\) be the collection of affine opens of \(X\) which map into an affine open of \(S\). For each \(U \in \mathcal{B}\) we have a unique isomorphism \(\alpha_U : K|_U \to L|_U\) mapping \(\xi\) to \(\eta\) by the discussion in the previous two paragraphs. Observe that \(\text{Ext}^i(K|_U, K|_U) = 0\) for \(i < 0\) and any open \(U\) of \(X\) by Lemma 0E2V. By Cohomology, Lemma 0D66 applied to \(\text{id} : X \to X\) we get a unique morphism \(\alpha : K \to L\) agreeing with \(\alpha_U\) for all \(U \in \mathcal{B}\). Then \(\alpha\) sends \(\xi\) to \(\eta\) as this is true locally.

Lemma

Let \(X \to S\) be a morphism of schemes which is flat and locally of finite presentation. There exists a relative dualizing complex \((K, \xi)\).

Proof

Let \(\mathcal{B}\) be the collection of affine opens of \(X\) which map into an affine open of \(S\). For each \(U\) we have a relative dualizing complex \((K_U, \xi_U)\) for \(U\) over \(S\). Namely, choose an affine open \(V \subset S\) such that \(U \to X \to S\) factors through \(V\). Write \(U = \Spec(A)\) and \(V = \Spec(R)\). By Dualizing Complexes, Lemma 0E2F there exists a relative dualizing complex \(K_A \in D(A)\) for \(R \to A\). Arguing backwards through the proof of Lemma 0E2U this determines an \(V\)-perfect object \(K_U \in D(\mathcal{O}_U)\) and a map \[\xi : \Delta_*\mathcal{O}_U \to L\text{pr}_1^*K_U\] in \(D(\mathcal{O}_{U \times_V U})\). Since being \(V\)-perfect is the same as being \(S\)-perfect and since \(U \times_V U = U \times_S U\) we find that \((K_U, \xi_U)\) is as desired.

If \(U' \subset U \subset X\) with \(U', U \in \mathcal{B}\), then we have a unique isomorphism \(\rho_{U'}^U : K_U|_{U'} \to K_{U'}\) in \(D(\mathcal{O}_{U'})\) sending \(\xi_U|_{U' \times_S U'}\) to \(\xi_{U'}\) by Lemma 0E2W (note that trivially the restriction of a relative dualizing complex to an open is a relative dualizing complex). The uniqueness guarantees that \(\rho^U_{U''} = \rho^V_{U''} \circ \rho ^U_{U'}|_{U''}\) for \(U'' \subset U' \subset U\) in \(\mathcal{B}\). Observe that \(\text{Ext}^i(K_U, K_U) = 0\) for \(i < 0\) for \(U \in \mathcal{B}\) by Lemma 0E2V applied to \(U/S\) and \(K_U\). Thus the BBD glueing lemma (Cohomology, Theorem 0D6C) tells us there is a unique solution, namely, an object \(K \in D(\mathcal{O}_X)\) and isomorphisms \(\rho_U : K|_U \to K_U\) such that we have \(\rho^U_{U'} \circ \rho_U|_{U'} = \rho_{U'}\) for all \(U' \subset U\), \(U, U' \in \mathcal{B}\).

To finish the proof we have to construct the map \[\xi : \Delta_*\mathcal{O}_X \longrightarrow L\text{pr}_1^*K|_W\] in \(D(\mathcal{O}_W)\) inducing an isomorphism from \(\Delta_*\mathcal{O}_X\) to \(R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)\). Since we may change \(W\), we choose \(W = \bigcup_{U \in \mathcal{B}} U \times_S U\). We can use \(\rho_U\) to get isomorphisms \[R\SheafHom_{\mathcal{O}_W}( \Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)|_{U \times_S U} \xrightarrow{\rho_U} R\SheafHom_{\mathcal{O}_{U \times_S U}}( \Delta_*\mathcal{O}_U, L\text{pr}_1^*K_U)\] As \(W\) is covered by the opens \(U \times_S U\) we conclude that the cohomology sheaves of \(R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)\) are zero except in degree \(0\). Moreover, we obtain isomorphisms \[H^0\left(U \times_S U, R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)\right) \xrightarrow{\rho_U} H^0\left((R\SheafHom_{\mathcal{O}_{U \times_S U}}( \Delta_*\mathcal{O}_U, L\text{pr}_1^*K_U)\right)\] Let \(\tau_U\) in the LHS be an element mapping to \(\xi_U\) under this map. The compatibilities between \(\rho^U_{U'}\), \(\xi_U\), \(\xi_{U'}\), \(\rho_U\), and \(\rho_{U'}\) for \(U' \subset U \subset X\) open \(U', U \in \mathcal{B}\) imply that \(\tau_U|_{U' \times_S U'} = \tau_{U'}\). Thus we get a global section \(\tau\) of the \(0\)th cohomology sheaf \(H^0(R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W))\). Since the other cohomology sheaves of \(R\SheafHom_{\mathcal{O}_W}(\Delta_*\mathcal{O}_X, L\text{pr}_1^*K|_W)\) are zero, this global section \(\tau\) determines a morphism \(\xi\) as desired. Since the restriction of \(\xi\) to \(U \times_S U\) gives \(\xi_U\), we see that it satisfies the final condition of Definition 0E2T.

Lemma

Consider a cartesian square \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] of schemes. Assume \(X \to S\) is flat and locally of finite presentation. Let \((K, \xi)\) be a relative dualizing complex for \(f\). Set \(K' = L(g')^*K\). Let \(\xi'\) be the derived base change of \(\xi\) (see proof). Then \((K', \xi')\) is a relative dualizing complex for \(f'\).

Proof

Consider the cartesian square \[\xymatrix{ X' \ar[d]_{\Delta_{X'/S'}} \ar[r] & X \ar[d]^{\Delta_{X/S}} \\ X' \times_{S'} X' \ar[r]^{g' \times g'} & X \times_S X }\] Choose \(W \subset X \times_S X\) open such that \(\Delta_{X/S}\) factors through a closed immersion \(\Delta : X \to W\). Choose \(W' \subset X' \times_{S'} X'\) open such that \(\Delta_{X'/S'}\) factors through a closed immersion \(\Delta' : X \to W'\) and such that \((g' \times g')(W') \subset W\). Let us still denote \(g' \times g' : W' \to W\) the induced morphism. We have \[L(g' \times g')^*\Delta_*\mathcal{O}_X = \Delta'_*\mathcal{O}_{X'} \quad\text{and}\quad L(g' \times g')^*L\text{pr}_1^*K|_W = L\text{pr}_1^*K'|_{W'}\] The first equality holds because \(X\) and \(X' \times_{S'} X'\) are tor independent over \(X \times_S X\) (see for example More on Morphisms, Lemma 0CTA). The second holds by transitivity of derived pullback (Cohomology, Lemma 0D5S). Thus \(\xi' = L(g' \times g')^*\xi\) can be viewed as a map \[\xi' : \Delta'_*\mathcal{O}_{X'} \longrightarrow L\text{pr}_1^*K'|_{W'}\] Having said this the proof of the lemma is straightforward. First, \(K'\) is \(S'\)-perfect by Derived Categories of Schemes, Lemma 0DI5. To check that \(\xi'\) induces an isomorphism of \(\Delta'_*\mathcal{O}_{X'}\) to \(R\SheafHom_{\mathcal{O}_{W'}}( \Delta'_*\mathcal{O}_{X'}, L\text{pr}_1^*K'|_{W'})\) we may work affine locally. By Lemma 0E2U we reduce to the corresponding statement in algebra which is proven in Dualizing Complexes, Lemma 0E2F.

Lemma

Let \(S\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to S\) be a proper, flat morphism of finite presentation. The relative dualizing complex \(\omega_{X/S}^\bullet\) of Remark 0B6S together with (0E2Q) is a relative dualizing complex in the sense of Definition 0E2T.

Proof

In Lemma 0E4L we proved that \(\omega_{X/S}^\bullet\) is \(S\)-perfect. Let \(c\) be the right adjoint of Lemma 0A9E for the diagonal \(\Delta : X \to X \times_S X\). Then we can apply \(\Delta_*\) to (0E2Q) to get an isomorphism \[\Delta_*\mathcal{O}_X \to \Delta_*(c(L\text{pr}_1^*\omega_{X/S}^\bullet)) = R\SheafHom_{\mathcal{O}_{X \times_S X}}( \Delta_*\mathcal{O}_X, L\text{pr}_1^*\omega_{X/S}^\bullet)\] The equality holds by Lemmas 0A9X and 0A77. This finishes the proof.

Remark

Let \(X \to S\) be a morphism of schemes which is flat, proper, and of finite presentation. By Lemma 0E2X there exists a relative dualizing complex \((\omega_{X/S}^\bullet, \xi)\) in the sense of Definition 0E2T. Consider any morphism \(g : S' \to S\) where \(S'\) is quasi-compact and quasi-separated (for example an affine open of \(S\)). By Lemma 0E2Y we see that \((L(g')^*\omega_{X/S}^\bullet, L(g')^*\xi)\) is a relative dualizing complex for the base change \(f' : X' \to S'\) in the sense of Definition 0E2T. Let \(\omega_{X'/S'}^\bullet\) be the relative dualizing complex for \(X' \to S'\) in the sense of Remark 0B6S. Combining Lemmas 0E2Z and 0E2W we see that there is a unique isomorphism \[\omega_{X'/S'}^\bullet \longrightarrow L(g')^*\omega_{X/S}^\bullet\] compatible with (0E2Q) and \(L(g')^*\xi\). These isomorphisms are compatible with morphisms between quasi-compact and quasi-separated schemes over \(S\) and the base change isomorphisms of Lemma 0AAB (if we ever need this compatibility we will carefully state and prove it here).

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). If \(f\) is flat, then \(f^!\mathcal{O}_Y\) is (the first component of) a relative dualizing complex for \(X\) over \(Y\) in the sense of Definition 0E2T.

Proof

By Lemma 0E9T we have that \(f^!\mathcal{O}_Y\) is \(Y\)-perfect. As \(f\) is separated the diagonal \(\Delta : X \to X \times_Y X\) is a closed immersion and \(\Delta_*\Delta^!(-) = R\SheafHom_{\mathcal{O}_{X \times_Y X}}(\mathcal{O}_X, -)\), see Lemmas 0A9X and 0A77. Hence to finish the proof it suffices to show \(\Delta^!(L\text{pr}_1^*f^!(\mathcal{O}_Y)) \cong \mathcal{O}_X\) where \(\text{pr}_1 : X \times_Y X \to X\) is the first projection. We have \[\mathcal{O}_X = \Delta^! \text{pr}_1^!\mathcal{O}_X = \Delta^! \text{pr}_1^! L\text{pr}_2^*\mathcal{O}_Y = \Delta^!(L\text{pr}_1^* f^!\mathcal{O}_Y)\] where \(\text{pr}_2 : X \times_Y X \to X\) is the second projection and where we have used the base change isomorphism \(\text{pr}_1^! \circ L\text{pr}_2^* = L\text{pr}_1^* \circ f^!\) of Lemma 0E9U.

Lemma

Let \(f : Y \to X\) and \(X \to S\) be morphisms of schemes which are flat and of finite presentation. Let \((K, \xi)\) and \((M, \eta)\) be a relative dualizing complex for \(X \to S\) and \(Y \to X\). Set \(E = M \otimes_{\mathcal{O}_Y}^\mathbf{L} Lf^*K\). Then \((E, \zeta)\) is a relative dualizing complex for \(Y \to S\) for a suitable \(\zeta\).

Proof

Using Lemma 0E2U and the algebraic version of this lemma (Dualizing Complexes, Lemma 0E2H) we see that \(E\) is affine locally the first component of a relative dualizing complex. In particular we see that \(E\) is \(S\)-perfect since this may be checked affine locally, see Derived Categories of Schemes, Lemma 0DI2.

Let us first prove the existence of \(\zeta\) in case the morphisms \(X \to S\) and \(Y \to X\) are separated so that \(\Delta_{X/S}\), \(\Delta_{Y/X}\), and \(\Delta_{Y/S}\) are closed immersions. Consider the following diagram \[\xymatrix{ & & Y \ar@{=}[r] & Y \ar[d]^f \\ Y \ar[r]_{\Delta_{Y/X}} & Y \times_X Y \ar[d]_m \ar[r]_\delta \ar[ru]_q & Y \times_S Y \ar[d]^{f \times f} \ar[ru]_p & X\\ & X \ar[r]^{\Delta_{X/S}} & X \times_S X \ar[ru]_r }\] where \(p\), \(q\), \(r\) are the first projections. By Lemma 0E2I we have \[R\SheafHom_{\mathcal{O}_{Y \times_S Y}}( \Delta_{Y/S, *}\mathcal{O}_Y, Lp^*E) = R\delta_*\left(R\SheafHom_{\mathcal{O}_{Y \times_X Y}}( \Delta_{Y/X, *}\mathcal{O}_Y, R\SheafHom(\mathcal{O}_{Y \times_X Y}, Lp^*E))\right)\] By Lemma 0E2N we have \[R\SheafHom(\mathcal{O}_{Y \times_X Y}, Lp^*E) = R\SheafHom(\mathcal{O}_{Y \times_X Y}, L(f \times f)^*Lr^*K) \otimes_{\mathcal{O}_{Y \times_S Y}}^\mathbf{L} Lq^*M\] By Lemma 0E2M we have \[R\SheafHom(\mathcal{O}_{Y \times_X Y}, L(f \times f)^*Lr^*K) = Lm^*R\SheafHom(\mathcal{O}_X, Lr^*K)\] The last expression is isomorphic (via \(\xi\)) to \(Lm^*\mathcal{O}_X = \mathcal{O}_{Y \times_X Y}\). Hence the expression preceding is isomorphic to \(Lq^*M\). Hence \[R\SheafHom_{\mathcal{O}_{Y \times_S Y}}( \Delta_{Y/S, *}\mathcal{O}_Y, Lp^*E) = R\delta_*\left(R\SheafHom_{\mathcal{O}_{Y \times_X Y}}( \Delta_{Y/X, *}\mathcal{O}_Y, Lq^*M)\right)\] The material inside the parentheses is isomorphic to \(\Delta_{Y/X, *}*\mathcal{O}_X\) via \(\eta\). This finishes the proof in the separated case.

In the general case we choose an open \(W \subset X \times_S X\) such that \(\Delta_{X/S}\) factors through a closed immersion \(\Delta : X \to W\) and we choose an open \(V \subset Y \times_X Y\) such that \(\Delta_{Y/X}\) factors through a closed immersion \(\Delta' : Y \to V\). Finally, choose an open \(W' \subset Y \times_S Y\) whose intersection with \(Y \times_X Y\) gives \(V\) and which maps into \(W\). Then we consider the diagram \[\xymatrix{ & & Y \ar@{=}[r] & Y \ar[d]^f \\ Y \ar[r]_{\Delta'} & V \ar[d]_m \ar[r]_\delta \ar[ru]_q & W' \ar[d]^{f \times f} \ar[ru]_p & X\\ & X \ar[r]^\Delta & W \ar[ru]_r }\] and we use exactly the same argument as before.

The fundamental class of an lci morphism

In this section we will use the computations made in Section 0BQV. Thus our result will suffer from the same kind of non-uniqueness as we have in that section.

Lemma

Let \(X\) be a locally ringed space. Let \[\mathcal{E}_1 \xrightarrow{\alpha} \mathcal{E}_0 \to \mathcal{F} \to 0\] be a short exact sequence of \(\mathcal{O}_X\)-modules. Assume \(\mathcal{E}_1\) and \(\mathcal{E}_0\) are locally free of ranks \(r_1, r_0\). Then there is a canonical map \[\wedge^{r_0 - r_1}\mathcal{F} \longrightarrow \wedge^{r_1}(\mathcal{E}_1^\vee) \otimes \wedge^{r_0}\mathcal{E}_0\] which is an isomorphism on the stalk at \(x \in X\) if and only if \(\mathcal{F}\) is locally free of rank \(r_0 - r_1\) in an open neighbourhood of \(x\).

Proof

If \(r_1 > r_0\) then \(\wedge^{r_0 - r_1}\mathcal{F} = 0\) by convention and the unique map cannot be an isomorphism. Thus we may assume \(r = r_0 - r_1 \geq 0\). Define the map by the formula \[s_1 \wedge \ldots \wedge s_r \mapsto t_1^\vee \wedge \ldots \wedge t_{r_1}^\vee \otimes \alpha(t_1) \wedge \ldots \wedge \alpha(t_{r_1}) \wedge \tilde s_1 \wedge \ldots \wedge \tilde s_r\] where \(t_1, \ldots, t_{r_1}\) is a local basis for \(\mathcal{E}_1\), correspondingly \(t_1^\vee, \ldots, t_{r_1}^\vee\) is the dual basis for \(\mathcal{E}_1^\vee\), and \(s'_i\) is a local lift of \(s_i\) to a section of \(\mathcal{E}_0\). We omit the proof that this is well defined.

If \(\mathcal{F}\) is locally free of rank \(r\), then it is straightforward to verify that the map is an isomorphism. Conversely, assume the map is an isomorphism on stalks at \(x\). Then \(\wedge^r\mathcal{F}_x\) is invertible. This implies that \(\mathcal{F}_x\) is generated by at most \(r\) elements. This can only happen if \(\alpha\) has rank \(r\) modulo \(\mathfrak m_x\), i.e., \(\alpha\) has maximal rank modulo \(\mathfrak m_x\). This implies that \(\alpha\) has maximal rank in a neighbourhood of \(x\) and hence \(\mathcal{F}\) is locally free of rank \(r\) in a neighbourhood as desired.

Lemma

Let \(Y\) be a Noetherian scheme. Let \(f : X \to Y\) be a local complete intersection morphism which factors as an immersion \(X \to P\) followed by a proper smooth morphism \(P \to Y\). Let \(r\) be the locally constant function on \(X\) such that \(\omega_{X/Y} = H^{-r}(f^!\mathcal{O}_Y)\) is the unique nonzero cohomology sheaf of \(f^!\mathcal{O}_Y\), see Lemma 0B6V. Then there is a map \[\wedge^r\Omega_{X/Y} \longrightarrow \omega_{X/Y}\] which is an isomorphism on the stalk at a point \(x\) if and only if \(f\) is smooth at \(x\).

Proof

The assumption implies that \(X\) is compactifiable over \(Y\) hence \(f^!\) is defined, see Section 0A9Y. Let \(j : W \to P\) be an open subscheme such that \(X \to P\) factors through a closed immersion \(i : X \to W\). Moreover, we have \(f^! = i^! \circ j^! \circ g^!\) where \(g : P \to Y\) is the given morphism. We have \(g^!\mathcal{O}_Y = \wedge^d\Omega_{P/Y}[d]\) by Lemma 0BRT where \(d\) is the locally constant function giving the relative dimension of \(P\) over \(Y\). We have \(j^! = j^*\). We have \(i^!\mathcal{O}_W = \wedge^c\mathcal{N}[-c]\) where \(c\) is the codimension of \(X\) in \(W\) (a locally constant function on \(X\)) and where \(\mathcal{N}\) is the normal sheaf of the Koszul-regular immersion \(i\), see Lemma 0BR0. Combining the above we find \[f^!\mathcal{O}_Y = \left(\wedge^c\mathcal{N} \otimes_{\mathcal{O}_X} \wedge^d\Omega_{P/Y}|_X\right)[d - c]\] where we have also used Lemma 0B6U. Thus \(r = d|_X - c\) as locally constant functions on \(X\). The conormal sheaf of \(X \to P\) is the module \(\mathcal{I}/\mathcal{I}^2\) where \(\mathcal{I} \subset \mathcal{O}_W\) is the ideal sheaf of \(i\), see Morphisms, Section 01R1. Consider the canonical exact sequence \[\mathcal{I}/\mathcal{I}^2 \to \Omega_{P/Y}|_X \to \Omega_{X/Y} \to 0\] of Morphisms, Lemma 01UZ. We obtain our map by an application of Lemma 0E9Y.

If \(f\) is smooth at \(x\), then the map is an isomorphism by an application of Lemma 0E9Y and the fact that \(\Omega_{X/Y}\) is locally free at \(x\) of rank \(r\). Conversely, assume that our map is an isomorphism on stalks at \(x\). Then the lemma shows that \(\Omega_{X/Y}\) is free of rank \(r\) after replacing \(X\) by an open neighbourhood of \(x\). On the other hand, we may also assume that \(X = \Spec(A)\) and \(Y = \Spec(R)\) where \(A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)\) and where \(f_1, \ldots, f_m\) is a Koszul regular sequence (this follows from the definition of local complete intersection morphisms). Clearly this implies \(r = n - m\). We conclude that the rank of the matrix of partials \(\partial f_j/\partial x_i\) in the residue field at \(x\) is \(m\). Thus after reordering the variables we may assume the determinant of \((\partial f_j/\partial x_i)_{1 \leq i, j \leq m}\) is invertible in an open neighbourhood of \(x\). It follows that \(R \to A\) is smooth at this point, see for example Algebra, Example 00T8.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(r \geq 0\). Assume

1. \(Y\) is Cohen-Macaulay (Properties, Definition 02IO),

2. \(f\) factors as \(X \to P \to Y\) where the first morphism is an immersion and the second is smooth and proper,

3. if \(x \in X\) and \(\dim(\mathcal{O}_{X, x}) \leq 1\), then \(f\) is Koszul at \(x\) (More on Morphisms, Definition 069F), and

4. if \(\xi\) is a generic point of an irreducible component of \(X\), then we have \(\text{trdeg}_{\kappa(f(\xi))} \kappa(\xi) = r\).

Then with \(\omega_{X/Y} = H^{-r}(f^!\mathcal{O}_Y)\) there is a map \[\wedge^r\Omega_{X/Y} \longrightarrow \omega_{X/Y}\] which is an isomorphism on the locus where \(f\) is smooth.

Proof

Let \(U \subset X\) be the open subscheme over which \(f\) is a local complete intersection morphism. Since \(f\) has relative dimension \(r\) at all generic points by assumption (4) we see that the locally constant function of Lemma 0E9Z is constant with value \(r\) and we obtain a map \[\wedge^r\Omega_{X/Y}|_U = \wedge^r \Omega_{U/Y} \longrightarrow \omega_{U/Y} = \omega_{X/Y}|_U\] which is an isomorphism in the smooth points of \(f\) (this locus is contained in \(U\) because a smooth morphism is a local complete intersection morphism). By Lemma 0E9V and the assumption that \(Y\) is Cohen-Macaulay the module \(\omega_{X/Y}\) is \((S_2)\). Since \(U\) contains all the points of codimension \(1\) by condition (3) and using Divisors, Lemma 0E9I we see that \(j_*\omega_{U/Y} = \omega_{X/Y}\). Hence the map over \(U\) extends to \(X\) and the proof is complete.

Extension by zero for coherent modules

The material in this section and the next few can be found in the appendix by Deligne of [RD].

In this section \(j : U \to X\) will be an open immersion of Noetherian schemes. We are going to consider inverse systems \((K_n)\) in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) constructed as follows. Let \(\mathcal{F}^\bullet\) be a bounded complex of coherent \(\mathcal{O}_X\)-modules. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals with \(V(\mathcal{I}) = X \setminus U\). Then we can set \[K_n = \mathcal{I}^n\mathcal{F}^\bullet\] More precisely, \(K_n\) is the object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) represented by the complex whose term in degree \(q\) is the coherent submodule \(\mathcal{I}^n\mathcal{F}^q\) of \(\mathcal{F}^q\). Observe that the maps \(\ldots \to K_3 \to K_2 \to K_1\) induce isomorphisms on restriction to \(U\). Let us call such a system a Deligne system.

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Let \((K_n)\) be a Deligne system and denote \(K \in D^b_{\textit{Coh}}(\mathcal{O}_U)\) the value of the constant system \((K_n|_U)\). Let \(L\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Then \(\colim \Hom_X(K_n, L) = \Hom_U(K, L|_U)\).

Proof

Let \(L \to M \to N \to L[1]\) be a distinguished triangle in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Then we obtain a commutative diagram \[\xymatrix{ \ldots \ar[r] & \colim \Hom_X(K_n, L) \ar[r] \ar[d] & \colim \Hom_X(K_n, M) \ar[r] \ar[d] & \colim \Hom_X(K_n, N) \ar[r] \ar[d] & \ldots \\ \ldots \ar[r] & \Hom_U(K, L|_U) \ar[r] & \Hom_U(K, M|_U) \ar[r] & \Hom_U(K, N|_U) \ar[r] & \ldots }\] whose rows are exact by Derived Categories, Lemma 0149 and Algebra, Lemma 00DB. Hence if the statement of the lemma holds for \(N[-1]\), \(L\), \(N\), and \(L[1]\) then it holds for \(M\) by the 5-lemma. Thus, using the distinguished triangles for the canonical truncations of \(L\) (see Derived Categories, Remark 08J5) we reduce to the case that \(L\) has only one nonzero cohomology sheaf.

Choose a bounded complex \(\mathcal{F}^\bullet\) of coherent \(\mathcal{O}_X\)-modules and a quasi-coherent ideal \(\mathcal{I} \subset \mathcal{O}_X\) cutting out \(X \setminus U\) such that \(K_n\) is represented by \(\mathcal{I}^n\mathcal{F}^\bullet\). Using “stupid” truncations we obtain compatible termwise split short exact sequences of complexes \[0 \to \sigma_{\geq a + 1} \mathcal{I}^n\mathcal{F}^\bullet \to \mathcal{I}^n\mathcal{F}^\bullet \to \sigma_{\leq a} \mathcal{I}^n\mathcal{F}^\bullet \to 0\] which in turn correspond to compatible systems of distinguished triangles in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Arguing as above we reduce to the case where \(\mathcal{F}^\bullet\) has only one nonzero term. This reduces us to the case discussed in the next paragraph.

Given a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) and a coherent \(\mathcal{O}_X\)-module \(\mathcal{G}\) we have to show that the canonical map \[\colim \Ext^i_X(\mathcal{I}^n\mathcal{F}, \mathcal{G}) \longrightarrow \Ext^i_U(\mathcal{F}|_U, \mathcal{G}|_U)\] is an isomorphism for all \(i \geq 0\). For \(i = 0\) this is Cohomology of Schemes, Lemma 01YB. Assume \(i > 0\).

Injectivity. Let \(\xi \in \Ext^i_X(\mathcal{I}^n\mathcal{F}, \mathcal{G})\) be an element whose restriction to \(U\) is zero. We have to show there exists an \(m \geq n\) such that the restriction of \(\xi\) to \(\mathcal{I}^m\mathcal{F} = \mathcal{I}^{m - n}\mathcal{I}^n\mathcal{F}\) is zero. After replacing \(\mathcal{F}\) by \(\mathcal{I}^n\mathcal{F}\) we may assume \(n = 0\), i.e., we have \(\xi \in \Ext^i_X(\mathcal{F}, \mathcal{G})\) whose restriction to \(U\) is zero. By Derived Categories of Schemes, Proposition 0FDB we have \(D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X))\). Hence we can compute the \(\Ext\) group in the abelian category of coherent \(\mathcal{O}_X\)-modules. This implies there exists an surjection \(\alpha : \mathcal{F}'' \to \mathcal{F}\) such that \(\xi \circ \alpha = 0\) (this is where we use that \(i > 0\)). Set \(\mathcal{F}' = \Ker(\alpha)\) so that we have a short exact sequence \[0 \to \mathcal{F}' \to \mathcal{F}'' \to \mathcal{F} \to 0\] It follows that \(\xi\) is the image of an element \(\xi' \in \Ext^{i - 1}_X(\mathcal{F}', \mathcal{G})\) whose restriction to \(U\) is in the image of \(\Ext^{i - 1}_U(\mathcal{F}''|_U, \mathcal{G}|_U) \to \Ext^{i - 1}_U(\mathcal{F}'|_U, \mathcal{G}|_U)\). By Artin-Rees the inverse systems \((\mathcal{I}^n\mathcal{F}')\) and \((\mathcal{I}^n \mathcal{F}'' \cap \mathcal{F}')\) are pro-isomorphic, see Cohomology of Schemes, Lemma 01YA. Since we have the compatible system of short exact sequences \[0 \to \mathcal{F}' \cap \mathcal{I}^n\mathcal{F}'' \to \mathcal{I}^n\mathcal{F}'' \to \mathcal{I}^n\mathcal{F} \to 0\] we obtain a commutative diagram \[\xymatrix{ \colim \Ext^{i - 1}_X(\mathcal{I}^n\mathcal{F}'', \mathcal{G}) \ar[r] \ar[d] & \colim \Ext^{i - 1}_X(\mathcal{F}' \cap \mathcal{I}^n\mathcal{F}'', \mathcal{G}) \ar[r] \ar[d] & \colim \Ext^i_X(\mathcal{I}^n\mathcal{F}, \mathcal{G}) \ar[d] \\ \Ext^{i - 1}_U(\mathcal{F}''|_U, \mathcal{G}|_U) \ar[r] & \Ext^{i - 1}_U(\mathcal{F}'|_U, \mathcal{G}|_U) \ar[r] & \Ext^{i - 1}_U(\mathcal{F}|_U, \mathcal{G}|_U) }\] with exact rows. By induction on \(i\) and the comment on inverse systems above we find that the left two vertical arrows are isomorphisms. Now \(\xi\) gives an element in the top right group which is the image of \(\xi'\) in the middle top group, which in turn maps to an element of the bottom middle group coming from some element in the left bottom group. We conclude that \(\xi\) maps to zero in \(\Ext^i_X(\mathcal{I}^n\mathcal{F}, \mathcal{G})\) for some \(n\) as desired.

Surjectivity. Let \(\xi \in \Ext^i_U(\mathcal{F}|_U, \mathcal{G}|_U)\). Arguing as above using that \(i > 0\) we can find an surjection \(\mathcal{H} \to \mathcal{F}|_U\) of coherent \(\mathcal{O}_U\)-modules such that \(\xi\) maps to zero in \(\Ext^i_U(\mathcal{H}, \mathcal{G}|_U)\). Then we can find a map \(\varphi : \mathcal{F}'' \to \mathcal{F}\) of coherent \(\mathcal{O}_X\)-modules whose restriction to \(U\) is \(\mathcal{H} \to \mathcal{F}|_U\), see Properties, Lemma 01PI. Observe that the lemma doesn’t guarantee \(\varphi\) is surjective but this won’t matter (it is possible to pick a surjective \(\varphi\) with a little bit of additional work). Denote \(\mathcal{F}' = \Ker(\varphi)\). The short exact sequence \[0 \to \mathcal{F}'|_U \to \mathcal{F}''|_U \to \mathcal{F}|_U \to 0\] shows that \(\xi\) is the image of \(\xi'\) in \(\Ext^{i - 1}_U(\mathcal{F}'|_U, \mathcal{G}|_U)\). By induction on \(i\) we can find an \(n\) such that \(\xi'\) is the image of some \(\xi'_n\) in \(\Ext^{i - 1}_X(\mathcal{I}^n\mathcal{F}', \mathcal{G})\). By Artin-Rees we can find an \(m \geq n\) such that \(\mathcal{F}' \cap \mathcal{I}^m\mathcal{F}'' \subset \mathcal{I}^n\mathcal{F}'\). Using the short exact sequence \[0 \to \mathcal{F}' \cap \mathcal{I}^m\mathcal{F}'' \to \mathcal{I}^m\mathcal{F}'' \to \mathcal{I}^m\Im(\varphi) \to 0\] the image of \(\xi'_n\) in \(\Ext^{i - 1}_X(\mathcal{F}' \cap \mathcal{I}^m\mathcal{F}'', \mathcal{G})\) maps by the boundary map to an element \(\xi_m\) of \(\Ext^i_X(\mathcal{I}^m\Im(\varphi), \mathcal{G})\) which maps to \(\xi\). Since \(\Im(\varphi)\) and \(\mathcal{F}\) agree over \(U\) we see that \(\mathcal{F}/\mathcal{I}^m\Im(\varphi)\) is supported on \(X \setminus U\). Hence there exists an \(l \geq m\) such that \(\mathcal{I}^l\mathcal{F} \subset \mathcal{I}^m\Im(\varphi)\), see Cohomology of Schemes, Lemma 01Y9. Taking the image of \(\xi_m\) in \(\Ext^i_X(\mathcal{I}^l\mathcal{F}, \mathcal{G})\) we win.

Lemma

The result of Lemma 0G2H holds even for \(L \in D^+_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

Namely, if \((K_n)\) is a Deligne system then there exists a \(b \in \mathbf{Z}\) such that \(H^i(K_n) = 0\) for \(i > b\). Then \(\Hom(K_n, L) = \Hom(K_n, \tau_{\leq b}L)\) and \(\Hom(K, L) = \Hom(K, \tau_{\leq b}L)\). Hence using the result of the lemma for \(\tau_{\leq b}L\) we win.

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes.

1. Let \((K_n)\) and \((L_n)\) be Deligne systems. Let \(K\) and \(L\) be the values of the constant systems \((K_n|_U)\) and \((L_n|_U)\). Given a morphism \(\alpha : K \to L\) of \(D(\mathcal{O}_U)\) there is a unique morphism of pro-systems \((K_n) \to (L_n)\) of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) whose restriction to \(U\) is \(\alpha\).

2. Given \(K \in D^b_{\textit{Coh}}(\mathcal{O}_U)\) there exists a Deligne system \((K_n)\) such that \((K_n|_U)\) is constant with value \(K\).

3. The pro-object \((K_n)\) of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) of (2) is unique up to unique isomorphism (as a pro-object).

Proof

Part (1) is an immediate consequence of Lemma 0G2H and the fact that morphisms between pro-systems are the same as morphisms between the functors they corepresent, see Categories, Remark 0G2X.

Let \(K\) be as in (2). We can choose \(K' \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) whose restriction to \(U\) is isomorphic to \(K\), see Derived Categories of Schemes, Lemma 0G48. By Derived Categories of Schemes, Proposition 0FDB we can represent \(K'\) by a bounded complex \(\mathcal{F}^\bullet\) of coherent \(\mathcal{O}_X\)-modules. Choose a quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) whose vanishing locus is \(X \setminus U\) (for example choose \(\mathcal{I}\) to correspond to the reduced induced subscheme structure on \(X \setminus U\)). Then we can set \(K_n\) equal to the object represented by the complex \(\mathcal{I}^n\mathcal{F}^\bullet\) as in the introduction to this section.

Part (3) is immediate from parts (1) and (2).

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Let \[K \to L \to M \to K[1]\] be a distinguished triangle of \(D^b_{\textit{Coh}}(\mathcal{O}_U)\). Then there exists an inverse system of distinguished triangles \[K_n \to L_n \to M_n \to K_n[1]\] in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) such that \((K_n)\), \((L_n)\), \((M_n)\) are Deligne systems and such that the restriction of these distinguished triangles to \(U\) is isomorphic to the distinguished triangle we started out with.

Proof

Let \((K_n)\) be as in Lemma 0G4L part (2). Choose an object \(L'\) of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) whose restriction to \(U\) is \(L\) (we can do this as the lemma shows). By Lemma 0G2H we can find an \(n\) and a morphism \(K_n \to L'\) on \(X\) whose restriction to \(U\) is the given arrow \(K \to L\). We conclude there is a morphism \(K' \to L'\) of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) whose restriction to \(U\) is the given arrow \(K \to L\).

By Derived Categories of Schemes, Proposition 0FDB we can find a morphism \(\alpha^\bullet : \mathcal{F}^\bullet \to \mathcal{G}^\bullet\) of bounded complexes of coherent \(\mathcal{O}_X\)-modules representing \(K' \to L'\). Choose a quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) whose vanishing locus is \(X \setminus U\). Then we let \(K_n = \mathcal{I}^n\mathcal{F}^\bullet\) and \(L_n = \mathcal{I}^n\mathcal{G}^\bullet\). Observe that \(\alpha^\bullet\) induces a morphism of complexes \(\alpha_n^\bullet : \mathcal{I}^n\mathcal{F}^\bullet \to \mathcal{I}^n\mathcal{G}^\bullet\). From the construction of cones in Derived Categories, Section 014D it is clear that \[C(\alpha_n)^\bullet = \mathcal{I}^nC(\alpha^\bullet)\] and hence we can set \(M_n = C(\alpha_n)^\bullet\). Namely, we have a compatible system of distinguished triangles (see discussion in Derived Categories, Section 014Z) \[K_n \to L_n \to M_n \to K_n[1]\] whose restriction to \(U\) is isomorphic to the distinguished triangle we started out with by axiom TR3 and Derived Categories, Lemma 014A.

Remark

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Sending \(K \in D^b_{\textit{Coh}}(\mathcal{O}_U)\) to a Deligne system whose restriction to \(U\) is \(K\) determines a functor \[Rj_! : D^b_{\textit{Coh}}(\mathcal{O}_U) \longrightarrow \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)\] which is “exact” by Lemma 0G4M and which is “left adjoint” to the functor \(j^* : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_U)\) by Lemma 0G2H.

Remark

Let \((A_n)\) and \((B_n)\) be inverse systems of a category \(\mathcal{C}\). Let us say a linear-pro-morphism from \((A_n)\) to \((B_n)\) is given by a compatible family of morphisms \(\varphi_n : A_{cn + d} \to B_n\) for all \(n \geq 1\) for some fixed integers \(c, d \geq 1\). We’ll say \((\varphi_n : A_{cn + d} \to B_n)\) and \((\psi_n : A_{c'n + d'} \to B_n)\) determine the same morphism if there exist \(c'' \geq \max(c, c')\) and \(d'' \geq \max(d, d')\) such that the two induced morphisms \(A_{c'' n + d''} \to B_n\) are the same for all \(n\). It seems likely that Deligne systems \((K_n)\) with given value on \(U\) are well defined up to linear-pro-isomorphisms. If we ever need this we will carefully formulate and prove this here.

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Let \[K_n \to L_n \to M_n \to K_n[1]\] be an inverse system of distinguished triangles in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). If \((K_n)\) and \((M_n)\) are pro-isomorphic to Deligne systems, then so is \((L_n)\).

Proof

Observe that the systems \((K_n|_U)\) and \((M_n|_U)\) are essentially constant as they are pro-isomorphic to constant systems. Denote \(K\) and \(M\) their values. By Derived Categories, Lemma 0G3A we see that the inverse system \(L_n|_U\) is essentially constant as well. Denote \(L\) its value. Let \(N \in D^b_{\textit{Coh}}(\mathcal{O}_X)\). Consider the commutative diagram \[\xymatrix{ \ldots \ar[r] & \colim \Hom_X(M_n, N) \ar[r] \ar[d] & \colim \Hom_X(L_n, N) \ar[r] \ar[d] & \colim \Hom_X(K_n, N) \ar[r] \ar[d] & \ldots \\ \ldots \ar[r] & \Hom_U(M, N|_U) \ar[r] & \Hom_U(L, N|_U) \ar[r] & \Hom_U(K, N|_U) \ar[r] & \ldots }\] By Lemma 0G2H and the fact that isomorphic ind-systems have the same colimit, we see that the vertical arrows two to the right and two to the left of the middle one are isomorphisms. By the 5-lemma we conclude that the middle vertical arrow is an isomorphism. Now, if \((L'_n)\) is a Deligne system whose restriction to \(U\) has constant value \(L\) (which exists by Lemma 0G4L), then we have \(\colim \Hom_X(L'_n, N) = \Hom_U(L, N|_U)\) as well. Hence the pro-systems \((L_n)\) and \((L'_n)\) are pro-isomorphic by Categories, Remark 0G2X.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(\mathcal{F}^\bullet\) be a complex of coherent \(\mathcal{O}_X\)-modules. Let \(p \in \mathbf{Z}\). Set \(\mathcal{H} = H^p(\mathcal{F}^\bullet)\) and \(\mathcal{H}_n = H^p(\mathcal{I}^n\mathcal{F}^\bullet)\). Then there are canonical \(\mathcal{O}_X\)-module maps \[\ldots \to \mathcal{H}_3 \to \mathcal{H}_2 \to \mathcal{H}_1 \to \mathcal{H}\] There exists a \(c > 0\) such that for \(n \geq c\) the image of \(\mathcal{H}_n \to \mathcal{H}\) is contained in \(\mathcal{I}^{n - c}\mathcal{H}\) and there is a canonical \(\mathcal{O}_X\)-module map \(\mathcal{I}^n\mathcal{H} \to \mathcal{H}_{n - c}\) such that the compositions \[\mathcal{I}^n \mathcal{H} \to \mathcal{H}_{n - c} \to \mathcal{I}^{n - 2c}\mathcal{H} \quad\text{and}\quad \mathcal{H}_n \to \mathcal{I}^{n - c}\mathcal{H} \to \mathcal{H}_{n - 2c}\] are the canonical ones. In particular, the inverse systems \((\mathcal{H}_n)\) and \((\mathcal{I}^n\mathcal{H})\) are isomorphic as pro-objects of \(\textit{Mod}(\mathcal{O}_X)\).

Proof

If \(X\) is affine, translated into algebra this is More on Algebra, Lemma 0G3K. In the general case, argue exactly as in the proof of that lemma replacing the reference to Artin-Rees in algebra with a reference to Cohomology of Schemes, Lemma 01YA. Details omitted.

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Let \(a \leq b\) be integers. Let \((K_n)\) be an inverse system of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) such that \(H^i(K_n) = 0\) for \(i \not \in [a, b]\). The following are equivalent

1. \((K_n)\) is pro-isomorphic to a Deligne system,

2. for every \(p \in \mathbf{Z}\) there exists a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) such that the pro-systems \((H^p(K_n))\) and \((\mathcal{I}^n\mathcal{F})\) are pro-isomorphic.

Proof

Assume (1). To prove (2) holds we may assume \((K_n)\) is a Deligne system. By definition we may choose a bounded complex \(\mathcal{F}^\bullet\) of coherent \(\mathcal{O}_X\)-modules and a quasi-coherent sheaf of ideals cutting out \(X \setminus U\) such that \(K_n\) is represented by \(\mathcal{I}^n\mathcal{F}^\bullet\). Thus the result follows from Lemma 0G4R.

Assume (2). We will prove that \((K_n)\) is as in (1) by induction on \(b - a\). If \(a = b\) then (1) holds essentially by assumption. If \(a < b\) then we consider the compatible system of distinguished triangles \[\tau_{\leq a}K_n \to K_n \to \tau_{\geq a + 1}K_n \to (\tau_{\leq a}K_n)[1]\] See Derived Categories, Remark 08J5. By induction on \(b - a\) we know that \(\tau_{\leq a}K_n\) and \(\tau_{\geq a + 1}K_n\) are pro-isomorphic to Deligne systems. We conclude by Lemma 0G4Q.

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Let \((K_n)\) be an inverse system in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Let \(X = W_1 \cup \ldots \cup W_r\) be an open covering. The following are equivalent

1. \((K_n)\) is pro-isomorphic to a Deligne system,

2. for each \(i\) the restriction \((K_n|_{W_i})\) is pro-isomorphic to a Deligne system with respect to the open immersion \(U \cap W_i \to W_i\).

Proof

By induction on \(r\). If \(r = 1\) then the result is clear. Assume \(r > 1\). Set \(V = W_1 \cup \ldots \cup W_{r - 1}\). By induction we see that \((K_n|_V)\) is a Deligne system. This reduces us to the discussion in the next paragraph.

Assume \(X = V \cup W\) is an open covering and \((K_n|_W)\) and \((K_n|_V)\) are pro-isomorphic to Deligne systems. We have to show that \((K_n)\) is pro-isomorphic to a Deligne system. Observe that \((K_n|_{V \cap W})\) is pro-isomorphic to a Deligne system (it follows immediately from the construction of Deligne systems that restrictions to opens preserves them). In particular the pro-systems \((K_n|_{U \cap V})\), \((K_n|_{U \cap W})\), and \((K_n|_{U \cap V \cap W})\) are essentially constant. It follows from the distinguished triangles in Cohomology, Lemma 08BV and Derived Categories, Lemma 0G3A that \((K_n|_U)\) is essentially constant. Denote \(K \in D^b_{\textit{Coh}}(\mathcal{O}_U)\) the value of this system. Let \(L\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Consider the diagram \[\xymatrix{ \colim \Ext^{-1}(K_n|_V, L|_V) \oplus \colim \Ext^{-1}(K_n|_W, L|_W) \ar[r] \ar[d] & \Ext^{-1}(K|_{U \cap V}, L|_{U \cap V}) \oplus \Ext^{-1}(K|_{U \cap W}, L|_{U \cap W}) \ar[d] \\ \colim \Ext^{-1}(K_n|_{V \cap W}, L|_{V \cap W}) \ar[r] \ar[d] & \Ext^{-1}(K|_{U \cap V \cap W}, L|_{U \cap V \cap W}) \ar[d] \\ \colim \Hom(K_n, L) \ar[d] \ar[r] & \Hom(K|_U, L|_U) \ar[d] \\ \colim \Hom(K_n|_V, L|_V) \oplus \colim \Hom(K_n|_W, L|_W) \ar[r] \ar[d] & \Hom(K|_{U \cap V}, L|_{U \cap V}) \oplus \Hom(K|_{U \cap W}, L|_{U \cap W}) \ar[d] \\ \colim \Hom(K_n|_{V \cap W}, L|_{V \cap W}) \ar[r] & \Hom(K|_{U \cap V \cap W}, L|_{U \cap V \cap W}) }\] The vertical sequences are exact by Cohomology, Lemma 08BW and the fact that filtered colimits are exact. All horizontal arrows except for the middle one are isomorphisms by Lemma 0G2H and the fact that pro-isomorphic systems have the same colimits. Hence the middle one is an isomorphism too by the 5-lemma. It follows that \((K_n)\) is pro-isomorphic to a Deligne system for \(K\). Namey, if \((K'_n)\) is a Deligne system whose restriction to \(U\) has constant value \(K\) (which exists by Lemma 0G4L), then we have \(\colim \Hom_X(K'_n, L) = \Hom_U(K, L|_U)\) as well. Hence the pro-systems \((K_n)\) and \((K'_n)\) are pro-isomorphic by Categories, Remark 0G2X.

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals with \(V(\mathcal{I}) = X \setminus U\). Let \(K\) be in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Then \[K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I}^n\] is pro-isomorphic to a Deligne system with constant value \(K|_U\) over \(U\).

Proof

By Lemma 0G4T the question is local on \(X\). Thus we may assume \(X\) is the spectrum of a Noetherian ring. In this case the statement follows from the algebra version which is More on Algebra, Lemma 0G3N.

Preliminaries to compactly supported cohomology

In Situation 0F42 let \(f : X \to Y\) be a morphism in the category \(\textit{FTS}_S\). Using the constructions in the previous section, we will construct a functor \[Rf_! : D^b_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)\] which reduces to the functor of Remark 0G4N if \(f\) is an open immersion and in general is constructed using a compactification of \(f\). Before we do this, we need the following lemmas to prove our construction is well defined.

Lemma

Let \(f : X \to Y\) be a proper morphism of Noetherian schemes. Let \(V \subset Y\) be an open subscheme and set \(U = f^{-1}(V)\). Picture \[\xymatrix{ U \ar[r]_j \ar[d]_g & X \ar[d]^f \\ V \ar[r]^{j'} & Y }\] Then we have a canonical isomorphism \(Rj'_! \circ Rg_* \to Rf_* \circ Rj_!\) of functors \(D^b_{\textit{Coh}}(\mathcal{O}_U) \to \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)\) where \(Rj_!\) and \(Rj'_!\) are as in Remark 0G4N.

Proof

Let \(K\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_U)\). Let \((K_n)\) be a Deligne system for \(U \to X\) whose restriction to \(U\) is constant with value \(K\). Of course this means that \((K_n)\) represents \(Rj_!K\) in \(\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)\). Observe that both \(Rj'_!Rg_*K\) and \(Rf_*Rj_!K\) restrict to the constant pro-object with value \(Rg_*K\) on \(V\). This is immediate for the first one and for the second one it follows from the fact that \((Rf_*K_n)|_V = Rg_*(K_n|_U) = Rg_*K\). By the uniqueness of Deligne systems in Lemma 0G4L it suffices to show that \((Rf_*K_n)\) is pro-isomorphic to a Deligne system. The lemma referenced will also show that the isomorphism we obtain is functorial.

Proof that \((Rf_*K_n)\) is pro-isomorphic to a Deligne system. First, we observe that the question is independent of the choice of the Deligne system \((K_n)\) corresponding to \(K\) (by the aforementioned uniqueness). By Lemmas 0G4M and 0G4Q if we have a distinguished triangle \[K \to L \to M \to K[1]\] in \(D^b_{\textit{Coh}}(\mathcal{O}_U)\) and the result holds for \(K\) and \(M\), then the result holds for \(L\). Using the distinguished triangles of canonical truncations (Derived Categories, Remark 08J5) we reduce to the problem studied in the next paragraph.

Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{J} \subset \mathcal{O}_Y\) be a quasi-coherent sheaf of ideals cutting out \(Y \setminus V\). Denote \(\mathcal{J}^n\mathcal{F}\) the image of \(f^*\mathcal{J}^n \otimes \mathcal{F} \to \mathcal{F}\). We have to show that \((Rf_*(\mathcal{J}^n\mathcal{F}))\) is a Deligne system. By Lemma 0G4T the question is local on \(Y\). Thus we may assume \(Y = \Spec(A)\) is affine and \(\mathcal{J}\) corresponds to an ideal \(I \subset A\). By Lemma 0G4S it suffices to show that the inverse system of cohomology modules \((H^p(X, I^n\mathcal{F}))\) is pro-isomorphic to the inverse system \((I^n M)\) for some finite \(A\)-module \(M\). This is shown in Cohomology of Schemes, Lemma 02OA.

Proof

Let \(K\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_U)\). Let \(L\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\). We will construct a bijection \[\Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rj'_!Rg_*K, L) \longrightarrow \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_*Rj_!K, L)\] functorial in \(K\) and \(L\). Fixing \(K\) this will determine an isomorphism of pro-objects \(Rf_*Rj_!K \to Rj'_!Rg_*K\) by Categories, Remark 0G2X and varying \(K\) we obtain that this determines an isomorphism of functors. To actually produce the isomorphism we use the sequence of functorial equalities \[\begin{align*} \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rj'_!Rg_*K, L) & = \Hom_V(Rg_*K, L|_V) \\ & = \Hom_U(K, g^!(L|_V)) \\ & = \Hom_U(K, f^!L|_U)) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)}(Rj_!K, f^!L) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_*Rj_!K, L) \end{align*}\] The first equality is true by Lemma 0G2H. The second equality is true because \(g\) is proper (as the base change of \(f\) to \(V\)) and hence \(g^!\) is the right adjoint of pushforward by construction, see Section 0A9Y. The third equality holds as \(g^!(L|_V) = f^!L|_U\) by Lemma 0G4J. Since \(f^!L\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) by Lemma 0AU1 the fourth equality follows from Lemma 0G4K. The fifth equality holds again because \(f^!\) is the right adjoint to \(Rf_*\) as \(f\) is proper.

Lemma

Let \(j : U \to X\) be an open immersion of Noetherian schemes. Let \(j' : U \to X'\) be a compactification of \(U\) over \(X\) (see proof) and denote \(f : X' \to X\) the structure morphism. Then we have a canonical isomorphism \(Rj_! \to Rf_* \circ R(j')_!\) of functors \(D^b_{\textit{Coh}}(\mathcal{O}_U) \to \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)\) where \(Rj_!\) and \(Rj'_!\) are as in Remark 0G4N.

Proof

The fact that \(X'\) is a compactification of \(U\) over \(X\) means precisely that \(f : X' \to X\) is proper, that \(j'\) is an open immersion, and \(j = f \circ j'\). See More on Flatness, Section 0ATT. If \(j'(U) = f^{-1}(j(U))\), then the lemma follows immediately from Lemma 0G4W. If \(j'(U) \not = f^{-1}(j(U))\), then denote \(X'' \subset X'\) the scheme theoretic closure of \(j' : U \to X'\) and denote \(j'' : U \to X''\) the corresponding open immersion. Picture \[\xymatrix{ & & X'' \ar[d]^{f'} \\ & & X' \ar[d]^f \\ U \ar[rr]^j \ar[rru]^{j'} \ar[rruu]^{j''} & & X }\] By More on Flatness, Lemma 0ATU part (c) and the discussion above we have isomorphisms \(Rf'_* \circ Rj''_! = Rj'_!\) and \(R(f \circ f')_* \circ Rj''_! = Rj_!\). Since \(R(f \circ f')_* = Rf_* \circ Rf'_*\) we conclude.

Remark

Let \(X \supset U \supset U'\) be open subschemes of a Noetherian scheme \(X\). Denote \(j : U \to X\) and \(j' : U' \to X\) the inclusion morphisms. We claim there is a canonical map \[Rj'_!(K|_{U'}) \longrightarrow Rj_!K\] functorial for \(K\) in \(D^b_{\textit{Coh}}(\mathcal{O}_U)\). Namely, by Lemma 0G2H we have for any \(L\) in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) the map \[\begin{align*} \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)}(Rj_!K, L) & = \Hom_U(K, L|_U) \\ & \to \Hom_{U'}(K|_{U'}, L|_{U'}) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)}(Rj'_!(K|_{U'}), L) \end{align*}\] functorial in \(L\) and \(K'\). The functoriality in \(L\) shows by Categories, Remark 0G2X that we obtain a canonical map \(Rj'_!(K|_{U'}) \to Rj_!K\) which is functorial in \(K\) by the functoriality of the arrow above in \(K\).

Here is an explicit construction of this arrow. Namely, suppose that \(\mathcal{F}^\bullet\) is a bounded complex of coherent \(\mathcal{O}_X\)-modules whose restriction to \(U\) represents \(K\) in the derived category. We have seen in the proof of Lemma 0G4L that such a complex always exists. Let \(\mathcal{I}\), resp. \(\mathcal{I}'\) be a quasi-coherent sheaf of ideals on \(X\) with \(V(\mathcal{I}) = X \setminus U\), resp. \(V(\mathcal{I}') = X \setminus U'\). After replacing \(\mathcal{I}\) by \(\mathcal{I} + \mathcal{I}'\) we may assume \(\mathcal{I}' \subset \mathcal{I}\). By construction \(Rj_!K\), resp. \(Rj'_!(K|_{U'})\) is represented by the inverse system \((K_n)\), resp. \((K'_n)\) of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) with \[K_n = \mathcal{I}^n\mathcal{F}^\bullet \quad\text{resp.}\quad K'_n = (\mathcal{I}')^n\mathcal{F}^\bullet\] Clearly the map constructed above is given by the maps \(K'_n \to K_n\) coming from the inclusions \((\mathcal{I}')^n \subset \mathcal{I}^n\).

Compactly supported cohomology for coherent modules

In Situation 0F42 given a morphism \(f : X \to Y\) in \(\textit{FTS}_S\), we will define a functor \[Rf_! : D^b_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)\] Namely, we choose a compactification \(j : X \to \overline{X}\) over \(Y\) which is possible by More on Flatness, Theorem 0F41 and Lemma 0A9Z. Denote \(\overline{f} : \overline{X} \to Y\) the structure morphism. Then we set \[Rf_!K = R\overline{f}_* Rj_! K\] for \(K \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) where \(Rj_!\) is as in Remark 0G4N.

Lemma

The functor \(Rf_!\) is, up to isomorphism, independent of the choice of the compactification.

In fact, the functor \(Rf_!\) will be characterized as a “left adjoint” to \(f^!\) which will determine it up to unique isomorphism.

Proof

Consider the category of compactifications of \(X\) over \(Y\), which is cofiltered according to More on Flatness, Theorem 0F41 and Lemmas 0ATU and 0A9Z. To every choice of a compactification \[j : X \to \overline{X},\quad \overline{f} : \overline{X} \to Y\] the construction above associates the functor \(R\overline{f}_* \circ Rj_!\). Suppose given a morphism \(g : \overline{X}_1 \to \overline{X}_2\) between compactifications \(j_i : X \to \overline{X}_i\) over \(Y\). Then we get an isomorphism \[R\overline{f}_{2, *} \circ Rj_{2, !} = R\overline{f}_{2, *} \circ Rg_* \circ j_{1, !} = R\overline{f}_{1, *} \circ Rj_{1, !}\] using Lemma 0G4X in the first equality. In this way we see our functor is independent of the choice of compactification up to isomorphism.

Proposition

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Then the functors \(Rf_!\) and \(f^!\) are adjoint in the following sense: for all \(K \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) and \(L \in D^+_{\textit{Coh}}(\mathcal{O}_Y)\) we have \[\Hom_X(K, f^!L) = \Hom_{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_!K, L)\] bifunctorially in \(K\) and \(L\).

Proof

Choose a compactification \(j : X \to \overline{X}\) over \(Y\) and denote \(\overline{f} : \overline{X} \to Y\) the structure morphism. Then we have \[\begin{align*} \Hom_X(K, f^!L) & = \Hom_X(K, j^*\overline{f}{}^!L) \\ & = \Hom_{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_{\overline{X}})} (Rj_!K, \overline{f}{}^!L) \\ & = \Hom_{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_*Rj_!K, L) \\ & = \Hom_{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_!K, L) \end{align*}\] The first equality follows immediately from the construction of \(f^!\) in Section 0A9Y. By Lemma 0AU1 we have \(\overline{f}{}^!L\) in \(D^+_{\textit{Coh}}(\mathcal{O}_{\overline{X}})\) hence the second equality follows from Lemma 0G4K. Since \(\overline{f}\) is proper the functor \(\overline{f}{}^!\) is the right adjoint of pushforward by construction. This is why we have the third equality. The fourth equality holds because \(Rf_! = Rf_* Rj_!\).

Lemma

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\). Let \[K \to L \to M \to K[1]\] be a distinguished triangle of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Then there exists an inverse system of distinguished triangles \[K_n \to L_n \to M_n \to K_n[1]\] in \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\) such that the pro-systems \((K_n)\), \((L_n)\), and \((M_n)\) give \(Rf_!K\), \(Rf_!L\), and \(Rf_!M\).

Proof

Choose a compactification \(j : X \to \overline{X}\) over \(Y\) and denote \(\overline{f} : \overline{X} \to Y\) the structure morphism. Choose an inverse system of distinguished triangles \[\overline{K}_n \to \overline{L}_n \to \overline{M}_n \to \overline{K}_n[1]\] in \(D^b_{\textit{Coh}}(\mathcal{O}_{\overline{X}})\) as in Lemma 0G4M corresponding to the open immersion \(j\) and the given distinguished triangle. Take \(K_n = R\overline{f}_*\overline{K}_n\) and similarly for \(L_n\) and \(M_n\). This works by the very definition of \(Rf_!\).

Remark

Let \(\mathcal{C}\) be a category. Suppose given an inverse system \[\ldots \xrightarrow{\alpha_4} (M_{3, n}) \xrightarrow{\alpha_3} (M_{2, n}) \xrightarrow{\alpha_2} (M_{1, n})\] of inverse systems in the category of pro-objects of \(\mathcal{C}\). In other words, the arrows \(\alpha_i\) are morphisms of pro-objects. By Categories, Example 0G2W we can represent each \(\alpha_i\) by a pair \((m_i, a_i)\) where \(m_i : \mathbf{N} \to \mathbf{N}\) is an increasing function and \(a_{i, n} : M_{i, m_i(n)} \to M_{i - 1, n}\) is a morphism of \(\mathcal{C}\) making the diagrams \[\xymatrix{ \ldots \ar[r] & M_{i, m_i(3)} \ar[d]^{a_{i, 3}} \ar[r] & M_{i, m_i(2)} \ar[d]^{a_{i, 2}} \ar[r] & M_{i, m_i(1)} \ar[d]^{a_{i, 1}} \\ \ldots \ar[r] & M_{i - 1, 3} \ar[r] & M_{i - 1, 2} \ar[r] & M_{i - 1, 1} }\] commute. By replacing \(m_i(n)\) by \(\max(n, m_i(n))\) and adjusting the morphisms \(a_i(n)\) accordingly (as in the example referenced) we may assume that \(m_i(n) \geq n\). In this situation consider the inverse system \[\ldots \to M_{4, m_4(m_3(m_2(4)))} \to M_{3, m_3(m_2(3))} \to M_{2, m_2(2)} \to M_{1, 1}\] with general term \[M_k = M_{k, m_k(m_{k - 1}(\ldots (m_2(k))\ldots))}\] For any object \(N\) of \(\mathcal{C}\) we have \[\colim_i \colim_n \Mor_\mathcal{C}(M_{i, n}, N) = \colim_k \Mor_\mathcal{C}(M_k, N)\] We omit the details. In other words, we see that the inverse system \((M_k)\) has the property \[\colim_i \Mor_{\text{Pro-}\mathcal{C}}((M_{i, n}), N) = \Mor_{\text{Pro-}\mathcal{C}}((M_k), N)\] This property determines the inverse system \((M_k)\) up to pro-isomorphism by the discussion in Categories, Remark 0G2X. In this way we can turn certain inverse systems in \(\text{Pro-}\mathcal{C}\) into pro-objects with countable index categories.

Remark

In Situation 0F42 let \(f : X \to Y\) and \(g : Y \to Z\) be composable morphisms of \(\textit{FTS}_S\). Let us define the composition \[Rg_! \circ Rf_! : D^b_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Z)\] Namely, by the very construction of \(Rf_!\) for \(K\) in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) the output \(Rf_!K\) is the pro-isomorphism class of an inverse system \((M_n)\) in \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\). Then, since \(Rg_!\) is constructed similarly, we see that \[\ldots \to Rg_!M_3 \to Rg_!M_2 \to Rg_!M_1\] is an inverse system of \(\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)\). By the discussion in Remark 0G53 there is a unique pro-isomorphism class, which we will denote \(Rg_! Rf_! K\), of inverse systems in \(D^b_{\textit{Coh}}(\mathcal{O}_Z)\) such that \[\Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Z)}(Rg_!Rf_!K, L) = \colim_n \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Z)}(Rg_!M_n, L)\] We omit the discussion necessary to see that this construction is functorial in \(K\) as it will immediately follow from the next lemma.

Lemma

In Situation 0F42 let \(f : X \to Y\) and \(g : Y \to Z\) be composable morphisms of \(\textit{FTS}_S\). With notation as in Remark 0G54 we have \(Rg_! \circ Rf_! = R(g \circ f)_!\).

Proof

By the discussion in Categories, Remark 0G2X it suffices to show that we obtain the same answer if we compute \(\Hom\) into \(L\) in \(D^b_{\textit{Coh}}(\mathcal{O}_Z)\). To do this we compute, using the notation in Remark 0G54, as follows \[\begin{align*} \Hom_Z(Rg_!Rf_!K, L) & = \colim_n \Hom_Z(Rg_!M_n, L) \\ & = \colim_n \Hom_Y(M_n, g^!L) \\ & = \Hom_Y(Rf_!K, g^!L) \\ & = \Hom_X(K, f^!g^!L) \\ & = \Hom_X(K, (g \circ f)^!L) \\ & = \Hom_Z(R(g \circ f)_!K, L) \end{align*}\] The first equality is the definition of \(Rg_!Rf_!K\). The second equality is Proposition 0G51 for \(g\). The third equality is the fact that \(Rf_!K\) is given by \((M_n)\). The fourth equality is Proposition 0G51 for \(f\). The fifth equality is Lemma 0ATX. The sixth is Proposition 0G51 for \(g \circ f\).

Remark

In Situation 0F42 let \(f : X \to Y\) be a morphism of \(\textit{FTS}_S\) and let \(U \subset X\) be an open. Set \(g = f|_U : U \to Y\). Then there is a canonical morphism \[Rg_!(K|_U) \longrightarrow Rf_!K\] functorial in \(K\) in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) which can be defined in at least 3 ways.

1. Denote \(i : U \to X\) the inclusion morphism. We have \(Rg_! = Rf_! \circ Ri_!\) by Lemma 0G55 and we can use \(Rf_!\) applied to the map \(Ri_!(K|_U) \to K\) which is a special case of Remark 0G4Y.

2. Choose a compactification \(j : X \to \overline{X}\) of \(X\) over \(Y\) with structure morphism \(\overline{f} : \overline{X} \to Y\). Set \(j' = j \circ i : U \to \overline{X}\). We can use that \(Rf_! = R\overline{f}_* \circ Rj_!\) and \(Rg_! = R\overline{f}_* \circ Rj'_!\) and we can use \(R\overline{f}_*\) applied to the map \(Rj'_!(K|_U) \to Rj_!K\) of Remark 0G4Y.

3. We can use \[\begin{align*} \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_!K, L) & = \Hom_X(K, f^!L) \\ & \to \Hom_U(K|_U, f^!L|_U) \\ & = \Hom_U(K|_U, g^!L) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rg_!(K|_U), L) \end{align*}\] functorial in \(L\) and \(K\). Here we have used Proposition 0G51 twice and the construction of upper shriek functors which shows that \(g^! = i^* \circ f^!\). The functoriality in \(L\) shows by Categories, Remark 0G2X that we obtain a canonical map \(Rg_!(K|_U) \to Rf_!K\) in \(\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)\) which is functorial in \(K\) by the functoriality of the arrow above in \(K\).

Each of these three constructions gives the same arrow; we omit the details.

Remark

Let us generalize the covariance of compactly supported cohomology given in Remark 0G56 to étale morphisms. Namely, in Situation 0F42 suppose given a commutative diagram \[\xymatrix{ U \ar[rr]_h \ar[rd]_g & & X \ar[ld]^f \\ & Y }\] of \(\textit{FTS}_S\) with \(h\) étale. Then there is a canonical morphism \[Rg_!(h^*K) \longrightarrow Rf_!K\] functorial in \(K\) in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). We define this transformation using the sequence of maps \[\begin{align*} \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_!K, L) & = \Hom_X(K, f^!L) \\ & \to \Hom_U(h^*K, h^*(f^!L)) \\ & = \Hom_U(h^*K, h^!f^!L) \\ & = \Hom_U(h^*K, g^!L) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rg_!(h^*K), L) \end{align*}\] functorial in \(L\) and \(K\). Here we have used Proposition 0G51 twice, we have used the equality \(h^* = h^!\) of Lemma 0FWI, and we have used the equality \(h^! \circ f^! = g^!\) of Lemma 0ATX. The functoriality in \(L\) shows by Categories, Remark 0G2X that we obtain a canonical map \(Rg_!(h^*K) \to Rf_!K\) in \(\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)\) which is functorial in \(K\) by the functoriality of the arrow above in \(K\).

Remark

In Remarks 0G56 and 0G57 we have seen that the construction of compactly supported cohomology is covariant with respect to open immersions and étale morphisms. In fact, the correct generality is that given a commutative diagram \[\xymatrix{ U \ar[rr]_h \ar[rd]_g & & X \ar[ld]^f \\ & Y }\] of \(\textit{FTS}_S\) with \(h\) flat and quasi-finite there exists a canonical transformation \[Rg_! \circ h^* \longrightarrow Rf_!\] As in Remark 0G57 this map can be constructed using a transformation of functors \(h^* \to h^!\) on \(D^+_{\textit{Coh}}(\mathcal{O}_X)\). Recall that \(h^!K = h^*K \otimes \omega_{U/X}\) where \(\omega_{U/X} = h^!\mathcal{O}_X\) is the relative dualizing sheaf of the flat quasi-finite morphism \(h\) (see Lemmas 0B6U and 0BV7). Recall that \(\omega_{U/X}\) is the same as the relative dualizing module which will be constructed in Discriminants, Remark 0BVG by Discriminants, Lemma 0BUL. Thus we can use the trace element \(\tau_{U/X} : \mathcal{O}_U \to \omega_{U/X}\) which will be constructed in Discriminants, Remark 0BVJ to define our transformation. If we ever need this, we will precisely formulate and prove the result here.

Duality for compactly supported cohomology

Let \(k\) be a field. Let \(U\) be a separated scheme of finite type over \(k\). Let \(K\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_U)\). Let us define the compactly supported cohomology \(H^i_c(U, K)\) of \(K\) as follows. Choose an open immersion \(j : U \to X\) into a scheme proper over \(k\) and a Deligne system \((K_n)\) for \(j : U \to X\) whose restriction to \(U\) is constant with value \(K\). Then we set \[H^i_c(U, K) = \lim H^i(X, K_n)\] We view this as a topological \(k\)-vector space using the limit topology (see More on Algebra, Section 07E7). There are several points to make here.

First, this definition is independent of the choice of \(X\) and \((K_n)\). Namely, if \(p : U \to \Spec(k)\) denotes the structure morphism, then we already know that \(Rp_!K = (R\Gamma(X, K_n))\) is well defined up to pro-isomorphism in \(D(k)\) hence so is the limit defining \(H^i_c(U, K)\).

Second, it may seem more natural to use the expression \[H^i(R\lim R\Gamma(X, K_n)) = R\Gamma(X, R\lim K_n)\] but this would give the same answer: since the \(k\)-vector spaces \(H^j(X, K_n)\) are finite dimensional, these inverse systems satisfy Mittag-Leffler and hence \(R^1\lim\) terms of Cohomology, Lemma 0D60 vanish.

If \(U' \subset U\) is an open subscheme, then there is a canonical map \[H^i_c(U', K|_{U'}) \longrightarrow H^i_c(U, K)\] functorial for \(K\) in \(D^b_{\textit{Coh}}(\mathcal{O}_U)\). See for example Remark 0G56. In fact, using Remark 0G57 we see that more generally such a map exists for an étale morphism \(U' \to U\) of separated schemes of finite type over \(k\).

If \(V\) is a \(k\)-vector space then we put a topology on \(\Hom_k(V, k)\) as follows: write \(V = \bigcup V_i\) as the filtered union of its finite dimensional \(k\)-subvector spaces and use the limit topology on \(\Hom_k(V, k) = \lim \Hom_k(V_i, k)\). If \(\dim_k V < \infty\) then the topology on \(\Hom_k(V, k)\) is discrete. More generally, if \(V = \colim_n V_n\) is written as a directed colimit of finite dimensional vector spaces, then \(\Hom_k(V, k) = \lim \Hom_k(V_n, k)\) as topological vector spaces.

Lemma

Let \(p : U \to \Spec(k)\) be separated of finite type where \(k\) is a field. Let \(\omega_{U/k}^\bullet = p^!\mathcal{O}_{\Spec(k)}\). There are canonical isomorphisms \[\Hom_k(H^i(U, K), k) = H^{-i}_c(U, R\SheafHom_{\mathcal{O}_U}(K, \omega_{U/k}^\bullet))\] of topological \(k\)-vector spaces functorial for \(K\) in \(D^b_{\textit{Coh}}(\mathcal{O}_U)\).

Proof

Choose a compactification \(j : U \to X\) over \(k\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent ideal sheaf with \(V(\mathcal{I}) = X \setminus U\). By Derived Categories of Schemes, Proposition 0FDB we may choose \(M \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) with \(K = M|_U\). We have \[H^i(U, K) = \Ext^i_U(\mathcal{O}_U, M|_U) = \colim \Ext^i_X(\mathcal{I}^n, M) = \colim H^i(X, R\SheafHom_{\mathcal{O}_X}(\mathcal{I}^n, M))\] by Lemma 0G2H. Since \(\mathcal{I}^n\) is a coherent \(\mathcal{O}_X\)-module, we have \(\mathcal{I}^n\) in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\), hence \(R\SheafHom_{\mathcal{O}_X}(\mathcal{I}^n, M)\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) by Derived Categories of Schemes, Lemma 0D0C.

Let \(\omega_{X/k}^\bullet = q^!\mathcal{O}_{\Spec(k)}\) where \(q : X \to \Spec(k)\) is the structure morphism, see Section 0FVU. We find that \[\begin{align*} \Hom_k( & H^i(X, R\SheafHom_{\mathcal{O}_X}(\mathcal{I}^n, M)), k) \\ & = \Ext^{-i}_X(R\SheafHom_{\mathcal{O}_X}(\mathcal{I}^n, M), \omega_{X/k}^\bullet) \\ & = H^{-i}(X, R\SheafHom_{\mathcal{O}_X}(R\SheafHom_{\mathcal{O}_X}( \mathcal{I}^n, M), \omega_{X/k}^\bullet)) \end{align*}\] by Lemma 0FVV. By Lemma 0G4I part (1) the canonical map \[R\SheafHom_{\mathcal{O}_X}(M, \omega_{X/k}^\bullet) \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I}^n \longrightarrow R\SheafHom_{\mathcal{O}_X}(R\SheafHom_{\mathcal{O}_X}( \mathcal{I}^n, M), \omega_{X/k}^\bullet)\] is an isomorphism. Observe that \(\omega^\bullet_{U/k} = \omega^\bullet_{X/k}|_U\) because \(p^!\) is constructed as \(q^!\) composed with restriction to \(U\). Hence \(R\SheafHom_{\mathcal{O}_X}(M, \omega_{X/k}^\bullet)\) is an object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) which restricts to \(R\SheafHom_{\mathcal{O}_U}(K, \omega_{U/k}^\bullet)\) on \(U\). Hence by Lemma 0G4U we conclude that \[\lim H^{-i}(X, R\SheafHom_{\mathcal{O}_X}(M, \omega_{X/k}^\bullet) \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I}^n)\] is an avatar for the right hand side of the equality of the lemma. Combining all the isomorphisms obtained in this manner we get the isomorphism of the lemma.

Lemma

With notation as in Lemma 0G5A suppose \(U' \subset U\) is an open subscheme. Then the diagram \[\xymatrix{ \Hom_k(H^i(U, K), k) \ar[rr] & & H^{-i}_c(U, R\SheafHom_{\mathcal{O}_U}(K, \omega_{U/k}^\bullet)) \\ \Hom_k(H^i(U', K|_{U'}), k) \ar[rr] \ar[u] & & H^{-i}_c(U', R\SheafHom_{\mathcal{O}_{U'}}(K, \omega_{U'/k}^\bullet)) \ar[u] }\] is commutative. Here the horizontal arrows are the isomorphisms of Lemma 0G5A, the vertical arrow on the left is the contragredient to the restriction map \(H^i(U, K) \to H^i(U', K|_{U'})\), and the right vertical arrow is Remark 0G56 (see discussion before the lemma).

Proof

We strongly urge the reader to skip this proof. Choose \(X\) and \(M\) as in the proof of Lemma 0G5A. We are going to drop the subscript \(\mathcal{O}_X\) from \(R\SheafHom\) and \(\otimes^\mathbf{L}\). We write \[H^i(U, K) = \colim H^i(X, R\SheafHom(\mathcal{I}^n, M))\] and \[H^i(U', K|_{U'}) = \colim H^i(X, R\SheafHom((\mathcal{I}')^n, M))\] as in the proof of Lemma 0G5A where we choose \(\mathcal{I}' \subset \mathcal{I}\) as in the discussion in Remark 0G4Y so that the map \(H^i(U, K) \to H^i(U', K|_{U'})\) is induced by the maps \((\mathcal{I}')^n \to \mathcal{I}^n\). We similarly write \[H^i_c(U, R\SheafHom(K, \omega_{U/k}^\bullet)) = \lim H^i(X, R\SheafHom(M, \omega_{X/k}^\bullet) \otimes^\mathbf{L} \mathcal{I}^n)\] and \[H^i_c(U', R\SheafHom(K|_{U'}, \omega_{U'/k}^\bullet)) = \lim H^i(X, R\SheafHom(M, \omega_{X/k}^\bullet) \otimes^\mathbf{L} (\mathcal{I}')^n)\] so that the arrow \(H^i_c(U', R\SheafHom(K|_{U'}, \omega_{U'/k}^\bullet)) \to H^i_c(U, R\SheafHom(K, \omega_{U/k}^\bullet))\) is similarly deduced from the maps \((\mathcal{I}')^n \to \mathcal{I}^n\). The diagrams \[\xymatrix{ R\SheafHom(M, \omega_{X/k}^\bullet) \otimes^\mathbf{L} \mathcal{I}^n \ar[rr] & & R\SheafHom(R\SheafHom(\mathcal{I}^n, M), \omega_{X/k}^\bullet) \\ R\SheafHom(M, \omega_{X/k}^\bullet) \otimes^\mathbf{L} (\mathcal{I}')^n \ar[rr] \ar[u] & & R\SheafHom(R\SheafHom((\mathcal{I}')^n, M), \omega_{X/k}^\bullet) \ar[u] }\] commute because the construction of the horizontal arrows in Cohomology, Lemma 0A8U is functorial in all three entries. Hence we finally come down to the assertion that the diagrams \[\xymatrix{ \Hom_k(H^i(X, R\SheafHom(\mathcal{I}^n, M)), k) \ar[r] & H^{-i}(X, R\SheafHom(R\SheafHom( \mathcal{I}^n, M), \omega_{X/k}^\bullet)) \\ \Hom_k(H^i(X, R\SheafHom((\mathcal{I}')^n, M)), k) \ar[r] \ar[u] & H^{-i}(X, R\SheafHom(R\SheafHom( (\mathcal{I}')^n, M), \omega_{X/k}^\bullet)) \ar[u] }\] commute. This is true because the duality isomorphism \[\Hom_k(H^i(X, L), k) = \Ext^{-i}_X(L, \omega_{X/k}^\bullet) = H^{-i}(X, R\SheafHom(L, \omega_{X/k}^\bullet))\] is functorial for \(L\) in \(D_\QCoh(\mathcal{O}_X)\).

Lemma

Let \(X\) be a proper scheme over a field \(k\). Let \(K \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) with \(H^i(K) = 0\) for \(i < 0\). Set \(\mathcal{F} = H^0(K)\). Let \(Z \subset X\) be closed with complement \(U = X \setminus U\). Then \[H^0_c(U, K|_U) \subset H^0(X, \mathcal{F})\] is given by those global sections of \(\mathcal{F}\) which vanish in an open neighbourhood of \(Z\).

Proof

Consider the map \(H^0_c(U, K|_U) \to H^0_X(X, K) = H^0(X, K) = H^0(X, \mathcal{F})\) of Remark 0G56. To study this we represent \(K\) by a bounded complex \(\mathcal{F}^\bullet\) with \(\mathcal{F}^i = 0\) for \(i < 0\). Then we have by definition \[H^0_c(U, K|_U) = \lim H^0(X, \mathcal{I}^n\mathcal{F}^\bullet) = \lim \Ker( H^0(X, \mathcal{I}^n\mathcal{F}^0) \to H^0(X, \mathcal{I}^n\mathcal{F}^1))\] By Artin-Rees (Cohomology of Schemes, Lemma 01YA) this is the same as \(\lim H^0(X, \mathcal{I}^n\mathcal{F})\). Thus the arrow \(H^0_c(U, K|_U) \to H^0(X, \mathcal{F})\) is injective and the image consists of those global sections of \(\mathcal{F}\) which are contained in the subsheaf \(\mathcal{I}^n\mathcal{F}\) for any \(n\). The characterization of these as the sections which vanish in a neighbourhood of \(Z\) comes from Krull’s intersection theorem (Algebra, Lemma 00IP) by looking at stalks of \(\mathcal{F}\). See discussion in Algebra, Remark 00IR for the case of functions.

Lichtenbaum’s theorem

The theorem below was conjectured by Lichtenbaum and proved by Grothendieck (see [Hartshorne-local-cohomology]). There is a very nice proof of the theorem by Kleiman in [Kleiman-Lichtenbaum]. A generalization of the theorem to the case of cohomology with supports can be found in [Lyubeznik-Lichtenbaum]. The most interesting part of the argument is contained in the proof of the following lemma.

Lemma

Let \(U\) be a variety. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_U\)-module. If \(H^d(U, \mathcal{F})\) is nonzero, then \(\dim(U) \geq d\) and if equality holds, then \(U\) is proper.

Proof

By the Grothendieck’s vanishing result in Cohomology, Proposition 02UZ we conclude that \(\dim(U) \geq d\). Assume \(\dim(U) = d\). Choose a compactification \(U \to X\) such that \(U\) is dense in \(X\). (This is possible by More on Flatness, Theorem 0F41 and Lemma 0A9Z.) After replacing \(X\) by its reduction we find that \(X\) is a proper variety of dimension \(d\) and we see that \(U\) is proper if and only if \(U = X\). Set \(Z = X \setminus U\). We will show that \(H^d(U, \mathcal{F})\) is zero if \(Z\) is nonempty.

Choose a coherent \(\mathcal{O}_X\)-module \(\mathcal{G}\) whose restriction to \(U\) is \(\mathcal{F}\), see Properties, Lemma 0G41. Let \(\omega_X^\bullet\) denote the dualizing complex of \(X\) as in Section 0FVU. Set \(\omega_U^\bullet = \omega_X^\bullet|_U\). Then \(H^d(U, \mathcal{F})\) is dual to \[H^{-d}_c(U, R\SheafHom_{\mathcal{O}_U}(\mathcal{F}, \omega_U^\bullet))\] by Lemma 0G5A. By Lemma 0FVV we see that the cohomology sheaves of \(\omega_X^\bullet\) vanish in degrees \(< -d\) and \(H^{-d}(\omega_X^\bullet) = \omega_X\) is a coherent \(\mathcal{O}_X\)-module which is \((S_2)\) and whose support is \(X\). In particular, \(\omega_X\) is torsion free, see Divisors, Lemma 0AXY. Thus we see that the cohomology sheaf \[H^{-d}(R\SheafHom_{\mathcal{O}_X}(\mathcal{G}, \omega_X^\bullet)) = \SheafHom(\mathcal{G}, \omega_X)\] is torsion free, see Divisors, Lemma 0AXZ. Consequently this sheaf has no nonzero sections vanishing on any nonempty open of \(X\) (those would be torsion sections). Thus it follows from Lemma 0G5C that \(H^{-d}_c(U, R\SheafHom_{\mathcal{O}_U}(\mathcal{F}, \omega_U^\bullet))\) is zero, and hence \(H^d(U, \mathcal{F})\) is zero as desired.

Theorem

Let \(X\) be a nonempty separated scheme of finite type over a field \(k\). Let \(d = \dim(X)\). The following are equivalent

1. \(H^d(X, \mathcal{F}) = 0\) for all coherent \(\mathcal{O}_X\)-modules \(\mathcal{F}\) on \(X\),

2. \(H^d(X, \mathcal{F}) = 0\) for all quasi-coherent \(\mathcal{O}_X\)-modules \(\mathcal{F}\) on \(X\), and

3. no irreducible component \(X' \subset X\) of dimension \(d\) is proper over \(k\).

Proof

Assume there exists an irreducible component \(X' \subset X\) (which we view as an integral closed subscheme) which is proper and has dimension \(d\). Let \(\omega_{X'}\) be a dualizing module of \(X'\) over \(k\), see Lemma 0FVV. Then \(H^d(X', \omega_{X'})\) is nonzero as it is dual to \(H^0(X', \mathcal{O}_{X'})\) by the lemma. Hence we see that \(H^d(X, \omega_{X'}) = H^d(X', \omega_{X'})\) is nonzero and we conclude that (1) does not hold. In this way we see that (1) implies (3).

Let us prove that (3) implies (1). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module such that \(H^d(X, \mathcal{F})\) is nonzero. Choose a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F}\] as in Cohomology of Schemes, Lemma 01YF. We obtain exact sequences \[H^d(X, \mathcal{F}_i) \to H^d(X, \mathcal{F}_{i + 1}) \to H^d(X, \mathcal{F}_{i + 1}/\mathcal{F}_i)\] Thus for some \(i \in \{1, \ldots, m\}\) we find that \(H^d(X, \mathcal{F}_{i + 1}/\mathcal{F}_i)\) is nonzero. By our choice of the filtration this means that there exists an integral closed subscheme \(Z \subset X\) and a nonzero coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\) such that \(H^d(Z, \mathcal{I})\) is nonzero. By Lemma 0G5E we conclude \(\dim(Z) = d\) and \(Z\) is proper over \(k\) contradicting (3). Hence (3) implies (1).

Finally, let us show that (1) and (2) are equivalent for any Noetherian scheme \(X\). Namely, (2) trivially implies (1). On the other hand, assume (1) and let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then we can write \(\mathcal{F} = \colim \mathcal{F}_i\) as the filtered colimit of its coherent submodules, see Properties, Lemma 01PG. Then we have \(H^d(X, \mathcal{F}) = \colim H^d(X, \mathcal{F}_i) = 0\) by Cohomology, Lemma 01FF. Thus (2) is true.