Introduction
In this chapter we continue the discussion started in Derived Categories of Schemes, Section 08CV. We will discuss Fourier-Mukai transforms, first studied by Mukai in [Mukai]. We will prove Orlov’s theorem on derived equivalences ([Orlov-K3]). We also discuss the countability of derived equivalence classes proved by Anel and Toën in [AT].
A good introduction to this material is the book [Huybrechts] by Daniel Huybrechts. Some other papers which helped popularize this topic are
the paper by Bondal and Kapranov, see [Bondal-Kapranov]
the paper by Bondal and Orlov, see [Bondal-Orlov]
the paper by Bondal and Van den Bergh, see [BvdB]
the papers by Beilinson, see [Beilinson] and [Beilinson-derived]
the paper by Orlov, see [Orlov-AV]
the paper by Orlov, see [Orlov-motives]
the paper by Rouquier, see [Rouquier-dimensions]
there are many more we could mention here.
Conventions and notation
Let \(k\) be a field. A \(k\)-linear triangulated category \(\mathcal{T}\) is a triangulated category (Derived Categories, Section 05QK) which is endowed with a \(k\)-linear structure (Differential Graded Algebra, Section 09MI) such that the translation functors \([n] : \mathcal{T} \to \mathcal{T}\) are \(k\)-linear for all \(n \in \mathbf{Z}\).
Let \(k\) be a field. We denote \(\text{Vect}_k\) the category of \(k\)-vector spaces. For a \(k\)-vector space \(V\) we denote \(V^\vee\) the \(k\)-linear dual of \(V\), i.e., \(V^\vee = \Hom_k(V, k)\).
Let \(X\) be a scheme. We denote \(D_{perf}(\mathcal{O}_X)\) the full subcategory of \(D(\mathcal{O}_X)\) consisting of perfect complexes (Cohomology, Section 08CL). If \(X\) is Noetherian then \(D_{perf}(\mathcal{O}_X) \subset D^b_{\textit{Coh}}(\mathcal{O}_X)\), see Derived Categories of Schemes, Lemma 0FXU. If \(X\) is Noetherian and regular, then \(D_{perf}(\mathcal{O}_X) = D^b_{\textit{Coh}}(\mathcal{O}_X)\), see Derived Categories of Schemes, Lemma 0FDC.
Let \(k\) be a field. Let \(X\) and \(Y\) be schemes over \(k\). In this situation we will write \(X \times Y\) instead of \(X \times_{\Spec(k)} Y\).
Let \(S\) be a scheme. Let \(X\), \(Y\) be schemes over \(S\). Let \(\mathcal{F}\) be a \(\mathcal{O}_X\)-module and let \(\mathcal{G}\) be a \(\mathcal{O}_Y\)-module. We set \[\mathcal{F} \boxtimes \mathcal{G} = \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} \text{pr}_2^*\mathcal{G}\] as \(\mathcal{O}_{X \times_S Y}\)-modules. If \(K \in D(\mathcal{O}_X)\) and \(M \in D(\mathcal{O}_Y)\) then we set \[K \boxtimes M = L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} L\text{pr}_2^*M\] as an object of \(D(\mathcal{O}_{X \times_S Y})\). Thus our notation is potentially ambiguous, but context should make it clear which of the two is meant.
Serre functors
The material in this section is taken from [Bondal-Kapranov].
Lemma
Let \(k\) be a field. Let \(\mathcal{T}\) be a \(k\)-linear triangulated category such that \(\dim_k \Hom_\mathcal{T}(X, Y) < \infty\) for all \(X, Y \in \Ob(\mathcal{T})\). The following are equivalent
there exists a \(k\)-linear equivalence \(S : \mathcal{T} \to \mathcal{T}\) and \(k\)-linear isomorphisms \(c_{X, Y} : \Hom_\mathcal{T}(X, Y) \to \Hom_\mathcal{T}(Y, S(X))^\vee\) functorial in \(X, Y \in \Ob(\mathcal{T})\),
for every \(X \in \Ob(\mathcal{T})\) the functor \(Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee\) is representable and the functor \(Y \mapsto \Hom_\mathcal{T}(Y, X)^\vee\) is corepresentable.
Proof
Condition (1) implies (2) since given \((S, c)\) and \(X \in \Ob(\mathcal{T})\) the object \(S(X)\) represents the functor \(Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee\) and the object \(S^{-1}(X)\) corepresents the functor \(Y \mapsto \Hom_\mathcal{T}(Y, X)^\vee\).
Assume (2). We will repeatedly use the Yoneda lemma, see Categories, Lemma 001P. For every \(X\) denote \(S(X)\) the object representing the functor \(Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee\). Given \(\varphi : X \to X'\), we obtain a unique arrow \(S(\varphi) : S(X) \to S(X')\) determined by the corresponding transformation of functors \(\Hom_\mathcal{T}(X, -)^\vee \to \Hom_\mathcal{T}(X', -)^\vee\). Thus \(S\) is a functor and we obtain the isomorphisms \(c_{X, Y}\) by construction. It remains to show that \(S\) is an equivalence. For every \(X\) denote \(S'(X)\) the object corepresenting the functor \(Y \mapsto \Hom_\mathcal{T}(Y, X)^\vee\). Arguing as above we find that \(S'\) is a functor. We claim that \(S'\) is quasi-inverse to \(S\). To see this observe that \[\Hom_\mathcal{T}(X, Y) = \Hom_\mathcal{T}(Y, S(X))^\vee = \Hom_\mathcal{T}(S'(S(X)), Y)\] bifunctorially, i.e., we find \(S' \circ S \cong \text{id}_\mathcal{T}\). Similarly, we have \[\Hom_\mathcal{T}(Y, X) = \Hom_\mathcal{T}(S'(X), Y)^\vee = \Hom_\mathcal{T}(Y, S(S'(X)))\] and we find \(S \circ S' \cong \text{id}_\mathcal{T}\).
Definition
Let \(k\) be a field. Let \(\mathcal{T}\) be a \(k\)-linear triangulated category such that \(\dim_k \Hom_\mathcal{T}(X, Y) < \infty\) for all \(X, Y \in \Ob(\mathcal{T})\). We say a Serre functor exists if the equivalent conditions of Lemma 0FY4 are satisfied. In this case a Serre functor is a \(k\)-linear equivalence \(S : \mathcal{T} \to \mathcal{T}\) endowed with \(k\)-linear isomorphisms \(c_{X, Y} : \Hom_\mathcal{T}(X, Y) \to \Hom_\mathcal{T}(Y, S(X))^\vee\) functorial in \(X, Y \in \Ob(\mathcal{T})\).
Remark
Let \(X^0 \to X^1 \to X^2 \to X^0[1]\) and \(Y^0 \to Y^1 \to Y^2 \to Y^0[1]\) be distinguished triangles in a triangulated category. For \(p \in \mathbf{Z}\) write \(p = 3n + i\) with \(i \in \{0, 1, 2\}\) and set we set \(X^p = X^i[n]\). Simlarly for \(Y^q\). Consider the double complex with terms \[K^{p, q} = \Hom(X^{-p}, Y^q)\] The differential \(d_1 : K^{p, q} \to K^{p + 1, q}\) is given by the map \(X^{-p - 1} \to X^{-p}\) (equal to the corresponding map in the first distinguished triangle up the a shift) and the differential \(d_2 : K^{p, q} \to K^{p, q + 1}\) likewise by the map \(Y^q \to Y^{q + 1}\). From Derived Categories, Lemma 0149 we see that the rows and columns of this double complex are exact. Furthermore, we see that \(K^{p, q} = K^{p + 3, q - 3}\) and these equalities are compatible with the differentials. Finally, axiom TR3 implies one additional property: given \(\alpha \in K^{p, q}\) and \(\beta \in K^{p - 1, q + 1}\) such that \(d_2 \alpha = d_1 \beta\), there exists a \(\gamma \in K^{p - 2, q + 2}\) such that \(d_1 \gamma = d_2 \beta\) in \(K^{p - 1, q + 2}\) and \(d_2 \gamma = d_1 \alpha\) in \(K^{p - 2, q + 3} = K^{p + 1, q}\). (Hint: for \(p = q = 0\) this is exactly the statement of TR3 and for other indices prove it by shifting.) A double complex with these properties is called a matress (see [Bondal-Kapranov]).
Remark
Let \(k\) be a field and let \(K^{\bullet, \bullet}\) be a double complex of finite dimensional \(k\)-vector spaces which is also a matress as in Remark 0H8B. We claim that the dual double complex \(L^{p, q} = \Hom_k(K^{-q, -p}, k)\) is also a matress. The exactness of rows and columns and the 3-periodicity are immediate. To see the additional condition holds, consider the linear map \[\partial : K^{p, q} \oplus K^{p - 1, q + 1} \oplus K^{p - 2, q + 2} \longrightarrow K^{p, q + 1} \oplus K^{p - 1, q + 2} \oplus K^{p - 2, q + 3}\] sending \((\alpha, \beta, \gamma)\) to \((d_2 \alpha - d_1 \beta, d_2 \gamma - d_1 \alpha, d_1 \gamma - d_2 \beta)\) with identifications as in Remark 0H8B. The condition of being a matress is that \[\Im(\partial) \cap \left( 0 \oplus K^{p - 1, q + 2} \oplus K^{p - 2, q + 3} \right) = \Im(\partial|_{0 \oplus 0 \oplus K^{p - 2, q + 2}})\] Denoting \({}^\wedge\) the dual of a vector space or map, taking duals we get \[\Ker(\partial^\wedge) + \left(L^{-p, -q - 1} \oplus 0 \oplus 0 \right) = \Ker(\text{pr}_{L^{-p + 2, -q - 2}} \circ \partial^\wedge)\] where the \(\text{pr}\) term indicates projection. This implies that \(\Im(\partial^\wedge) \cap (L^{-p, -q} \oplus L^{-p + 1, -q - 1} \oplus 0)\) is equal to \(\Im(\partial^\wedge|_{L^{-p, -q - 1} \oplus 0 \oplus 0})\) and this translates into the matress condition for \(L^{\bullet, \bullet}\). Some details omitted.
Lemma
In the situation of Definition 0FY5. If a Serre functor exists, then it is unique up to unique isomorphism and it is an exact functor of triangulated categories.
Proof
Given a Serre functor \(S\) the object \(S(X)\) represents the functor \(Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee\). Thus the object \(S(X)\) together with the functorial identification \(\Hom_\mathcal{T}(X, Y)^\vee = \Hom_\mathcal{T}(Y, S(X))\) is determined up to unique isomorphism by the Yoneda lemma (Categories, Lemma 001P). Moreover, for \(\varphi : X \to X'\), the arrow \(S(\varphi) : S(X) \to S(X')\) is uniquely determined by the corresponding transformation of functors \(\Hom_\mathcal{T}(X, -)^\vee \to \Hom_\mathcal{T}(X', -)^\vee\).
For objects \(X, Y\) of \(\mathcal{T}\) we have \[\begin{align*} \Hom(Y, S(X)[1])^\vee & = \Hom(Y[-1], S(X))^\vee \\ & = \Hom(X, Y[-1]) \\ & = \Hom(X[1], Y) \\ & = \Hom(Y, S(X[1]))^\vee \end{align*}\] By the Yoneda lemma we conclude that there is a unique isomorphism \(S(X[1]) \to S(X)[1]\) inducing the isomorphism from top left to bottom right. Since each of the isomorphisms above is functorial in both \(X\) and \(Y\) we find that this defines an isomorphism of functors \(S \circ [1] \to [1] \circ S\).
Let \((A, B, C, f, g, h)\) be a distinguished triangle in \(\mathcal{T}\). We have to show that the triangle \((S(A), S(B), S(C), S(f), S(g), S(h))\) is distinguished. Here we use the canonical isomorphism \(S(A[1]) \to S(A)[1]\) constructed above to identify the target \(S(A[1])\) of \(S(h)\) with \(S(A)[1]\). We first observe that for any \(X\) in \(\mathcal{T}\) the triangle \((S(A), S(B), S(C), S(f), S(g), S(h))\) induces a long exact sequence \[\ldots \to \Hom(X, S(A)) \to \Hom(X, S(B)) \to \Hom(X, S(C)) \to \Hom(X, S(A)[1]) \to \ldots\] of finite dimensional \(k\)-vector spaces. Namely, this sequence is \(k\)-linear dual of the sequence \[\ldots \leftarrow \Hom(A, X) \leftarrow \Hom(B, X) \leftarrow \Hom(C, X) \leftarrow \Hom(A[1], X) \leftarrow \ldots\] which is exact by Derived Categories, Lemma 0149. Next, we choose a distinguished triangle \((S(A), E, S(C), i, p, S(h))\) which is possible by axioms TR1 and TR2. We want to construct the dotted arrow making following diagram commute \[\xymatrix{ S(C)[-1] \ar[r]_-{S(h[-1])} & S(A) \ar[r]_{S(f)} & S(B) \ar[r]_{S(g)} & S(C) \ar[r]_{S(h)} & S(A)[1] \\ S(C)[-1] \ar[r]^-{S(h[-1])} \ar@{=}[u] & S(A) \ar[r]^i \ar@{=}[u] & E \ar[r]^p \ar@{..>}[u]^\varphi & S(C) \ar[r]^{S(h)} \ar@{=}[u] & S(A)[1] \ar@{=}[u] }\] Namely, if we have \(\varphi\), then we claim for any \(X\) the resulting map \(\Hom(X, E) \to \Hom(X, S(B))\) will be an isomorphism of \(k\)-vector spaces. Namely, we will obtain a commutative diagram \[\xymatrix{ \Hom(X, S(C)[-1]) \ar[r] & \Hom(X, S(A)) \ar[r] & \Hom(X, S(B)) \ar[r] & \Hom(X, S(C)) \ar[r] & \Hom(X, S(A)[1]) \\ \Hom(X, S(C)[-1]) \ar[r] \ar@{=}[u] & \Hom(X, S(A)) \ar[r] \ar@{=}[u] & \Hom(X, E) \ar[r] \ar[u]^\varphi & \Hom(X, S(C)) \ar[r] \ar@{=}[u] & \Hom(X, S(A)[1]) \ar@{=}[u] }\] with exact rows (see above) and we can apply the 5 lemma (Homology, Lemma 05QB) to see that the middle arrow is an isomorphism. By the Yoneda lemma we conclude that \(\varphi\) is an isomorphism.
To find \(\varphi\), let us form the matress of Remark 0H8B from the distinguished triangles \(A \to B \to C \to A[1]\) and \(S(A) \to E \to S(C) \to S(A)[1]\). By Remark 0H8C the \(k\)-linear dual of this is also a matress. Using that \(S\) is a Serre functor, we see as above that this dual matress is the double complex made from the triangles (the second of which we do not yet know to be distinguished) \(S(A) \to E \to S(C) \to S(A)[1]\) and \(S(A) \to S(B) \to S(C) \to S(A)[1]\). But the matress condition exactly tells us that the maps \(S(A) \to S(A)\) and \(S(C) \to S(C)\) fit into a morphism of triangles, i.e., that we have a \(\varphi\) making the diagram above commute.
Examples of Serre functors
The lemma below is the standard example.
Lemma
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\) which is Gorenstein. Consider the complex \(\omega_X^\bullet\) of Duality for Schemes, Lemmas 0FVV. Then the functor \[S : D_{perf}(\mathcal{O}_X) \longrightarrow D_{perf}(\mathcal{O}_X),\quad K \longmapsto S(K) = \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K\] is a Serre functor.
Proof
The statement make sense because \(\dim \Hom_X(K, L) < \infty\) for \(K, L \in D_{perf}(\mathcal{O}_X)\) by Derived Categories of Schemes, Lemma 0D0D. Since \(X\) is Gorenstein the dualizing complex \(\omega_X^\bullet\) is an invertible object of \(D(\mathcal{O}_X)\), see Duality for Schemes, Lemma 0BFQ. In particular, locally on \(X\) the complex \(\omega_X^\bullet\) has one nonzero cohomology sheaf which is an invertible module, see Cohomology, Lemma 0FPG. Thus \(S(K)\) lies in \(D_{perf}(\mathcal{O}_X)\). On the other hand, the invertibility of \(\omega_X^\bullet\) clearly implies that \(S\) is a self-equivalence of \(D_{perf}(\mathcal{O}_X)\). Finally, we have to find an isomorphism \[c_{K, L} : \Hom_X(K, L) \longrightarrow \Hom_X(L, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K)^\vee\] bifunctorially in \(K, L\). To do this we use the canonical isomorphisms \[\Hom_X(K, L) = H^0(X, L \otimes_{\mathcal{O}_X}^\mathbf{L} K^\vee)\] and \[\Hom_X(L, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K) = H^0(X, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K \otimes_{\mathcal{O}_X}^\mathbf{L} L^\vee)\] given in Cohomology, Lemma 08DQ. Since \((L \otimes_{\mathcal{O}_X}^\mathbf{L} K^\vee)^\vee = (K^\vee)^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L^\vee\) and since there is a canonical isomorphism \(K \to (K^\vee)^\vee\) we find these \(k\)-vector spaces are canonically dual by Duality for Schemes, Lemma 0FVY. This produces the isomorphisms \(c_{K, L}\). We omit the proof that these isomorphisms are functorial.
Characterizing coherent modules
This section is in some sense a continuation of the discussion in Derived Categories of Schemes, Section 0CSE and More on Morphisms, Section 0CSI.
Before we can state the result we need some notation. Let \(k\) be a field. Let \(n \geq 0\) be an integer. Let \(S = k[X_0, \ldots, X_n]\). For an integer \(e\) denote \(S_e \subset S\) the homogeneous polynomials of degree \(e\). Consider the (noncommutative) \(k\)-algebra \[R = \left( \begin{matrix} S_0 & S_1 & S_2 & \ldots & \ldots \\ 0 & S_0 & S_1 & \ldots & \ldots\\ 0 & 0 & S_0 & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & \ldots & S_0 \end{matrix} \right)\] (with \(n + 1\) rows and columns) with obvious multiplication and addition.
Lemma
With \(k\), \(n\), and \(R\) as above, for an object \(K\) of \(D(R)\) the following are equivalent
\(\sum_{i \in \mathbf{Z}} \dim_k H^i(K) < \infty\), and
\(K\) is a compact object.
Proof
If \(K\) is a compact object, then \(K\) can be represented by a complex \(M^\bullet\) which is finite projective as a graded \(R\)-module, see Differential Graded Algebra, Lemma 09RB. Since \(\dim_k R < \infty\) we conclude \(\sum \dim_k M^i < \infty\) and a fortiori \(\sum \dim_k H^i(M^\bullet) < \infty\). (One can also easily deduce this implication from the easier Differential Graded Algebra, Proposition 09R3.)
Assume \(K\) satisfies (1). Consider the distinguished triangle of trunctions \(\tau_{\leq m}K \to K \to \tau_{\geq m + 1}K\), see Derived Categories, Remark 08J5. It is clear that both \(\tau_{\leq m}K\) and \(\tau_{\geq m + 1} K\) satisfy (1). If we can show both are compact, then so is \(K\), see Derived Categories, Lemma 09QH. Hence, arguing on the number of nonzero cohomology modules of \(K\) we may assume \(H^i(K)\) is nonzero only for one \(i\). Shifting, we may assume \(K\) is given by the complex consisting of a single finite dimensional \(R\)-module \(M\) sitting in degree \(0\).
Since \(\dim_k(M) < \infty\) we see that \(M\) is Artinian as an \(R\)-module. Thus it suffices to show that every simple \(R\)-module represents a compact object of \(D(R)\). Observe that \[I = \left( \begin{matrix} 0 & S_1 & S_2 & \ldots & \ldots \\ 0 & 0 & S_1 & \ldots & \ldots\\ 0 & 0 & 0 & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & \ldots & 0 \end{matrix} \right)\] is a nilpotent two sided ideal of \(R\) and that \(R/I\) is a commutative \(k\)-algebra isomorphic to a product of \(n + 1\) copies of \(k\) (placed along the diagonal in the matrix, i.e., \(R/I\) can be lifted to a \(k\)-subalgebra of \(R\)). It follows that \(R\) has exactly \(n + 1\) isomorphism classes of simple modules \(M_0, \ldots, M_n\) (sitting along the diagonal). Consider the right \(R\)-module \(P_i\) of row vectors \[P_i = \left( \begin{matrix} 0 & \ldots & 0 & S_0 & \ldots & S_{i - 1} & S_i \end{matrix} \right)\] with obvious multiplication \(P_i \times R \to P_i\). Then we see that \(R \cong P_0 \oplus \ldots \oplus P_n\) as a right \(R\)-module. Since clearly \(R\) is a compact object of \(D(R)\), we conclude each \(P_i\) is a compact object of \(D(R)\). (We of course also conclude each \(P_i\) is projective as an \(R\)-module, but this isn’t what we have to show in this proof.) Clearly, \(P_0 = M_0\) is the first of our simple \(R\)-modules. For \(P_1\) we have a short exact sequence \[0 \to P_0^{\oplus n + 1} \to P_1 \to M_1 \to 0\] which proves that \(M_1\) fits into a distinguished triangle whose other members are compact objects and hence \(M_1\) is a compact object of \(D(R)\). More generally, there exists a short exact sequence \[0 \to C_i \to P_i \to M_i \to 0\] where \(C_i\) is a finite dimensional \(R\)-module whose simple constituents are isomorphic to \(M_j\) for \(j < i\). By induction, we first conclude that \(C_i\) determines a compact object of \(D(R)\) whereupon we conclude that \(M_i\) does too as desired.
Lemma
Let \(k\) be a field. Let \(n \geq 0\). Let \(K \in D_\QCoh(\mathcal{O}_{\mathbf{P}^n_k})\). The following are equivalent
\(K\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^n_k})\),
\(\sum_{i \in \mathbf{Z}} \dim_k H^i(\mathbf{P}^n_k, E \otimes^\mathbf{L} K) < \infty\) for each perfect object \(E\) of \(D(\mathcal{O}_{\mathbf{P}^n_k})\),
\(\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_{\mathbf{P}^n_k}(E, K) < \infty\) for each perfect object \(E\) of \(D(\mathcal{O}_{\mathbf{P}^n_k})\),
\(\sum_{i \in \mathbf{Z}} \dim_k H^i(\mathbf{P}^n_k, K \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_k}(d)) < \infty\) for \(d = 0, 1, \ldots, n\).
Proof
Parts (2) and (3) are equivalent by Cohomology, Lemma 08DQ. If (1) is true, then for \(E\) perfect the derived tensor product \(E \otimes^\mathbf{L} K\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^n_k})\) and we see that (2) holds by Derived Categories of Schemes, Lemma 08E2. It is clear that (2) implies (4) as \(\mathcal{O}_{\mathbf{P}^n_k}(d)\) can be viewed as a perfect object of the derived category of \(\mathbf{P}^n_k\). Thus it suffices to prove that (4) implies (1).
Assume (4). Let \(R\) be as in Lemma 0FYA. Let \(P = \bigoplus_{d = 0, \ldots, n} \mathcal{O}_{\mathbf{P}^n_k}(-d)\). Recall that \(R = \text{End}_{\mathbf{P}^n_k}(P)\) whereas all other self-Exts of \(P\) are zero and that \(P\) determines an equivalence \(- \otimes^\mathbf{L} P : D(R) \to D_\QCoh(\mathcal{O}_{\mathbf{P}^n_k})\) by Derived Categories of Schemes, Lemma 0BQU. Say \(K\) corresponds to \(L\) in \(D(R)\). Then \[\begin{align*} H^i(L) & = \Ext^i_{D(R)}(R, L) \\ & = \Ext^i_{\mathbf{P}^n_k}(P, K) \\ & = H^i(\mathbf{P}^n_k, K \otimes P^\vee) \\ & = \bigoplus\nolimits_{d = 0, \ldots, n} H^i(\mathbf{P}^n_k, K \otimes \mathcal{O}(d)) \end{align*}\] by Differential Graded Algebra, Lemma 0CS6 (and the fact that \(- \otimes^\mathbf{L} P\) is an equivalence) and Cohomology, Lemma 08DQ. Thus our assumption (4) implies that \(L\) satisfies condition (2) of Lemma 0FYA and hence is a compact object of \(D(R)\). Therefore \(K\) is a compact object of \(D_\QCoh(\mathcal{O}_{\mathbf{P}^n_k})\). Thus \(K\) is perfect by Derived Categories of Schemes, Proposition 09M1. Since \(D_{perf}(\mathcal{O}_{\mathbf{P}^n_k}) = D^b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^n_k})\) by Derived Categories of Schemes, Lemma 0FDC we conclude (1) holds.
Lemma
Let \(X\) be a scheme proper over a field \(k\). Let \(K \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) and let \(E\) in \(D(\mathcal{O}_X)\) be perfect. Then \(\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty\).
Proof
This follows for example by combining Derived Categories of Schemes, Lemmas 0D0D and 09M4. Alternative proof: combine Derived Categories of Schemes, Lemmas 0FXU and 08E2.
Lemma
Let \(X\) be a proper scheme over a field \(k\). Let \(K \in \Ob(D_\QCoh(\mathcal{O}_X))\). The following are equivalent
\(K \in D^b_{\textit{Coh}}(\mathcal{O}_X)\), and
\(\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty\) for all perfect \(E\) in \(D(\mathcal{O}_X)\).
Proof
The implication (1) \(\Rightarrow\) (2) follows from Lemma 0FYC. The implication (2) \(\Rightarrow\) (1) follows from More on Morphisms, Lemma 0GET (see Derived Categories of Schemes, Example 0DI1 for the meaning of a relatively perfect object over a field); the easier proof in the projective case is in the next paragraph.
Assume (2) and \(X\) projective over \(k\). Choose a closed immersion \(i : X \to \mathbf{P}^n_k\). It suffices to show that \(Ri_*K\) is in \(D^b_{\textit{Coh}}(\mathbf{P}^n_k)\) since a quasi-coherent module \(\mathcal{F}\) on \(X\) is coherent, resp. zero if and only if \(i_*\mathcal{F}\) is coherent, resp. zero. For a perfect object \(E\) of \(D(\mathcal{O}_{\mathbf{P}^n_k})\), \(Li^*E\) is a perfect object of \(D(\mathcal{O}_X)\) and \[\Ext^q_{\mathbf{P}^n_k}(E, Ri_*K) = \Ext^q_X(Li^*E, K)\] Hence by our assumption we see that \(\sum_{q \in \mathbf{Z}} \dim_k \Ext^q_{\mathbf{P}^n_k}(E, Ri_*K) < \infty\). We conclude by Lemma 0FYB.
A representability theorem
The material in this section is taken from [BvdB].
Let \(\mathcal{T}\) be a \(k\)-linear triangulated category. In this section we consider \(k\)-linear cohomological functors \(H\) from \(\mathcal{T}\) to the category of \(k\)-vector spaces. This will mean \(H\) is a functor \[H : \mathcal{T}^{opp} \longrightarrow \text{Vect}_k\] which is \(k\)-linear such that for any distinguished triangle \(X \to Y \to Z\) in \(\mathcal{T}\) the sequence \(H(Z) \to H(Y) \to H(X)\) is an exact sequence of \(k\)-vector spaces. See Derived Categories, Definition 0147 and Differential Graded Algebra, Section 09MI.
Lemma
Let \(\mathcal{D}\) be a triangulated category. Let \(\mathcal{D}' \subset \mathcal{D}\) be a full triangulated subcategory. Let \(X \in \Ob(\mathcal{D})\). The category of arrows \(E \to X\) with \(E \in \Ob(\mathcal{D}')\) is filtered.
Proof
We check the conditions of Categories, Definition 002V. The category is nonempty because it contains \(0 \to X\). If \(E_i \to X\), \(i = 1, 2\) are objects, then \(E_1 \oplus E_2 \to X\) is an object and there are morphisms \((E_i \to X) \to (E_1 \oplus E_2 \to X)\). Finally, suppose that \(a, b : (E \to X) \to (E' \to X)\) are morphisms. Choose a distinguished triangle \(E \xrightarrow{a - b} E' \to E''\) in \(\mathcal{D}'\). By Axiom TR3 we obtain a morphism of triangles \[\xymatrix{ E \ar[r]_{a - b} \ar[d] & E' \ar[d] \ar[r] & E'' \ar[d] \\ 0 \ar[r] & X \ar[r] & X }\] and we find that the resulting arrow \((E' \to X) \to (E'' \to X)\) equalizes \(a\) and \(b\).
Lemma
Let \(k\) be a field. Let \(\mathcal{D}\) be a \(k\)-linear triangulated category which has direct sums and is compactly generated. Denote \(\mathcal{D}_c\) the full subcategory of compact objects. Let \(H : \mathcal{D}_c^{opp} \to \text{Vect}_k\) be a \(k\)-linear cohomological functor such that \(\dim_k H(X) < \infty\) for all \(X \in \Ob(\mathcal{D}_c)\). Then \(H\) is isomorphic to the functor \(X \mapsto \Hom(X, Y)\) for some \(Y \in \Ob(\mathcal{D})\).
Proof
We will use Derived Categories, Lemma 09QH without further mention. Denote \(G : \mathcal{D}_c \to \text{Vect}_k\) the \(k\)-linear homological functor which sends \(X\) to \(H(X)^\vee\). For any object \(Y\) of \(\mathcal{D}\) we set \[G'(Y) = \colim_{X \to Y, X \in \Ob(\mathcal{D}_c)} G(X)\] The colimit is filtered by Lemma 0FYF. We claim that \(G'\) is a \(k\)-linear homological functor, the restriction of \(G'\) to \(\mathcal{D}_c\) is \(G\), and \(G'\) sends direct sums to direct sums.
Namely, suppose that \(Y_1 \to Y_2 \to Y_3\) is a distinguished triangle. Let \(\xi \in G'(Y_2)\) map to zero in \(G'(Y_3)\). Since the colimit is filtered \(\xi\) is represented by some \(X \to Y_2\) with \(X \in \Ob(\mathcal{D}_c)\) and \(g \in G(X)\). The fact that \(\xi\) maps to zero in \(G'(Y_3)\) means the composition \(X \to Y_2 \to Y_3\) factors as \(X \to X' \to Y_3\) with \(X' \in \mathcal{D}_c\) and \(g\) mapping to zero in \(G(X')\). Choose a distinguished triangle \(X'' \to X \to X'\). Then \(X'' \in \Ob(\mathcal{D}_c)\). Since \(G\) is homological we find that \(g\) is the image of some \(g'' \in G'(X'')\). By Axiom TR3 the maps \(X \to Y_2\) and \(X' \to Y_3\) fit into a morphism of distinguished triangles \((X'' \to X \to X') \to (Y_1 \to Y_2 \to Y_3)\) and we find that indeed \(\xi\) is the image of the element of \(G'(Y_1)\) represented by \(X'' \to Y_1\) and \(g'' \in G(X'')\).
If \(Y \in \Ob(\mathcal{D}_c)\), then \(\text{id} : Y \to Y\) is the final object in the category of arrows \(X \to Y\) with \(X \in \Ob(\mathcal{D}_c)\). Hence we see that \(G'(Y) = G(Y)\) in this case and the statement on restriction holds. Let \(Y = \bigoplus_{i \in I} Y_i\) be a direct sum. Let \(a : X \to Y\) with \(X \in \Ob(\mathcal{D}_c)\) and \(g \in G(X)\) represent an element \(\xi\) of \(G'(Y)\). The morphism \(a : X \to Y\) can be uniquely written as a sum of morphisms \(a_i : X \to Y_i\) almost all zero as \(X\) is a compact object of \(\mathcal{D}\). Let \(I' = \{i \in I \mid a_i \not = 0\}\). Then we can factor \(a\) as the composition \[X \xrightarrow{(1, \ldots, 1)} \bigoplus\nolimits_{i \in I'} X \xrightarrow{\bigoplus_{i \in I'} a_i} \bigoplus\nolimits_{i \in I} Y_i = Y\] We conclude that \(\xi = \sum_{i \in I'} \xi_i\) is the sum of the images of the elements \(\xi_i \in G'(Y_i)\) corresponding to \(a_i : X \to Y_i\) and \(g \in G(X)\). Hence \(\bigoplus G'(Y_i) \to G'(Y)\) is surjective. We omit the (trivial) verification that it is injective.
It follows that the functor \(Y \mapsto G'(Y)^\vee\) is cohomological and sends direct sums to direct products. Hence by Brown representability, see Derived Categories, Proposition 0A8G we conclude that there exists a \(Y \in \Ob(\mathcal{D})\) and an isomorphism \(G'(Z)^\vee = \Hom(Z, Y)\) functorially in \(Z\). For \(X \in \Ob(\mathcal{D}_c)\) we have \(G'(X)^\vee = G(X)^\vee = (H(X)^\vee)^\vee = H(X)\) because \(\dim_k H(X) < \infty\) and the proof is complete.
Theorem
Let \(X\) be a proper scheme over a field \(k\). Let \(F : D_{perf}(\mathcal{O}_X)^{opp} \to \text{Vect}_k\) be a \(k\)-linear cohomological functor such that \[\sum\nolimits_{n \in \mathbf{Z}} \dim_k F(E[n]) < \infty\] for all \(E \in D_{perf}(\mathcal{O}_X)\). Then \(F\) is isomorphic to a functor of the form \(E \mapsto \Hom_X(E, K)\) for some \(K \in D^b_{\textit{Coh}}(\mathcal{O}_X)\).
Proof
The derived category \(D_\QCoh(\mathcal{O}_X)\) has direct sums, is compactly generated, and \(D_{perf}(\mathcal{O}_X)\) is the full subcategory of compact objects, see Derived Categories of Schemes, Lemma 08DT, Theorem 09IS, and Proposition 09M1. By Lemma 0FYG we may assume \(F(E) = \Hom_X(E, K)\) for some \(K \in \Ob(D_\QCoh(\mathcal{O}_X))\). Then it follows that \(K\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) by Lemma 0FYD.
Lemma
Let \(X\) be a proper scheme over a field \(k\) which is regular. Let \(G : D_{perf}(\mathcal{O}_X) \to \text{Vect}_k\) be a \(k\)-linear homological functor such that \[\sum\nolimits_{n \in \mathbf{Z}} \dim_k G(E[n]) < \infty\] for all \(E \in D_{perf}(\mathcal{O}_X)\). Then \(G\) is isomorphic to a functor of the form \(E \mapsto \Hom_X(K, E)\) for some \(K \in D_{perf}(\mathcal{O}_X)\).
Proof
Consider the contravariant functor \(E \mapsto E^\vee\) on \(D_{perf}(\mathcal{O}_X)\), see Cohomology, Lemma 08DQ. This functor is an exact anti-self-equivalence of \(D_{perf}(\mathcal{O}_X)\). Hence we may apply Theorem 0FYH to the functor \(F(E) = G(E^\vee)\) to find \(K \in D_{perf}(\mathcal{O}_X)\) such that \(G(E^\vee) = \Hom_X(E, K)\). It follows that \(G(E) = \Hom_X(E^\vee, K) = \Hom_X(K^\vee, E)\) and we conclude that taking \(K^\vee\) works.
Existence of adjoints
As a consequence of the results in the paper of Bondal and van den Bergh we get the following automatic existence of adjoints.
Lemma
Let \(k\) be a field. Let \(X\) and \(Y\) be proper schemes over \(k\). If \(X\) is regular, then any \(k\)-linear exact functor \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) has an exact right adjoint and an exact left adjoint.
Proof
If an adjoint exists it is an exact functor by the very general Derived Categories, Lemma 0A8D.
Let us prove the existence of a right adjoint. To see existence, it suffices to show that for \(M \in D_{perf}(\mathcal{O}_Y)\) the contravariant functor \(K \mapsto \Hom_Y(F(K), M)\) is representable. This functor is contravariant, \(k\)-linear, and cohomological. Hence by Theorem 0FYH it suffices to show that \[\sum\nolimits_{i \in \mathbf{Z}} \dim_k \Ext^i_Y(F(K), M) < \infty\] This follows from Lemma 0FYC.
For the existence of the left adjoint we argue in the same manner using Lemma 0H4A in stead of Theorem 0FYH.
Fourier-Mukai functors
These functors were first introduced in [Mukai].
Definition
Let \(S\) be a scheme. Let \(X\) and \(Y\) be schemes over \(S\). Let \(K \in D(\mathcal{O}_{X \times_S Y})\). The exact functor \[\Phi_K : D(\mathcal{O}_X) \longrightarrow D(\mathcal{O}_Y),\quad M \longmapsto R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K)\] of triangulated categories is called a Fourier-Mukai functor and \(K\) is called a Fourier-Mukai kernel for this functor. Moreover,
if \(\Phi_K\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\) then the resulting exact functor \(\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) is called a Fourier-Mukai functor,
if \(\Phi_K\) sends \(D_{perf}(\mathcal{O}_X)\) into \(D_{perf}(\mathcal{O}_Y)\) then the resulting exact functor \(\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) is called a Fourier-Mukai functor, and
if \(X\) and \(Y\) are Noetherian and \(\Phi_K\) sends \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) into \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\) then the resulting exact functor \(\Phi_K : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)\) is called a Fourier-Mukai functor. Similarly for \(D_{\textit{Coh}}\), \(D^+_{\textit{Coh}}\), \(D^-_{\textit{Coh}}\).
Lemma
Let \(S\) be a scheme. Let \(X\) and \(Y\) be schemes over \(S\). Let \(K \in D(\mathcal{O}_{X \times_S Y})\). The corresponding Fourier-Mukai functor \(\Phi_K\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\) if \(K\) is in \(D_\QCoh(\mathcal{O}_{X \times_S Y})\) and \(X \to S\) is quasi-compact and quasi-separated.
Proof
This follows from the fact that derived pullback preserves \(D_\QCoh\) (Derived Categories of Schemes, Lemma 08DW), derived tensor products preserve \(D_\QCoh\) (Derived Categories of Schemes, Lemma 08DX), the projection \(\text{pr}_2 : X \times_S Y \to Y\) is quasi-compact and quasi-separated (Schemes, Lemmas 01K5 and 01KU), and total direct image along a quasi-separated and quasi-compact morphism preserves \(D_\QCoh\) (Derived Categories of Schemes, Lemma 08D5).
Lemma
Let \(S\) be a scheme. Let \(X, Y, Z\) be schemes over \(S\). Assume \(X \to S\), \(Y \to S\), and \(Z \to S\) are quasi-compact and quasi-separated. Let \(K \in D_\QCoh(\mathcal{O}_{X \times_S Y})\). Let \(K' \in D_\QCoh(\mathcal{O}_{Y \times_S Z})\). Consider the Fourier-Mukai functors \(\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) and \(\Phi_{K'} : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_Z)\). If \(X\) and \(Z\) are tor independent over \(S\) and \(Y \to S\) is flat, then \[\Phi_{K'} \circ \Phi_K = \Phi_{K''} : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(\mathcal{O}_Z)\] where \[K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K')\] in \(D_\QCoh(\mathcal{O}_{X \times_S Z})\).
Proof
The statement makes sense by Lemma 0FYR. We are going to use Derived Categories of Schemes, Lemmas 08DW, 08DX, and 08D5 and Schemes, Lemmas 01K5 and 01KU without further mention. By Derived Categories of Schemes, Lemma 0FXW we see that \(X \times_S Y\) and \(Y \times_S Z\) are tor independent over \(Y\). This means that we have base change for the cartesian diagram \[\xymatrix{ X \times_S Y \times_S Z \ar[d] \ar[r] & Y \times_S Z \ar[d]^{p^{YZ}_Y} \\ X \times_S Y \ar[r]^{p^{XY}_Y} & Y }\] for complexes with quasi-coherent cohomology sheaves, see Derived Categories of Schemes, Lemma 08IB. Abbreviating \(p^* = Lp^*\), \(p_* = Rp_*\) and \(\otimes = \otimes^\mathbf{L}\) we have for \(M \in D_\QCoh(\mathcal{O}_X)\) the sequence of equalities \[\begin{align*} \Phi_{K'}(\Phi_K(M)) & = p^{YZ}_{Z, *}(p^{YZ, *}_Y p^{XY}_{Y, *}(p^{XY, *}_X M \otimes K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *} \text{pr}_{12}^*(p^{XY, *}_X M \otimes K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K')) \\ & = \text{pr}_{3, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K') \\ & = p^{XZ}_{Z, *}\text{pr}_{13, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K') \\ & = p^{XZ}_{Z, *} (p^{XZ, *}_X M \otimes \text{pr}_{13, *}(\text{pr}_{12}^*K \otimes \text{pr}_{23}^*K')) \end{align*}\] as desired. Here we have used the remark on base change in the second equality and we have use Derived Categories of Schemes, Lemma 08EU in the \(4\)th and last equality.
Lemma
Let \(S\) be a scheme. Let \(X\) and \(Y\) be schemes over \(S\). Let \(K \in D(\mathcal{O}_{X \times_S Y})\). The corresponding Fourier-Mukai functor \(\Phi_K\) sends \(D_{perf}(\mathcal{O}_X)\) into \(D_{perf}(\mathcal{O}_Y)\) if at least one of the following conditions is satisfied:
\(S\) is Noetherian, \(X \to S\) and \(Y \to S\) are of finite type, \(K \in D^b_{\textit{Coh}}(\mathcal{O}_{X \times_S Y})\), the support of \(H^i(K)\) is proper over \(Y\) for all \(i\), and \(K\) has finite tor dimension as an object of \(D(\text{pr}_2^{-1}\mathcal{O}_Y)\),
\(X \to S\) is of finite presentation and \(K\) can be represented by a bounded complex \(\mathcal{K}^\bullet\) of finitely presented \(\mathcal{O}_{X \times_S Y}\)-modules, flat over \(Y\), with support proper over \(Y\),
\(X \to S\) is a proper flat morphism of finite presentation and \(K\) is perfect,
\(S\) is Noetherian, \(X \to S\) is flat and proper, and \(K\) is perfect
\(X \to S\) is a proper flat morphism of finite presentation and \(K\) is \(Y\)-perfect,
\(S\) is Noetherian, \(X \to S\) is flat and proper, and \(K\) is \(Y\)-perfect.
Proof
If \(M\) is perfect on \(X\), then \(L\text{pr}_1^*M\) is perfect on \(X \times_S Y\), see Cohomology, Lemma 09UA. We will use this without further mention below. We will also use that if \(X \to S\) is of finite type, or proper, or flat, or of finite presentation, then the same thing is true for the base change \(\text{pr}_2 : X \times_S Y \to Y\), see Morphisms, Lemmas 01T4, 01W4, 01U9, and 01TS.
Part (1) follows from Derived Categories of Schemes, Lemma 08EV combined with Derived Categories of Schemes, Lemma 0FXU.
Part (2) follows from Derived Categories of Schemes, Lemma 0A1H.
Part (3) follows from Derived Categories of Schemes, Lemma 0B91.
Part (4) follows from part (3) and the fact that a finite type morphism of Noetherian schemes is of finite presentation by Morphisms, Lemma 01TX.
Part (5) follows from Derived Categories of Schemes, Lemma 0DJT combined with Derived Categories of Schemes, Lemma 0DI4.
Part (6) follows from part (5) in the same way that part (4) follows from part (3).
Lemma
Let \(S\) be a Noetherian scheme. Let \(X\) and \(Y\) be schemes of finite type over \(S\). Let \(K \in D^b_{\textit{Coh}}(\mathcal{O}_{X \times_S Y})\). The corresponding Fourier-Mukai functor \(\Phi_K\) sends \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) into \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\) if at least one of the following conditions is satisfied:
the support of \(H^i(K)\) is proper over \(Y\) for all \(i\), and \(K\) has finite tor dimension as an object of \(D(\text{pr}_1^{-1}\mathcal{O}_X)\),
\(K\) can be represented by a bounded complex \(\mathcal{K}^\bullet\) of coherent \(\mathcal{O}_{X \times_S Y}\)-modules, flat over \(X\), with support proper over \(Y\),
the support of \(H^i(K)\) is proper over \(Y\) for all \(i\) and \(X\) is a regular scheme,
\(K\) is perfect, the support of \(H^i(K)\) is proper over \(Y\) for all \(i\), and \(Y \to S\) is flat.
Furthermore in each case the support condition is automatic if \(X \to S\) is proper.
Proof
Let \(M\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). In each case we will use Derived Categories of Schemes, Lemma 08E2 to show that \[\Phi_K(M) = R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K)\] is in \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\). The derived tensor product \(L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K\) is a pseudo-coherent object of \(D(\mathcal{O}_{X \times_S Y})\) (by Cohomology, Lemma 09U7, Derived Categories of Schemes, Lemma 08E8, and Cohomology, Lemma 09J3) whence has coherent cohomology sheaves (by Derived Categories of Schemes, Lemma 08E8 again). In each case the supports of the cohomology sheaves \(H^i(L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K)\) is proper over \(Y\) as these supports are contained in the union of the supports of the \(H^i(K)\). Hence in each case it suffices to prove that this tensor product is bounded below.
Case (1). By Cohomology, Lemma 08DE we have \[L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K \cong \text{pr}_1^{-1}M \otimes_{\text{pr}_1^{-1}\mathcal{O}_X}^\mathbf{L} K\] with obvious notation. Hence the assumption on tor dimension and the fact that \(M\) has only a finite number of nonzero cohomology sheaves, implies the bound we want.
Case (2) follows because here the assumption implies that \(K\) has finite tor dimension as an object of \(D(\text{pr}_1^{-1}\mathcal{O}_X)\) hence the argument in the previous paragraph applies.
In Case (3) it is also the case that \(K\) has finite tor dimension as an object of \(D(\text{pr}_1^{-1}\mathcal{O}_X)\). Namely, choose affine opens \(U = \Spec(A)\) and \(V = \Spec(B)\) of \(X\) and \(Y\) mapping into the affine open \(W = \Spec(R)\) of \(S\). Then \(K|_{U \times V}\) is given by a bounded complex of finite \(A \otimes_R B\)-modules \(M^\bullet\). Since \(A\) is a regular ring of finite dimension we see that each \(M^i\) has finite projective dimension as an \(A\)-module (Algebra, Lemma 00OE) and hence finite tor dimension as an \(A\)-module. Thus \(M^\bullet\) has finite tor dimension as a complex of \(A\)-modules (More on Algebra, Lemma 066H). Since \(X \times Y\) is quasi-compact we conclude there exist \([a, b]\) such that for every point \(z \in X \times Y\) the stalk \(K_z\) has tor amplitude in \([a, b]\) over \(\mathcal{O}_{X, \text{pr}_1(z)}\). This implies \(K\) has bounded tor dimension as an object of \(D(\text{pr}_1^{-1}\mathcal{O}_X)\), see Cohomology, Lemma 09U9. We conclude as in the previous to paragraphs.
Case (4). With notation as above, the ring map \(R \to B\) is flat. Hence the ring map \(A \to A \otimes_R B\) is flat. Hence any projective \(A \otimes_R B\)-module is \(A\)-flat. Thus any perfect complex of \(A \otimes_R B\)-modules has finite tor dimension as a complex of \(A\)-modules and we conclude as before.
Example
Let \(X \to S\) be a separated morphism of schemes. Then the diagonal \(\Delta : X \to X \times_S X\) is a closed immersion and hence \(\mathcal{O}_\Delta = \Delta_*\mathcal{O}_X = R\Delta_*\mathcal{O}_X\) is a quasi-coherent \(\mathcal{O}_{X \times_S X}\)-module of finite type which is flat over \(X\) (under either projection). The Fourier-Mukai functor \(\Phi_{\mathcal{O}_\Delta}\) is equal to the identity in this case. Namely, for any \(M \in D(\mathcal{O}_X)\) we have \[\begin{align*} L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S X}}^\mathbf{L} \mathcal{O}_\Delta & = L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S X}}^\mathbf{L} R\Delta_*\mathcal{O}_X \\ & = R\Delta_*( L\Delta^*L\text{pr}_1^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X) \\ & = R\Delta_*(M) \end{align*}\] The first equality we discussed above. The second equality is Cohomology, Lemma 0B55. The third because \(\text{pr}_1 \circ \Delta = \text{id}_X\) and we have Cohomology, Lemma 0D5S. If we push this to \(X\) using \(R\text{pr}_{2, *}\) we obtain \(M\) by Cohomology, Lemma 0D5T and the fact that \(\text{pr}_2 \circ \Delta = \text{id}_X\).
Lemma
Let \(X \to S\) and \(Y \to S\) be morphisms of quasi-compact and quasi-separated schemes. Let \(\Phi : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) be a Fourier-Mukai functor with pseudo-coherent kernel \(K \in D_\QCoh(\mathcal{O}_{X \times_S Y})\). Let \(a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_{X \times_S Y})\) be the right adjoint to \(R\text{pr}_{2, *}\), see Duality for Schemes, Lemma 0A9E. Denote \[K' = (Y \times_S X \to X \times_S Y)^* R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y)) \in D_\QCoh(\mathcal{O}_{Y \times_S X})\] and denote \(\Phi' : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)\) the corresponding Fourier-Mukai transform. There is a canonical map \[\Hom_X(M, \Phi'(N)) \longrightarrow \Hom_Y(\Phi(M), N)\] functorial in \(M\) in \(D_\QCoh(\mathcal{O}_X)\) and \(N\) in \(D_\QCoh(\mathcal{O}_Y)\) which is an isomorphism if
\(N\) is perfect, or
\(K\) is perfect and \(X \to S\) is proper flat and of finite presentation.
Proof
By Lemma 0FYR we obtain a functor \(\Phi\) as in the statement. Observe that \(a(\mathcal{O}_Y)\) is in \(D^+_\QCoh(\mathcal{O}_{X \times_S Y})\) by Duality for Schemes, Lemma 0A9I. Hence for \(K\) pseudo-coherent we have \(K' \in D_\QCoh(\mathcal{O}_{Y \times_S X})\) by Derived Categories of Schemes, Lemma 0A6H we we obtain \(\Phi'\) as indicated.
We abbreviate \(\otimes^\mathbf{L} = \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L}\) and \(\SheafHom = R\SheafHom_{\mathcal{O}_{X \times_S Y}}\). Let \(M\) be in \(D_\QCoh(\mathcal{O}_X)\) and let \(N\) be in \(D_\QCoh(\mathcal{O}_Y)\). We have \[\begin{align*} \Hom_Y(\Phi(M), N) & = \Hom_Y(R\text{pr}_{2, *}(L\text{pr}_1^*M \otimes^\mathbf{L} K), N) \\ & = \Hom_{X \times_S Y}(L\text{pr}_1^*M \otimes^\mathbf{L} K, a(N)) \\ & = \Hom_{X \times_S Y}(L\text{pr}_1^*M, R\SheafHom(K, a(N))) \\ & = \Hom_X(M, R\text{pr}_{1, *}R\SheafHom(K, a(N))) \end{align*}\] where we have used Cohomology, Lemmas 08DJ and 079W. There are canonical maps \[L\text{pr}_2^*N \otimes^\mathbf{L} R\SheafHom(K, a(\mathcal{O}_Y)) \xrightarrow{\alpha} R\SheafHom(K, L\text{pr}_2^*N \otimes^\mathbf{L} a(\mathcal{O}_Y)) \xrightarrow{\beta} R\SheafHom(K, a(N))\] Here \(\alpha\) is Cohomology, Lemma 0BYS and \(\beta\) is Duality for Schemes, Equation (0A9S). Combining all of these arrows we obtain the functorial displayed arrow in the statement of the lemma.
The arrow \(\alpha\) is an isomorphism by Derived Categories of Schemes, Lemma 0ATN as soon as either \(K\) or \(N\) is perfect. The arrow \(\beta\) is an isomorphism if \(N\) is perfect by Duality for Schemes, Lemma 0A9T or in general if \(X \to S\) is flat proper of finite presentation by Duality for Schemes, Lemma 0E4K.
Lemma
Let \(S\) be a Noetherian scheme. Let \(Y \to S\) be a flat proper Gorenstein morphism and let \(X \to S\) be a finite type morphism. Denote \(\omega^\bullet_{Y/S}\) the relative dualizing complex of \(Y\) over \(S\). Let \(\Phi : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) be a Fourier-Mukai functor with perfect kernel \(K \in D_\QCoh(\mathcal{O}_{X \times_S Y})\). Denote \[K' = (Y \times_S X \to X \times_S Y)^*(K^\vee \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} L\text{pr}_2^*\omega^\bullet_{Y/S}) \in D_\QCoh(\mathcal{O}_{Y \times_S X})\] and denote \(\Phi' : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)\) the corresponding Fourier-Mukai transform. There is a canonical isomorphism \[\Hom_Y(N, \Phi(M)) \longrightarrow \Hom_X(\Phi'(N), M)\] functorial in \(M\) in \(D_\QCoh(\mathcal{O}_X)\) and \(N\) in \(D_\QCoh(\mathcal{O}_Y)\).
Proof
By Lemma 0FYR we obtain a functor \(\Phi\) as in the statement.
Observe that formation of the relative dualizing complex commutes with base change in our setting, see Duality for Schemes, Remark 0B6S. Thus \(L\text{pr}_2^*\omega^\bullet_{Y/S} = \omega^\bullet_{X \times_S Y/X}\). Moreover, we observe that \(\omega^\bullet_{Y/S}\) is an invertible object of the derived category, see Duality for Schemes, Lemma 0C08, and a fortiori perfect.
To actually prove the lemma we’re going to cheat. Namely, we will show that if we replace the roles of \(X\) and \(Y\) and \(K\) and \(K'\) then these are as in Lemma 0FYW and we get the result. It is clear that \(K'\) is perfect as a tensor product of perfect objects so that the discussion in Lemma 0FYW applies to it. To show that the procedure of Lemma 0FYW applied to \(K'\) on \(Y \times_S X\) produces a complex isomorphic to \(K\) it suffices (details omitted) to show that \[R\SheafHom(R\SheafHom(K, \omega^\bullet_{X \times_S Y/X}), \omega^\bullet_{X \times_S Y/X}) = K\] This is clear because \(K\) is perfect and \(\omega^\bullet_{X \times_S Y/X}\) is invertible; details omitted. Thus Lemma 0FYW produces a map \[\Hom_Y(N, \Phi(M)) \longrightarrow \Hom_X(\Phi'(N), M)\] functorial in \(M\) in \(D_\QCoh(\mathcal{O}_X)\) and \(N\) in \(D_\QCoh(\mathcal{O}_Y)\) which is an isomorphism because \(K'\) is perfect. This finishes the proof.
Lemma
Let \(S\) be a Noetherian scheme.
For \(X\), \(Y\) proper and flat over \(S\) and \(K\) in \(D_{perf}(\mathcal{O}_{X \times_S Y})\) we obtain a Fourier-Mukai functor \(\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\).
For \(X\), \(Y\), \(Z\) proper and flat over \(S\), \(K \in D_{perf}(\mathcal{O}_{X \times_S Y})\), \(K' \in D_{perf}(\mathcal{O}_{Y \times_S Z})\) the composition \(\Phi_{K'} \circ \Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Z)\) is equal to \(\Phi_{K''}\) with \(K'' \in D_{perf}(\mathcal{O}_{X \times_S Z})\) computed as in Lemma 0FYS,
For \(X\), \(Y\), \(K\), \(\Phi_K\) as in (1) if \(X \to S\) is Gorenstein, then \(\Phi_{K'} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_X)\) is a right adjoint to \(\Phi_K\) where \(K' \in D_{perf}(\mathcal{O}_{Y \times_S X})\) is the pullback of \(L\text{pr}_1^*\omega_{X/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee\) by \(Y \times_S X \to X \times_S Y\).
For \(X\), \(Y\), \(K\), \(\Phi_K\) as in (1) if \(Y \to S\) is Gorenstein, then \(\Phi_{K''} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_X)\) is a left adjoint to \(\Phi_K\) where \(K'' \in D_{perf}(\mathcal{O}_{Y \times_S X})\) is the pullback of \(L\text{pr}_2^*\omega_{Y/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee\) by \(Y \times_S X \to X \times_S Y\).
Proof
Part (1) is immediate from Lemma 0FYT part (4).
Part (2) follows from Lemma 0FYS and the fact that \(K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K')\) is perfect for example by Derived Categories of Schemes, Lemma 0B6F.
The adjointness in part (3) on all complexes with quasi-coherent cohomology sheaves follows from Lemma 0FYW with \(K'\) equal to the pullback of \(R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y))\) by \(Y \times_S X \to X \times_S Y\) where \(a\) is the right adjoint to \(R\text{pr}_{2, *} : D_\QCoh(\mathcal{O}_{X \times_S Y}) \to D_\QCoh(\mathcal{O}_Y)\). Denote \(f : X \to S\) the structure morphism of \(X\). Since \(f\) is proper the functor \(f^! : D_\QCoh^+(\mathcal{O}_S) \to D_\QCoh^+(\mathcal{O}_X)\) is the restriction to \(D_\QCoh^+(\mathcal{O}_S)\) of the right adjoint to \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)\), see Duality for Schemes, Section 0A9Y. Hence the relative dualizing complex \(\omega_{X/S}^\bullet\) as defined in Duality for Schemes, Remark 0B6S is equal to \(\omega_{X/S}^\bullet = f^!\mathcal{O}_S\). Since formation of the relative dualizing complex commutes with base change (see Duality for Schemes, Remark 0B6S) we see that \(a(\mathcal{O}_Y) = L\text{pr}_1^*\omega_{X/S}^\bullet\). Thus \[R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y)) \cong L\text{pr}_1^*\omega_{X/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee\] by Cohomology, Lemma 08DQ. Finally, since \(X \to S\) is assumed Gorenstein the relative dualizing complex is invertible: this follows from Duality for Schemes, Lemma 0C08. We conclude that \(\omega_{X/S}^\bullet\) is perfect (Cohomology, Lemma 0FPG) and hence \(K'\) is perfect. Therefore \(\Phi_{K'}\) does indeed map \(D_{perf}(\mathcal{O}_Y)\) into \(D_{perf}(\mathcal{O}_X)\) which finishes the proof of (3).
The proof of (4) is the same as the proof of (3) except one uses Lemma 0FYX instead of Lemma 0FYW.
Resolutions and bounds
The diagonal of a smooth proper scheme has a nice resolution.
Lemma
Let \(R\) be a Noetherian ring. Let \(X\), \(Y\) be finite type schemes over \(R\) having the resolution property. For any coherent \(\mathcal{O}_{X \times_R Y}\)-module \(\mathcal{F}\) there exist a surjection \(\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}\) where \(\mathcal{E}\) is a finite locally free \(\mathcal{O}_X\)-module and \(\mathcal{G}\) is a finite locally free \(\mathcal{O}_Y\)-module.
Proof
Let \(U \subset X\) and \(V \subset Y\) be affine open subschemes. Let \(\mathcal{I} \subset \mathcal{O}_X\) be the ideal sheaf of the reduced induced closed subscheme structure on \(X \setminus U\). Similarly, let \(\mathcal{I}' \subset \mathcal{O}_Y\) be the ideal sheaf of the reduced induced closed subscheme structure on \(Y \setminus V\). Then the ideal sheaf \[\mathcal{J} = \Im(\text{pr}_1^*\mathcal{I} \otimes_{\mathcal{O}_{X \times_R Y}} \text{pr}_2^*\mathcal{I}' \to \mathcal{O}_{X \times_R Y})\] satisfies \(V(\mathcal{J}) = X \times_R Y \setminus U \times_R V\). For any section \(s \in \mathcal{F}(U \times_R V)\) we can find an integer \(n > 0\) and a map \(\mathcal{J}^n \to \mathcal{F}\) whose restriction to \(U \times_R V\) gives \(s\), see Cohomology of Schemes, Lemma 01YB. By assumption we can choose surjections \(\mathcal{E} \to \mathcal{I}\) and \(\mathcal{G} \to \mathcal{I}'\). These produce corresponding surjections \[\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{J} \quad\text{and}\quad \mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{J}^n\] and hence a map \(\mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{F}\) whose image contains the section \(s\) over \(U \times_R V\). Since we can cover \(X \times_R Y\) by a finite number of affine opens of the form \(U \times_R V\) and since \(\mathcal{F}|_{U \times_R V}\) is generated by finitely many sections (Properties, Lemma 01PB) we conclude that there exists a surjection \[\bigoplus\nolimits_{j = 1, \ldots, N} \mathcal{E}_j^{\otimes n_j} \boxtimes \mathcal{G}_j^{\otimes n_j} \to \mathcal{F}\] where \(\mathcal{E}_j\) is finite locally free on \(X\) and \(\mathcal{G}_j\) is finite locally free on \(Y\). Setting \(\mathcal{E} = \bigoplus \mathcal{E}_j^{\otimes n_j}\) and \(\mathcal{G} = \bigoplus \mathcal{G}_j^{\otimes n_j}\) we conclude that the lemma is true.
Lemma
Let \(R\) be a ring. Let \(X\), \(Y\) be quasi-compact and quasi-separated schemes over \(R\) having the resolution property. For any finite type quasi-coherent \(\mathcal{O}_{X \times_R Y}\)-module \(\mathcal{F}\) there exist a surjection \(\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}\) where \(\mathcal{E}\) is a finite locally free \(\mathcal{O}_X\)-module and \(\mathcal{G}\) is a finite locally free \(\mathcal{O}_Y\)-module.
Proof
Follows from Lemma 0FZ0 by a limit argument. We urge the reader to skip the proof. Since \(X \times_R Y\) is a closed subscheme of \(X \times_\mathbf{Z} Y\) it is harmless if we replace \(R\) by \(\mathbf{Z}\). We can write \(\mathcal{F}\) as the quotient of a finitely presented \(\mathcal{O}_{X \times_R Y}\)-module by Properties, Lemma 086M. Hence we may assume \(\mathcal{F}\) is of finite presentation. Next we can write \(X = \lim X_i\) with \(X_i\) of finite presentation over \(\mathbf{Z}\) and similarly \(Y = \lim Y_j\), see Limits, Proposition 01ZA. Then \(\mathcal{F}\) will descend to \(\mathcal{F}_{ij}\) on some \(X_i \times_R Y_j\) (Limits, Lemma 01ZR) and so does the property of having the resolution property (Derived Categories of Schemes, Lemma 0F8B). Then we apply Lemma 0FZ0 to \(\mathcal{F}_{ij}\) and we pullback.
Lemma
Let \(R\) be a Noetherian ring. Let \(X\) be a separated finite type scheme over \(R\) which has the resolution property. Set \(\mathcal{O}_\Delta = \Delta_*(\mathcal{O}_X)\) where \(\Delta : X \to X \times_R X\) is the diagonal of \(X/k\). There exists a resolution \[\ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0\] where each \(\mathcal{E}_i\) and \(\mathcal{G}_i\) is a finite locally free \(\mathcal{O}_X\)-module.
Proof
Since \(X\) is separated, the diagonal morphism \(\Delta\) is a closed immersion and hence \(\mathcal{O}_\Delta\) is a coherent \(\mathcal{O}_{X \times_R X}\)-module (Cohomology of Schemes, Lemma 087T). Thus the lemma follows immediately from Lemma 0FZ0.
Lemma
Let \(X\) be a regular Noetherian scheme of dimension \(d < \infty\). Then
for \(\mathcal{F}\), \(\mathcal{G}\) coherent \(\mathcal{O}_X\)-modules we have \(\Ext^n_X(\mathcal{F}, \mathcal{G}) = 0\) for \(n > d\), and
for \(K, L \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) and \(a \in \mathbf{Z}\) if \(H^i(K) = 0\) for \(i < a + d\) and \(H^i(L) = 0\) for \(i \geq a\) then \(\Hom_X(K, L) = 0\).
Proof
To prove (1) we use the spectral sequence \[H^p(X, \SheafExt^q(\mathcal{F}, \mathcal{G})) \Rightarrow \Ext^{p + q}_X(\mathcal{F}, \mathcal{G})\] of Cohomology, Section 0BQP. Let \(x \in X\). We have \[\SheafExt^q(\mathcal{F}, \mathcal{G})_x = \SheafExt^q_{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)\] see Cohomology, Lemma 0GM8 (this also uses that \(\mathcal{F}\) is pseudo-coherent by Derived Categories of Schemes, Lemma 08E8). Set \(d_x = \dim(\mathcal{O}_{X, x})\). Since \(\mathcal{O}_{X, x}\) is regular the ring \(\mathcal{O}_{X, x}\) has global dimension \(d_x\), see Algebra, Proposition 00O7. Thus \(\SheafExt^q_{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)\) is zero for \(q > d_x\). It follows that the modules \(\SheafExt^q(\mathcal{F}, \mathcal{G})\) have support of dimension at most \(d - q\). Hence we have \(H^p(X, \SheafExt^q(\mathcal{F}, \mathcal{G})) = 0\) for \(p > d - q\) by Cohomology, Proposition 02UZ. This proves (1).
Proof of (2). We may use induction on the number of nonzero cohomology sheaves of \(K\) and \(L\). The case where these numbers are \(0, 1\) follows from (1). If the number of nonzero cohomology sheaves of \(K\) is \(> 1\), then we let \(i \in \mathbf{Z}\) be minimal such that \(H^i(K)\) is nonzero. We obtain a distinguished triangle \[H^i(K)[-i] \to K \to \tau_{\geq i + 1}K\] (Derived Categories, Remark 08J5) and we get the vanishing of \(\Hom(K, L)\) from the vanishing of \(\Hom(H^i(K)[-i], L)\) and \(\Hom(\tau_{\geq i + 1}K, L)\) by Derived Categories, Lemma 0149. Similarly if \(L\) has more than one nonzero cohomology sheaf.
Lemma
Let \(X\) be a regular Noetherian scheme of dimension \(d < \infty\). Let \(K \in D^b_{\textit{Coh}}(\mathcal{O}_X)\) and \(a \in \mathbf{Z}\). If \(H^i(K) = 0\) for \(a < i < a + d\), then \(K = \tau_{\leq a}K \oplus \tau_{\geq a + d}K\).
Proof
We have \(\tau_{\leq a}K = \tau_{\leq a + d - 1}K\) by the assumed vanishing of cohomology sheaves. By Derived Categories, Remark 08J5 we have a distinguished triangle \[\tau_{\leq a}K \to K \to \tau_{\geq a + d}K \xrightarrow{\delta} (\tau_{\leq a}K)[1]\] By Derived Categories, Lemma 05QT it suffices to show that the morphism \(\delta\) is zero. This follows from Lemma 0FZ3.
Lemma
Let \(k\) be a field. Let \(X\) be a quasi-compact separated smooth scheme over \(k\). There exist finite locally free \(\mathcal{O}_X\)-modules \(\mathcal{E}\) and \(\mathcal{G}\) such that \[\mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle\] in \(D(\mathcal{O}_{X \times X})\) where the notation is as in Derived Categories, Section 09SI.
Proof
Recall that \(X\) is regular by Varieties, Lemma 056S. Hence \(X\) has the resolution property by Derived Categories of Schemes, Lemma 0F8A. Hence we may choose a resolution as in Lemma 0FZ2. Say \(\dim(X) = d\). Since \(X \times X\) is smooth over \(k\) it is regular. Hence \(X \times X\) is a regular Noetherian scheme with \(\dim(X \times X) = 2d\). The object \[K = (\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0)\] of \(D_{perf}(\mathcal{O}_{X \times X})\) has cohomology sheaves \(\mathcal{O}_\Delta\) in degree \(0\) and \(\Ker(\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \mathcal{E}_{2d-1} \boxtimes \mathcal{G}_{2d-1})\) in degree \(-2d\) and zero in all other degrees. Hence by Lemma 0FZ4 we see that \(\mathcal{O}_\Delta\) is a summand of \(K\) in \(D_{perf}(\mathcal{O}_{X \times X})\). Clearly, the object \(K\) is in \[\left\langle \bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{E}_i \boxtimes \mathcal{G}_i \right\rangle \subset \left\langle \left(\bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{E}_i\right) \boxtimes \left(\bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{G}_i\right) \right\rangle\] which finishes the proof. (The reader may consult Derived Categories, Lemmas 0FX9 and 0FX7 to see that our object is contained in this category.)
Lemma
Let \(k\) be a field. Let \(X\) be a scheme proper and smooth over \(k\). Then \(D_{perf}(\mathcal{O}_X)\) has a strong generator.
Proof
Using Lemma 0FZ5 choose finite locally free \(\mathcal{O}_X\)-modules \(\mathcal{E}\) and \(\mathcal{G}\) such that \(\mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle\) in \(D(\mathcal{O}_{X \times X})\). We claim that \(\mathcal{G}\) is a strong generator for \(D_{perf}(\mathcal{O}_X)\). With notation as in Derived Categories, Section 0FX0 choose \(m, n \geq 1\) such that \[\mathcal{O}_\Delta \in smd(add(\mathcal{E} \boxtimes \mathcal{G}[-m, m])^{\star n})\] This is possible by Derived Categories, Lemma 0ATG. Let \(K\) be an object of \(D_{perf}(\mathcal{O}_X)\). Since \(L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} -\) is an exact functor and since \[L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} (\mathcal{E} \boxtimes \mathcal{G}) = (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G}\] we conclude from Derived Categories, Remark 0FX5 that \[L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} \mathcal{O}_\Delta \in smd(add( (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G}[-m, m])^{\star n})\] Applying the exact functor \(R\text{pr}_{2, *}\) and observing that \[R\text{pr}_{2, *} \left((K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G}\right) = R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \otimes_k \mathcal{G}\] by Derived Categories of Schemes, Lemma 08EU we conclude that \[K = R\text{pr}_{2, *}(L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} \mathcal{O}_\Delta) \in smd(add(R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \otimes_k \mathcal{G}[-m, m])^{\star n})\] The equality follows from the discussion in Example 0FYV. Since \(K\) is perfect, there exist \(a \leq b\) such that \(H^i(X, K)\) is nonzero only for \(i \in [a, b]\). Since \(X\) is proper, each \(H^i(X, K)\) is finite dimensional. We conclude that the right hand side is contained in \(smd(add(\mathcal{G}[-m + a, m + b])^{\star n})\) which is itself contained in \(\langle \mathcal{G} \rangle_n\) by one of the references given above. This finishes the proof.
Lemma
Let \(k\) be a field. Let \(X\) be a proper smooth scheme over \(k\). There exists integers \(m, n \geq 1\) and a finite locally free \(\mathcal{O}_X\)-module \(\mathcal{G}\) such that every coherent \(\mathcal{O}_X\)-module is contained in \(smd(add(\mathcal{G}[-m, m])^{\star n})\) with notation as in Derived Categories, Section 0FX0.
Proof
In the proof of Lemma 0FZ6 we have shown that there exist \(m', n \geq 1\) such that for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), \[\mathcal{F} \in smd(add(\mathcal{G}[-m' + a, m' + b])^{\star n})\] for any \(a \leq b\) such that \(H^i(X, \mathcal{F})\) is nonzero only for \(i \in [a, b]\). Thus we can take \(a = 0\) and \(b = \dim(X)\). Taking \(m = \max(m', m' + b)\) finishes the proof.
The following lemma is the boundedness result referred to in the title of this section.
Lemma
Let \(k\) be a field. Let \(X\) be a smooth proper scheme over \(k\). Let \(\mathcal{A}\) be an abelian category. Let \(H : D_{perf}(\mathcal{O}_X) \to \mathcal{A}\) be a homological functor (Derived Categories, Definition 0147) such that for all \(K\) in \(D_{perf}(\mathcal{O}_X)\) the object \(H^i(K)\) is nonzero for only a finite number of \(i \in \mathbf{Z}\). Then there exists an integer \(m \geq 1\) such that \(H^i(\mathcal{F}) = 0\) for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) and \(i \not \in [-m, m]\). Similarly for cohomological functors.
Proof
Lemma
Let \(k\) be a field. Let \(X\), \(Y\) be finite type schemes over \(k\). Let \(K_0 \to K_1 \to K_2 \to \ldots\) be a system of objects of \(D_{perf}(\mathcal{O}_{X \times Y})\) and \(m \geq 0\) an integer such that
\(H^q(K_i)\) is nonzero only for \(q \leq m\),
for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) with \(\dim(\text{Supp}(\mathcal{F})) = 0\) the object \[R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}}^\mathbf{L} K_n)\] has vanishing cohomology sheaves in degrees outside \([-m, m] \cup [-m - n, m - n]\) and for \(n > 2m\) the transition maps induce isomorphisms on cohomology sheaves in degrees in \([-m, m]\).
Then \(K_n\) has vanishing cohomology sheaves in degrees outside \([-m, m] \cup [-m - n, m - n]\) and for \(n > 2m\) the transition maps induce isomorphisms on cohomology sheaves in degrees in \([-m, m]\). Moreover, if \(X\) and \(Y\) are smooth over \(k\), then for \(n\) large enough we find \(K_n = K \oplus C_n\) in \(D_{perf}(\mathcal{O}_{X \times Y})\) where \(K\) has cohomology only indegrees \([-m, m]\) and \(C_n\) only in degrees \([-m - n, m - n]\) and the transition maps define isomorphisms between various copies of \(K\).
Proof
Let \(Z\) be the scheme theoretic support of an \(\mathcal{F}\) as in (2). Then \(Z \to \Spec(k)\) is finite, hence \(Z \times Y \to Y\) is finite. It follows that for an object \(M\) of \(D_\QCoh(\mathcal{O}_{X \times Y})\) with cohomology sheaves supported on \(Z \times Y\) we have \(H^i(R\text{pr}_{2, *}(M)) = \text{pr}_{2, *}H^i(M)\) and the functor \(\text{pr}_{2, *}\) is faithful on quasi-coherent modules supported on \(Z \times Y\); details omitted. Hence we see that the objects \[\text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}}^\mathbf{L} K_n\] in \(D_{perf}(\mathcal{O}_{X \times Y})\) have vanishing cohomology sheaves outside \([-m, m] \cup [-m - n, m - n]\) and for \(n > 2m\) the transition maps induce isomorphisms on cohomology sheaves in \([-m, m]\). Let \(z \in X \times Y\) be a closed point mapping to the closed point \(x \in X\). Then we know that \[K_{n, z} \otimes_{\mathcal{O}_{X \times Y, z}}^\mathbf{L} \mathcal{O}_{X \times Y, z}/\mathfrak m_x^t\mathcal{O}_{X \times Y, z}\] has nonzero cohomology only in the intervals \([-m, m] \cup [-m - n, m - n]\). We conclude by More on Algebra, Lemma 0EGV that \(K_{n, z}\) only has nonzero cohomology in degrees \([-m, m] \cup [-m - n, m - n]\). Since this holds for all closed points of \(X \times Y\), we conclude \(K_n\) only has nonzero cohomology sheaves in degrees \([-m, m] \cup [-m - n, m - n]\). In exactly the same way we see that the maps \(K_n \to K_{n + 1}\) are isomorphisms on cohomology sheaves in degrees \([-m, m]\) for \(n > 2m\).
If \(X\) and \(Y\) are smooth over \(k\), then \(X \times Y\) is smooth over \(k\) and hence regular by Varieties, Lemma 056S. Thus we will obtain the direct sum decomposition of \(K_n\) as soon as \(n > 2m + \dim(X \times Y)\) from Lemma 0FZ4. The final statement is clear from this.
Sibling functors
In this section we prove some categorical result on the following notion.
Definition
Let \(\mathcal{A}\) be an abelian category. Let \(\mathcal{D}\) be a triangulated category. We say two exact functors of triangulated categories \[F, F' : D^b(\mathcal{A}) \longrightarrow \mathcal{D}\] are siblings, or we say \(F'\) is a sibling of \(F\), if the following two conditions are satisfied
the functors \(F \circ i\) and \(F' \circ i\) are isomorphic where \(i : \mathcal{A} \to D^b(\mathcal{A})\) is the inclusion functor, and
\(F(K) \cong F'(K)\) for any \(K\) in \(D^b(\mathcal{A})\).
Sometimes the second condition is a consequence of the first.
Lemma
Let \(\mathcal{A}\) be an abelian category. Let \(\mathcal{D}\) be a triangulated category. Let \(F, F' : D^b(\mathcal{A}) \longrightarrow \mathcal{D}\) be exact functors of triangulated categories. Assume
the functors \(F \circ i\) and \(F' \circ i\) are isomorphic where \(i : \mathcal{A} \to D^b(\mathcal{A})\) is the inclusion functor, and
for all \(X, Y \in \Ob(\mathcal{A})\) we have \(\Ext^q_\mathcal{D}(F(X), F(Y)) = 0\) for \(q < 0\) (for example if \(F\) is fully faithful).
Then \(F\) and \(F'\) are siblings.
Proof
Let \(K \in D^b(\mathcal{A})\). We will show \(F(K)\) is isomorphic to \(F'(K)\). We can represent \(K\) by a bounded complex \(A^\bullet\) of objects of \(\mathcal{A}\). After replacing \(K\) by a translation we may assume \(A^i = 0\) for \(i > 0\). Choose \(n \geq 0\) such that \(A^{-i} = 0\) for \(i > n\). The objects \[M_i = (A^{-i} \to \ldots \to A^0)[-i],\quad i = 0, \ldots, n\] form a Postnikov system in \(D^b(\mathcal{A})\) for the complex \(A^\bullet = A^{-n} \to \ldots \to A^0\) in \(D^b(\mathcal{A})\). See Derived Categories, Example 0D8Z. Since both \(F\) and \(F'\) are exact functors of triangulated categories both \[F(M_i) \quad\text{and}\quad F'(M_i)\] form a Postnikov system in \(\mathcal{D}\) for the complex \[F(A^{-n}) \to \ldots \to F(A^0) = F'(A^{-n}) \to \ldots \to F'(A^0)\] Since all negative \(\Ext\)s between these objects vanish by assumption we conclude by uniqueness of Postnikov systems (Derived Categories, Lemma 0D83) that \(F(K) = F(M_n[n]) \cong F'(M_n[n]) = F'(K)\).
Lemma
Let \(F\) and \(F'\) be siblings as in Definition 0FZT. Then
if \(F\) is essentially surjective, then \(F'\) is essentially surjective,
if \(F\) is fully faithful, then \(F'\) is fully faithful.
Proof
Part (1) is immediate from property (2) for siblings.
Assume \(F\) is fully faithful. Denote \(\mathcal{D}' \subset \mathcal{D}\) the essential image of \(F\) so that \(F : D^b(\mathcal{A}) \to \mathcal{D}'\) is an equivalence. Since the functor \(F'\) factors through \(\mathcal{D}'\) by property (2) for siblings, we can consider the functor \(H = F^{-1} \circ F' : D^b(\mathcal{A}) \to D^b(\mathcal{A})\). Observe that \(H\) is a sibling of the identity functor. Since it suffices to prove that \(H\) is fully faithful, we reduce to the problem discussed in the next paragraph.
Set \(\mathcal{D} = D^b(\mathcal{A})\). We have to show a sibling \(F : \mathcal{D} \to \mathcal{D}\) of the identity functor is fully faithful. Denote \(a_X : X \to F(X)\) the functorial isomorphism for \(X \in \Ob(\mathcal{A})\) given to us by Definition 0FZT. For any \(K\) in \(\mathcal{D}\) and distinguished triangle \(K_1 \to K_2 \to K_3\) of \(\mathcal{D}\) if the maps \[F : \Hom(K, K_i[n]) \to \Hom(F(K), F(K_i[n]))\] are isomorphisms for all \(n \in \mathbf{Z}\) and \(i = 1, 3\), then the same is true for \(i = 2\) and all \(n \in \mathbf{Z}\). This uses the \(5\)-lemma Homology, Lemma 05QB and Derived Categories, Lemma 0149; details omitted. Similarly, if the maps \[F : \Hom(K_i[n], K) \to \Hom(F(K_i[n]), F(K))\] are isomorphisms for all \(n \in \mathbf{Z}\) and \(i = 1, 3\), then the same is true for \(i = 2\) and all \(n \in \mathbf{Z}\). Using the canonical truncations and induction on the number of nonzero cohomology objects, we see that it is enough to show \[F : \Ext^q(X, Y) \to \Ext^q(F(X), F(Y))\] is bijective for all \(X, Y \in \Ob(\mathcal{A})\) and all \(q \in \mathbf{Z}\). Since \(F\) is a sibling of \(\text{id}\) we have \(F(X) \cong X\) and \(F(Y) \cong Y\) hence the right hand side is zero for \(q < 0\). The case \(q = 0\) is OK by our assumption that \(F\) is a sibling of the identity functor. It remains to prove the cases \(q > 0\).
The case \(q = 1\): Injectivity. An element \(\xi\) of \(\Ext^1(X, Y)\) gives rise to a distinguished triangle \[Y \to E \to X \xrightarrow{\xi} Y[1]\] Observe that \(E \in \Ob(\mathcal{A})\). Since \(F\) is a sibling of the identity functor we obtain a commutative diagram \[\xymatrix{ E \ar[d] \ar[r] & X \ar[d] \\ F(E) \ar[r] & F(X) }\] whose vertical arrows are the isomorphisms \(a_E\) and \(a_X\). By TR3 the distinguished triangle associated to \(\xi\) we started with is isomorphic to the distinguished triangle \[F(Y) \to F(E) \to F(X) \xrightarrow{F(\xi)} F(Y[1]) = F(Y)[1]\] Thus \(\xi = 0\) if and only if \(F(\xi)\) is zero, i.e., we see that \(F : \Ext^1(X, Y) \to \Ext^1(F(X), F(Y))\) is injective.
The case \(q = 1\): Surjectivity. Let \(\theta\) be an element of \(\Ext^1(F(X), F(Y))\). This defines an extension of \(F(X)\) by \(F(Y)\) in \(\mathcal{A}\) which we may write as \(F(E)\) as \(F\) is a sibling of the identity functor. We thus get a distinguished triangle \[F(Y) \xrightarrow{F(\alpha)} F(E) \xrightarrow{F(\beta)} F(X) \xrightarrow{\theta} F(Y[1]) = F(Y)[1]\] for some morphisms \(\alpha : Y \to E\) and \(\beta : E \to X\). Since \(F\) is a sibling of the identity functor, the sequence \(0 \to Y \to E \to X \to 0\) is a short exact sequence in \(\mathcal{A}\)! Hence we obtain a distinguished triangle \[Y \xrightarrow{\alpha} E \xrightarrow{\beta} X \xrightarrow{\delta} Y[1]\] for some morphism \(\delta : X \to Y[1]\). Applying the exact functor \(F\) we obtain the distinguished triangle \[F(Y) \xrightarrow{F(\alpha)} F(E) \xrightarrow{F(\beta)} F(X) \xrightarrow{F(\delta)} F(Y)[1]\] Arguing as above, we see that these triangles are isomorphic. Hence there exists a commutative diagram \[\xymatrix{ F(X) \ar[d]^\gamma \ar[r]_{F(\delta)} & F(Y[1]) \ar[d]_\epsilon \\ F(X) \ar[r]^\theta & F(Y[1]) }\] for some isomorphisms \(\gamma\), \(\epsilon\) (we can say more but we won’t need more information). We may write \(\gamma = F(\gamma')\) and \(\epsilon = F(\epsilon')\). Then we have \(\theta = F(\epsilon' \circ \delta \circ (\gamma')^{-1})\) and we see the surjectivity holds.
The case \(q > 1\): surjectivity. Using Yoneda extensions, see Derived Categories, Section 06XP, we find that for any element \(\xi\) in \(\Ext^q(F(X), F(Y))\) we can find \(F(X) = B_0, B_1, \ldots, B_{q - 1}, B_q = F(Y) \in \Ob(\mathcal{A})\) and elements \[\xi_i \in \Ext^1(B_{i - 1}, B_i)\] such that \(\xi\) is the composition \(\xi_q \circ \ldots \circ \xi_1\). Write \(B_i = F(A_i)\) (of course we have \(A_i = B_i\) but we don’t need to use this) so that \[\xi_i = F(\eta_i) \in \Ext^1(F(A_{i - 1}), F(A_i)) \quad\text{with}\quad \eta_i \in \Ext^1(A_{i - 1}, A_i)\] by surjectivity for \(q = 1\). Then \(\eta = \eta_q \circ \ldots \circ \eta_1\) is an element of \(\Ext^q(X, Y)\) with \(F(\eta) = \xi\).
The case \(q > 1\): injectivity. An element \(\xi\) of \(\Ext^q(X, Y)\) gives rise to a distinguished triangle \[Y[q - 1] \to E \to X \xrightarrow{\xi} Y[q]\] Applying \(F\) we obtain a distinguished triangle \[F(Y)[q - 1] \to F(E) \to F(X) \xrightarrow{F(\xi)} F(Y)[q]\] If \(F(\xi) = 0\), then \(F(E) \cong F(Y)[q - 1] \oplus F(X)\) in \(\mathcal{D}\), see Derived Categories, Lemma 05QT. Since \(F\) is a sibling of the identity functor we have \(E \cong F(E)\) and hence \[E \cong F(E) \cong F(Y)[q - 1] \oplus F(X) \cong Y[q - 1] \oplus X\] In other words, \(E\) is isomorphic to the direct sum of its cohomology objects. This implies that the initial distinguished triangle is split, i.e., \(\xi = 0\).
Let us make a nonstandard definition. Let \(\mathcal{A}\) be an abelian category. Let us say \(\mathcal{A}\) has enough negative objects if given any \(X \in \Ob(\mathcal{A})\) there exists an object \(N\) such that
there is a surjection \(N \to X\) and
\(\Hom(X, N) = 0\).
Let us prove a couple of lemmas about this notion in order to help with the proof of Proposition 0FZW.
Lemma
Let \(\mathcal{A}\) be an abelian category with enough negative objects. Let \(X \in D^b(\mathcal{A})\). Let \(b \in \mathbf{Z}\) with \(H^i(X) = 0\) for \(i > b\). Then there exists a map \(N[-b] \to X\) such that the induced map \(N \to H^b(X)\) is surjective and \(\Hom(H^b(X), N) = 0\).
Proof
Using the truncation functors we can represent \(X\) by a complex \(A^a \to A^{a + 1} \to \ldots \to A^b\) of objects of \(\mathcal{A}\). Choose \(N\) in \(\mathcal{A}\) such that there exists a surjection \(t : N \to A^b\) and such that \(\Hom(A^b, N) = 0\). Then the surjection \(t\) defines a map \(N[-b] \to X\) as desired.
Lemma
Let \(\mathcal{A}\) be an abelian category with enough negative objects. Let \(f : X \to X'\) be a morphism of \(D^b(\mathcal{A})\). Let \(b \in \mathbf{Z}\) such that \(H^i(X) = 0\) for \(i > b\) and \(H^i(X') = 0\) for \(i \geq b\). Then there exists a map \(N[-b] \to X\) such that the induced map \(N \to H^b(X)\) is surjective, such that \(\Hom(H^b(X), N) = 0\), and such that the composition \(N[-b] \to X \to X'\) is zero.
Proof
We can represent \(f\) by a map \(f^\bullet : A^\bullet \to B^\bullet\) of bounded complexes of objects of \(\mathcal{A}\), see for example Derived Categories, Lemma 05RW. Consider the object \[C = \Ker(A^b \to A^{b + 1}) \times_{\Ker(B^b \to B^{b + 1})} B^{b - 1}\] of \(\mathcal{A}\). Since \(H^b(B^\bullet) = 0\) we see that \(C \to H^b(A^\bullet)\) is surjective. On the other hand, the map \(C \to A^b \to B^b\) is the same as the map \(C \to B^{b - 1} \to B^b\) and hence the composition \(C[-b] \to X \to X'\) is zero. Since \(\mathcal{A}\) has enough negative objects, we can find an object \(N\) which has a surjection \(N \to C \oplus H^b(X)\) such that \(\Hom(C \oplus H^b(X), N) = 0\). Then \(N\) together with the map \(N[-b] \to X\) is a solution to the problem posed by the lemma.
We encourage the reader to read the original [Orlov-K3, Proposition 2.16] for the marvellous ideas that go into the proof of the following proposition.
Proposition
Let \(F\) and \(F'\) be siblings as in Definition 0FZT. Assume that \(F\) is fully faithful and that \(\mathcal{A}\) has enough negative objects (see above). Then \(F\) and \(F'\) are isomorphic functors.
Proof
By part (2) of Definition 0FZT the image of the functor \(F'\) is contained in the essential image of the functor \(F\). Hence the functor \(H = F^{-1} \circ F'\) is a sibling of the identity functor. This reduces us to the case described in the next paragraph.
Let \(\mathcal{D} = D^b(\mathcal{A})\). We have to show a sibling \(F : \mathcal{D} \to \mathcal{D}\) of the identity functor is isomorphic to the identity functor. Given an object \(X\) of \(\mathcal{D}\) let us say \(X\) has width \(w = w(X)\) if \(w \geq 0\) is minimal such that there exists an integer \(a \in \mathbf{Z}\) with \(H^i(X) = 0\) for \(i \not \in [a, a + w - 1]\). Since \(F\) is a sibling of the identity and since \(F \circ [n] = [n] \circ F\) we are already given isomorphisms \[c_X : X \to F(X)\] for \(w(X) \leq 1\) compatible with shifts. Moreover, if \(X = A[-a]\) and \(X' = A'[-a]\) for some \(A, A' \in \Ob(\mathcal{A})\) then for any morphism \(f : X \to X'\) the diagram [0FZX]\[\begin{equation} \vcenter{ \xymatrix{ X \ar[d]_{c_X} \ar[r]_f & X' \ar[d]^{c_{X'}} \\ F(X) \ar[r]^{F(f)} & F(X') } } \end{equation}\] is commutative.
Next, let us show that for any morphism \(f : X \to X'\) with \(w(X), w(X') \leq 1\) the diagram (0FZX) commutes. If \(X\) or \(X'\) is zero, this is clear. If not then we can write \(X = A[-a]\) and \(X' = A'[-a']\) for unique \(A, A'\) in \(\mathcal{A}\) and \(a, a' \in \mathbf{Z}\). The case \(a = a'\) was discussed above. If \(a' > a\), then \(f = 0\) (Derived Categories, Lemma 06XS) and the result is clear. If \(a' < a\) then \(f\) corresponds to an element \(\xi \in \Ext^q(A, A')\) with \(q = a - a'\). Using Yoneda extensions, see Derived Categories, Section 06XP, we can find \(A = A_0, A_1, \ldots, A_{q - 1}, A_q = A' \in \Ob(\mathcal{A})\) and elements \[\xi_i \in \Ext^1(A_{i - 1}, A_i)\] such that \(\xi\) is the composition \(\xi_q \circ \ldots \circ \xi_1\). In other words, setting \(X_i = A_i[-a + i]\) we obtain morphisms \[X = X_0 \xrightarrow{f_1} X_1 \to \ldots \to X_{q - 1} \xrightarrow{f_q} X_q = X'\] whose compostion is \(f\). Since the commutativity of (0FZX) for \(f_1, \ldots, f_q\) implies it for \(f\), this reduces us to the case \(q = 1\). In this case after shifting we may assume we have a distinguished triangle \[A' \to E \to A \xrightarrow{f} A'[1]\] Observe that \(E\) is an object of \(\mathcal{A}\). Consider the following diagram \[\xymatrix{ E \ar[d]_{c_E} \ar[r] & A \ar[d]_{c_A} \ar[r]_f & A'[1] \ar[d]^{c_{A'}[1]} \ar@{..>}@<-1ex>[d]_\gamma \ar@{..>}[ld]^\epsilon \ar[r] & E[1] \ar[d]^{c_E[1]} \\ F(E) \ar[r] & F(A) \ar[r]^{F(f)} & F(A')[1] \ar[r] & F(E)[1] }\] whose rows are distinguished triangles. The square on the right commutes already but we don’t yet know that the middle square does. By the axioms of a triangulated category we can find a morphism \(\gamma\) which does make the diagram commute. Then \(\gamma - c_{A'}[1]\) composed with \(F(A')[1] \to F(E)[1]\) is zero hence we can find \(\epsilon : A'[1] \to F(A)\) such that \(\gamma - c_{A'}[1] = F(f) \circ \epsilon\). However, any arrow \(A'[1] \to F(A)\) is zero as it is a negative ext class between objects of \(\mathcal{A}\). Hence \(\gamma = c_{A'}[1]\) and we conclude the middle square commutes too which is what we wanted to show.
To finish the proof we are going to argue by induction on \(w\) that there exist isomorphisms \(c_X : X \to F(X)\) for all \(X\) with \(w(X) \leq w\) compatible with all morphisms between such objects. The base case \(w = 1\) was shown above. Assume we know the result for some \(w \geq 1\).
Let \(X\) be an object with \(w(X) = w + 1\). Pick \(a \in \mathbf{Z}\) with \(H^i(X) = 0\) for \(i \not \in [a, a + w]\). Set \(b = a + w\) so that \(H^b(X)\) is nonzero. Choose \(N[-b] \to X\) as in Lemma 0GWF. Choose a distinguished diagram \[N[-b] \to X \to Y \to N[-b + 1]\] Computing the long exact cohomology sequence we find \(w(Y) \leq w\). Hence by induction we find the solid arrows in the following diagram \[\xymatrix{ N[-b] \ar[r] \ar[d]_{c_N[-b]} & X \ar[r] \ar@{..>}[d]_{c_{N[-b] \to X}} & Y \ar[r] \ar[d]^{c_Y} & N[-b + 1] \ar[d]^{c_N[-b + 1]} \\ F(N)[-b] \ar[r] & F(X) \ar[r] & F(Y) \ar[r] & F(N)[-b + 1] }\] We obtain the dotted arrow \(c_{N[-b] \to X}\). By Derived Categories, Lemma 0FWZ the dotted arrow is unique because \(\Hom(X, F(N)[-b]) \cong \Hom(X, N[-b]) = 0\) by our choice of \(N\). In fact, \(c_{N[-b] \to X}\) is the unique dotted arrow making the square with vertices \(X, Y, F(X), F(Y)\) commute.
Let \(N'[-b] \to X\) be another map as in Lemma 0GWF and let us prove that \(c_{N[-b] \to X} = c_{N'[-b] \to X}\). Observe that the map \((N \oplus N')[-b] \to X\) also satisfies the conditions of Lemma 0GWF. Thus we may assume \(N'[-b] \to X\) factors as \(N'[-b] \to N[-b] \to X\) for some morphism \(N' \to N\). Choose distinguished triangles \(N[-b] \to X \to Y \to N[-b + 1]\) and \(N'[-b] \to X \to Y' \to N'[-b + 1]\). By axiom TR3 we can find a morphism \(g : Y' \to Y\) which joint with \(\text{id}_X\) and \(N' \to N\) forms a morphism of triangles. Since we have (0FZX) for \(g\) we conclude that \[(F(X) \to F(Y)) \circ c_{N'[-b] \to X} = (F(X) \to F(Y)) \circ c_{N[-b] \to X}\] The uniqueness of \(c_{N[-b] \to X}\) pointed out in the construction above now shows that \(c_{N'[-b] \to X} = c_{N[-b] \to X}\).
Thus we can now define for \(X\) of width \(w + 1\) the isomorphism \(c_X : X \to F(X)\) as the common value of the maps \(c_{N[-b] \to X}\) where \(N[-b] \to X\) is as in Lemma 0GWF. To finish the proof, we have to show that the diagrams (0FZX) commute for all morphisms \(f : X \to X'\) between objects with \(w(X) \leq w + 1\) and \(w(X') \leq w + 1\). Choose \(a \leq b \leq a + w\) such that \(H^i(X) = 0\) for \(i \not \in [a, b]\) and \(a' \leq b' \leq a' + w\) such that \(H^i(X') = 0\) for \(i \not \in [a', b']\). We will use induction on \((b' - a') + (b - a)\) to show the claim. (The base case is when this number is zero which is OK because \(w \geq 1\).) We distinguish two cases.
Case I: \(b' < b\). In this case, by Lemma 0GWG we may choose \(N[-b] \to X\) as in Lemma 0GWF such that the composition \(N[-b] \to X \to X'\) is zero. Choose a distuiguished triangle \(N[-b] \to X \to Y \to N[-b + 1]\). Since \(N[-b] \to X'\) is zero, we find that \(f\) factors as \(X \to Y \to X'\). Since \(H^i(Y)\) is nonzero only for \(i \in [a, b - 1]\) we see by induction that (0FZX) commutes for \(Y \to X'\). The diagram (0FZX) commutes for \(X \to Y\) by construction if \(w(X) = w + 1\) and by our first induction hypothesis if \(w(X) \leq w\). Hence (0FZX) commutes for \(f\).
Case II: \(b' \geq b\). In this case we choose \(N'[-b'] \to X'\) as in Lemma 0GWF. We may also assume that \(\Hom(H^{b'}(X), N') = 0\) (this is relevant only if \(b' = b\)), for example because we can replace \(N'\) by an object \(N''\) which surjects onto \(N' \oplus H^{b'}(X)\) and such that \(\Hom(N' \oplus H^{b'}(X), N'') = 0\). We choose a distinguished triangle \(N'[-b'] \to X' \to Y' \to N'[-b' + 1]\). Since \(\Hom(X, X') \to \Hom(X, Y')\) is injective by our choice of \(N'\) (details omitted) the same is true for \(\Hom(X, F(X')) \to \Hom(X, F(Y'))\). Hence it suffices in this case to check that (0FZX) commutes for the composition \(X \to Y'\) of the morphisms \(X \to X' \to Y'\). Since \(H^i(Y')\) is nonzero only for \(i \in [a', b' - 1]\) we conclude by induction hypothesis.
Deducing fully faithfulness
It will be useful for us to know when a functor is fully faithful we offer the following variant of [Orlov-K3, Lemma 2.15].
Lemma
Let \(F : \mathcal{D} \to \mathcal{D}'\) be an exact functor of triangulated categories. Let \(S \subset \Ob(\mathcal{D})\) be a set of objects. Assume
\(F\) has both right and left adjoints,
for \(K \in \mathcal{D}\) if \(\Hom(E, K[i]) = 0\) for all \(E \in S\) and \(i \in \mathbf{Z}\) then \(K = 0\),
for \(K \in \mathcal{D}\) if \(\Hom(K, E[i]) = 0\) for all \(E \in S\) and \(i \in \mathbf{Z}\) then \(K = 0\),
the map \(\Hom(E, E'[i]) \to \Hom(F(E), F(E')[i])\) induced by \(F\) is bijective for all \(E, E' \in S\) and \(i \in \mathbf{Z}\).
Then \(F\) is fully faithful.
Proof
Denote \(F_r\) and \(F_l\) the right and left adjoints of \(F\). For \(E \in S\) choose a distinguished triangle \[E \to F_r(F(E)) \to C \to E[1]\] where the first arrow is the unit of the adjunction. For \(E' \in S\) we have \[\Hom(E', F_r(F(E))[i]) = \Hom(F(E'), F(E)[i]) = \Hom(E', E[i])\] The last equality holds by assumption (4). Hence applying the homological functor \(\Hom(E', -)\) (Derived Categories, Lemma 0149) to the distinguished triangle above we conclude that \(\Hom(E', C[i]) = 0\) for all \(i \in \mathbf{Z}\) and \(E' \in S\). By assumption (2) we conclude that \(C = 0\) and \(E = F_r(F(E))\).
For \(K \in \Ob(\mathcal{D})\) choose a distinguished triangle \[F_l(F(K)) \to K \to C \to F_l(F(K))[1]\] where the first arrow is the counit of the adjunction. For \(E \in S\) we have \[\Hom(F_l(F(K)), E[i]) = \Hom(F(K), F(E)[i]) = \Hom(K, F_r(F(E))[i]) = \Hom(K, E[i])\] where the last equality holds by the result of the first paragraph. Thus we conclude as before that \(\Hom(C, E[i]) = 0\) for all \(E \in S\) and \(i \in \mathbf{Z}\). Hence \(C = 0\) by assumption (3). Thus \(F\) is fully faithful by Categories, Lemma 07RB.
Lemma
Let \(k\) be a field. Let \(X\) be a scheme of finite type over \(k\) which is regular. Let \(x \in X\) be a closed point. For a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) supported at \(x\) choose a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}'\) supported at \(x\) such that \(\mathcal{F}_x\) and \(\mathcal{F}'_x\) are Matlis dual. Then there is an isomorphism \[\Hom_X(\mathcal{F}, M) = H^0(X, M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}'[-d_x])\] where \(d_x = \dim(\mathcal{O}_{X, x})\) functorial in \(M\) in \(D_{perf}(\mathcal{O}_X)\).
Proof
Since \(\mathcal{F}\) is supported at \(x\) we have \[\Hom_X(\mathcal{F}, M) = \Hom_{\mathcal{O}_{X, x}}(\mathcal{F}_x, M_x)\] and similarly we have \[H^0(X, M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}'[-d_x]) = \text{Tor}^{\mathcal{O}_{X, x}}_{d_x}(M_x, \mathcal{F}'_x)\] Thus it suffices to show that given a Noetherian regular local ring \(A\) of dimension \(d\) and a finite length \(A\)-module \(N\), if \(N'\) is the Matlis dual to \(N\), then there exists a functorial isomorphism \[\Hom_A(N, K) = \text{Tor}^A_d(K, N')\] for \(K\) in \(D_{perf}(A)\). We can write the left hand side as \(H^0(R\Hom_A(N, A) \otimes_A^\mathbf{L} K)\) by More on Algebra, Lemma 07VI and the fact that \(N\) determines a perfect object of \(D(A)\). Hence the formula holds because \[R\Hom_A(N, A) = R\Hom_A(N, A[d])[-d] = N'[-d]\] by Dualizing Complexes, Lemma 0A7Q and the fact that \(A[d]\) is a normalized dualizing complex over \(A\) (\(A\) is Gorenstein by Dualizing Complexes, Lemma 0AWX).
Lemma
Let \(k\) be a field. Let \(X\) be a scheme of finite type over \(k\) which is regular. Let \(x \in X\) be a closed point and denote \(\mathcal{O}_x\) the skyscraper sheaf at \(x\) with value \(\kappa(x)\). Let \(K\) in \(D_{perf}(\mathcal{O}_X)\).
If \(\Ext^i_X(\mathcal{O}_x, K) = 0\) then there exists an open neighbourhood \(U\) of \(x\) such that \(H^{i - d_x}(K)|_U = 0\) where \(d_x = \dim(\mathcal{O}_{X, x})\).
If \(\Hom_X(\mathcal{O}_x, K[i]) = 0\) for all \(i \in \mathbf{Z}\), then \(K\) is zero in an open neighbourhood of \(x\).
If \(\Ext^i_X(K, \mathcal{O}_x) = 0\) then there exists an open neighbourhood \(U\) of \(x\) such that \(H^i(K^\vee)|_U = 0\).
If \(\Hom_X(K, \mathcal{O}_x[i]) = 0\) for all \(i \in \mathbf{Z}\), then \(K\) is zero in an open neighbourhood of \(x\).
If \(H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x) = 0\) then there exists an open neighbourhood \(U\) of \(x\) such that \(H^i(K)|_U = 0\).
If \(H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x) = 0\) for \(i \in \mathbf{Z}\) then \(K\) is zero in an open neighbourhood of \(x\).
Proof
Observe that \(H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x)\) is equal to \(K_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \kappa(x)\). Hence part (5) follows from More on Algebra, Lemma 068U. Part (6) follows from part (5). Part (1) follows from part (5), Lemma 0G02, and the fact that the Matlis dual of \(\kappa(x)\) is \(\kappa(x)\). Part (2) follows from part (1). Part (3) follows from part (5) and the fact that \(\Ext^i(K, \mathcal{O}_x) = H^i(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x)\) by Cohomology, Lemma 08DQ. Part (4) follows from part (3) and the fact that \(K \cong (K^\vee)^\vee\) by the lemma just cited.
Lemma
Let \(X\) be a Noetherian scheme. Let \(x \in X\) be a closed point and denote \(\mathcal{O}_x\) the skyscraper sheaf at \(x\) with value \(\kappa(x)\). Let \(K\) in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Let \(b \in \mathbf{Z}\). The following are equivalent
\(H^i(K)_x = 0\) for all \(i > b\) and
\(\Hom_X(K, \mathcal{O}_x[-i]) = 0\) for all \(i > b\).
Proof
Consider the complex \(K_x\) in \(D^b_{\textit{Coh}}(\mathcal{O}_{X, x})\). There exist an integer \(b_x \in \mathbf{Z}\) such that \(K_x\) can be represented by a bounded above complex \[\ldots \to \mathcal{O}_{X, x}^{\oplus n_{b_x - 2}} \to \mathcal{O}_{X, x}^{\oplus n_{b_x - 1}} \to \mathcal{O}_{X, x}^{\oplus n_{b_x}} \to 0 \to \ldots\] with \(\mathcal{O}_{X, x}^{\oplus n_i}\) sitting in degree \(i\) where all the transition maps are given by matrices whose coefficients are in \(\mathfrak m_x\). See More on Algebra, Lemma 0BCC. The result follows easily from this (and the equivalent conditions hold if and only if \(b \geq b_x\)).
Lemma
Let \(k\) be a field. Let \(X\) and \(Y\) be proper schemes over \(k\). Assume \(X\) is regular. Then a \(k\)-linear exact functor \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) is fully faithful if and only if for any closed points \(x, x' \in X\) the maps \[F : \Ext^i_X(\mathcal{O}_x, \mathcal{O}_{x'}) \longrightarrow \Ext^i_Y(F(\mathcal{O}_x), F(\mathcal{O}_{x'}))\] are isomorphisms for all \(i \in \mathbf{Z}\). Here \(\mathcal{O}_x\) is the skyscraper sheaf at \(x\) with value \(\kappa(x)\).
Proof
By Lemma 0FYN the functor \(F\) has both a left and a right adjoint. Thus we may apply the criterion of Lemma 0G24 because assumptions (2) and (3) of that lemma follow from Lemma 0G03.
Lemma
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\) which is regular. Let \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)\) be a \(k\)-linear exact functor. Assume for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) with \(\dim(\text{Supp}(\mathcal{F})) = 0\) there is an isomorphism \(\mathcal{F} \cong F(\mathcal{F})\). Then \(F\) is fully faithful.
Proof
By Lemma 0G25 it suffices to show that the maps \[F : \Ext^i_X(\mathcal{O}_x, \mathcal{O}_{x'}) \longrightarrow \Ext^i_X(F(\mathcal{O}_x), F(\mathcal{O}_{x'}))\] are isomorphisms for all \(i \in \mathbf{Z}\) and all closed points \(x, x' \in X\). By assumption, the source and the target are isomorphic. If \(x \not = x'\), then both sides are zero and the result is true. If \(x = x'\), then it suffices to prove that the map is either injective or surjective. For \(i < 0\) both sides are zero and the result is true. For \(i = 0\) any nonzero map \(\alpha : \mathcal{O}_x \to \mathcal{O}_x\) of \(\mathcal{O}_X\)-modules is an isomorphism. Hence \(F(\alpha)\) is an isomorphism too and so \(F(\alpha)\) is nonzero. Thus the result for \(i = 0\). For \(i = 1\) a nonzero element \(\xi\) in \(\Ext^1(\mathcal{O}_x, \mathcal{O}_x)\) corresponds to a nonsplit short exact sequence \[0 \to \mathcal{O}_x \to \mathcal{F} \to \mathcal{O}_x \to 0\] Since \(F(\mathcal{F}) \cong \mathcal{F}\) we see that \(F(\mathcal{F})\) is a nonsplit extension of \(\mathcal{O}_x\) by \(\mathcal{O}_x\) as well. Since \(\mathcal{O}_x \cong F(\mathcal{O}_x)\) is a simple \(\mathcal{O}_X\)-module and \(\mathcal{F} \cong F(\mathcal{F})\) has length \(2\), we see that in the distinguished triangle \[F(\mathcal{O}_x) \to F(\mathcal{F}) \to F(\mathcal{O}_x) \xrightarrow{F(\xi)} F(\mathcal{O}_x)[1]\] the first two arrows must form a short exact sequence which must be isomorphic to the above short exact sequence and hence is nonsplit. It follows that \(F(\xi)\) is nonzero and we conclude for \(i = 1\). For \(i > 1\) composition of ext classes defines a surjection \[\Ext^1(F(\mathcal{O}_x), F(\mathcal{O}_x)) \otimes \ldots \otimes \Ext^1(F(\mathcal{O}_x), F(\mathcal{O}_x)) \longrightarrow \Ext^i(F(\mathcal{O}_x), F(\mathcal{O}_x))\] See Duality for Schemes, Lemma 0BQY. Hence surjectivity in degree \(1\) implies surjectivity for \(i > 0\). This finishes the proof.
Special functors
In this section we prove some results on functors of a special type that we will use later in this chapter.
Definition
Let \(k\) be a field. Let \(X\), \(Y\) be finite type schemes over \(k\). Recall that \(D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X))\) by Derived Categories of Schemes, Proposition 0FDB. We say two \(k\)-linear exact functors \[F, F' : D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^b_{\textit{Coh}}(\mathcal{O}_Y)\] are siblings, or we say \(F'\) is a sibling of \(F\) if \(F\) and \(F'\) are siblings in the sense of Definition 0FZT with abelian category being \(\textit{Coh}(\mathcal{O}_X)\). If \(X\) is regular then \(D_{perf}(\mathcal{O}_X) = D^b_{\textit{Coh}}(\mathcal{O}_X)\) by Derived Categories of Schemes, Lemma 0FXU and we use the same terminology for \(k\)-linear exact functors \(F, F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\).
Lemma
Let \(k\) be a field. Let \(X\), \(Y\) be finite type schemes over \(k\) with \(X\) separated. Let \(F : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)\) be a \(k\)-linear exact functor sending \(\textit{Coh}(\mathcal{O}_X) \subset D^b_{\textit{Coh}}(\mathcal{O}_X)\) into \(\textit{Coh}(\mathcal{O}_Y) \subset D^b_{\textit{Coh}}(\mathcal{O}_Y)\). Then there exists a Fourier-Mukai functor \(F' : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)\) whose kernel is a coherent \(\mathcal{O}_{X \times Y}\)-module \(\mathcal{K}\) flat over \(X\) and with support finite over \(Y\) which is a sibling of \(F\).
Proof
Denote \(H : \textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_Y)\) the restriction of \(F\). Since \(F\) is an exact functor of triangulated categories, we see that \(H\) is an exact functor of abelian categories. Of course \(H\) is \(k\)-linear as \(F\) is. By Functors and Morphisms, Lemma 0FZN we obtain a coherent \(\mathcal{O}_{X \times Y}\)-module \(\mathcal{K}\) which is flat over \(X\) and has support finite over \(Y\). Let \(F'\) be the Fourier-Mukai functor defined using \(\mathcal{K}\) so that \(F'\) restricts to \(H\) on \(\textit{Coh}(\mathcal{O}_X)\). The functor \(F'\) sends \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) into \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\) by Lemma 0FYU. Observe that \(F\) and \(F'\) satisfy the first and second condition of Lemma 0FZU and hence are siblings.
Remark
If \(F, F' : D^b_{\textit{Coh}}(\mathcal{O}_X) \to \mathcal{D}\) are siblings, \(F\) is fully faithful, and \(X\) is reduced and projective over \(k\) then \(F \cong F'\); this follows from Proposition 0FZW via the argument given in the proof of Theorem 0G0C. However, in general we do not know whether siblings are isomorphic. Even in the situation of Lemma 0G00 it seems difficult to prove that the siblings \(F\) and \(F'\) are isomorphic functors. If \(X\) is smooth and proper over \(k\) and \(F\) is fully faithful, then \(F \cong F'\) as is shown in [Noah]. If you have a proof or a counter example in more general situations, please email stacks.project@gmail.com.
Lemma
Let \(k\) be a field. Let \(X\), \(Y\) be proper schemes over \(k\). Assume \(X\) is regular. Let \(F, G : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) be \(k\)-linear exact functors such that
\(F(\mathcal{F}) \cong G(\mathcal{F})\) for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) with \(\dim(\text{Supp}(\mathcal{F})) = 0\),
\(F\) is fully faithful.
Then the essential image of \(G\) is contained in the essential image of \(F\).
Proof
Recall that \(F\) and \(G\) have both adjoints, see Lemma 0FYN. In particular the essential image \(\mathcal{A} \subset D_{perf}(\mathcal{O}_Y)\) of \(F\) satisfies the equivalent conditions of Derived Categories, Lemma 0CQS. We claim that \(G\) factors through \(\mathcal{A}\). Since \(\mathcal{A} = {}^\perp(\mathcal{A}^\perp)\) by Derived Categories, Lemma 0CQS it suffices to show that \(\Hom_Y(G(M), N) = 0\) for all \(M\) in \(D_{perf}(\mathcal{O}_X)\) and \(N \in \mathcal{A}^\perp\). We have \[\Hom_Y(G(M), N) = \Hom_X(M, G_r(N))\] where \(G_r\) is the right adjoint to \(G\). Thus it suffices to prove that \(G_r(N) = 0\). Since \(G(\mathcal{F}) \cong F(\mathcal{F})\) for \(\mathcal{F}\) as in (1) we see that \[\Hom_X(\mathcal{F}, G_r(N)) = \Hom_Y(G(\mathcal{F}), N) = \Hom_Y(F(\mathcal{F}), N) = 0\] as \(N\) is in the right orthogonal to the essential image \(\mathcal{A}\) of \(F\). Of course, the same vanishing holds for \(\Hom_X(\mathcal{F}, G_r(N)[i])\) for any \(i \in \mathbf{Z}\). Thus \(G_r(N) = 0\) by Lemma 0G03 and we win.
Lemma
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\) which is regular. Let \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)\) be a \(k\)-linear exact functor. Assume for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) with \(\dim(\text{Supp}(\mathcal{F})) = 0\) there is an isomorphism \(\mathcal{F} \cong F(\mathcal{F})\). Then there exists an automorphism \(f : X \to X\) over \(k\) which induces the identity on the underlying topological space1 and an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) such that \(F\) and \(F'(M) = f^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}\) are siblings.
Proof
By Lemma 0G26 the functor \(F\) is fully faithful. By Lemma 0GX0 the essential image of the identity functor is contained in the essential image of \(F\), i.e., we see that \(F\) is essentially surjective. Thus \(F\) is an equivalence. Observe that the quasi-inverse \(F^{-1}\) satisfies the same assumptions as \(F\).
Let \(M \in D_{perf}(\mathcal{O}_X)\) and say \(H^i(M) = 0\) for \(i > b\). Since \(F\) is fully faithful, we see that \[\Hom_X(M, \mathcal{O}_x[-i]) = \Hom_X(F(M), F(\mathcal{O}_x)[-i]) \cong \Hom_X(F(M), \mathcal{O}_x[-i])\] for any \(i \in \mathbf{Z}\) for any closed point \(x\) of \(X\). Thus by Lemma 0GWZ we see that \(F(M)\) has vanishing cohomology sheaves in degrees \(> b\).
Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. By the above \(F(\mathcal{F})\) has nonzero cohomology sheaves only in degrees \(\leq 0\). Set \(\mathcal{G} = H^0(F(\mathcal{F}))\). Choose a distinguished triangle \[K \to F(\mathcal{F}) \to \mathcal{G} \to K[1]\] Then \(K\) has nonvanishing cohomology sheaves only in degrees \(\leq -1\). Applying \(F^{-1}\) we obtain a distinguished triangle \[F^{-1}(K) \to \mathcal{F} \to F^{-1}(\mathcal{G}) \to F^{-1}(K')[1]\] Since \(F^{-1}(K)\) has nonvanishing cohomology sheaves only in degrees \(\leq -1\) (by the previous paragraph applied to \(F^{-1}\)) we see that the arrow \(F^{-1}(K) \to \mathcal{F}\) is zero (Derived Categories, Lemma 06XS). Hence \(K \to F(\mathcal{F})\) is zero, which implies that \(F(\mathcal{F}) = \mathcal{G}\) by our choice of the first distinguished triangle.
From the preceding paragraph, we deduce that \(F\) preserves \(\textit{Coh}(\mathcal{O}_X)\) and indeed defines an equivalence \(H : \textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_X)\). By Functors and Morphisms, Lemma 0FZR we get an automorphism \(f : X \to X\) over \(k\) and an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) such that \(H(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}\). Set \(F'(M) = f^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}\). Using Lemma 0FZU we see that \(F\) and \(F'\) are siblings. To see that \(f\) is the identity on the underlying topological space of \(X\), we use that \(F(\mathcal{O}_x) \cong \mathcal{O}_x\) and that the support of \(\mathcal{O}_x\) is \(\{x\}\). This finishes the proof.
Lemma
Let \(k\) be a field. Let \(X\), \(Y\) be proper schemes over \(k\). Assume \(X\) regular. Let \(F, G : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) be \(k\)-linear exact functors such that
\(F(\mathcal{F}) \cong G(\mathcal{F})\) for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) with \(\dim(\text{Supp}(\mathcal{F})) = 0\),
\(F\) is fully faithful, and
\(G\) is a Fourier-Mukai functor whose kernel is in \(D_{perf}(\mathcal{O}_{X \times Y})\).
Then there exists a Fourier-Mukai functor \(F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) whose kernel is in \(D_{perf}(\mathcal{O}_{X \times Y})\) such that \(F\) and \(F'\) are siblings.
Proof
The essential image of \(G\) is contained in the essential image of \(F\) by Lemma 0GX0. Consider the functor \(H = F^{-1} \circ G\) which makes sense as \(F\) is fully faithful. By Lemma 0G27 we obtain an automorphism \(f : X \to X\) and an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) such that the functor \(H' : K \mapsto f^*K \otimes \mathcal{L}\) is a sibling of \(H\). In particular \(H\) is an auto-equivalence by Lemma 0FZV and \(H\) induces an auto-equivalence of \(\textit{Coh}(\mathcal{O}_X)\) (as this is true for its sibling functor \(H'\)). Thus the quasi-inverses \(H^{-1}\) and \((H')^{-1}\) exist, are siblings (small detail omitted), and \((H')^{-1}\) sends \(M\) to \((f^{-1})^*(M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}^{\otimes -1})\) which is a Fourier-Mukai functor (details omitted). Then of course \(F = G \circ H^{-1}\) is a sibling of \(G \circ (H')^{-1}\). Since compositions of Fourier-Mukai functors are Fourier-Mukai by Lemma 0FYS we conclude.
Fully faithful functors
Our goal is to prove fully faithful functors between derived categories are siblings of Fourier-Mukai functors, following [Orlov-K3] and [Ballard].
Situation
Here \(k\) is a field. We have proper smooth schemes \(X\) and \(Y\) over \(k\). We have a \(k\)-linear, exact, fully faithful functor \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\).
Before reading on, it makes sense to read at least some of Derived Categories, Section 0D7Y.
Recall that \(X\) is regular and hence has the resolution property (Varieties, Lemma 056S and Derived Categories of Schemes, Lemma 0F8A). Thus on \(X \times X\) we may choose a resolution \[\ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0\] where each \(\mathcal{E}_i\) and \(\mathcal{G}_i\) is a finite locally free \(\mathcal{O}_X\)-module, see Lemma 0FZ2. Using the complex [0G09]\[\begin{equation} \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \end{equation}\] in \(D_{perf}(\mathcal{O}_{X \times X})\) as in Derived Categories, Example 0D8Z if for each \(n\) we denote \[M_n = (\mathcal{E}_n \boxtimes \mathcal{G}_n \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0)[-n]\] we obtain an infinite Postnikov system for the complex (0G09). This means the morphisms \(M_0 \to M_1[1] \to M_2[2] \to \ldots\) and \(M_n \to \mathcal{E}_n \boxtimes \mathcal{G}_n\) and \(\mathcal{E}_n \boxtimes \mathcal{G}_n \to M_{n - 1}\) satisfy certain conditions documented in Derived Categories, Definition 0D7Z. Set \[\mathcal{F}_n = \Ker(\mathcal{E}_n \boxtimes \mathcal{G}_n \to \mathcal{E}_{n - 1} \boxtimes \mathcal{G}_{n - 1})\] Observe that since \(\mathcal{O}_\Delta\) is flat over \(X\) via \(\text{pr}_1\) the same is true for \(\mathcal{F}_n\) for all \(n\) (this is a convenient though not essential observation). We have \[H^q(M_n[n]) = \left\{ \begin{matrix} \mathcal{O}_\Delta & \text{if} & q = 0 \\ \mathcal{F}_n & \text{if} & q = -n \\ 0 & \text{if} & q \not = 0, -n \end{matrix} \right.\] Thus for \(n \geq \dim(X \times X)\) we have \[M_n[n] \cong \mathcal{O}_\Delta \oplus \mathcal{F}_n[n]\] in \(D_{perf}(\mathcal{O}_{X \times X})\) by Lemma 0FZ4.
We are interested in the complex [0G0A]\[\begin{equation} \ldots \to \mathcal{E}_2 \boxtimes F(\mathcal{G}_2) \to \mathcal{E}_1 \boxtimes F(\mathcal{G}_1) \to \mathcal{E}_0 \boxtimes F(\mathcal{G}_0) \end{equation}\] in \(D_{perf}(\mathcal{O}_{X \times Y})\) as the “totalization” of this complex should give us the kernel of the Fourier-Mukai functor we are trying to construct. For all \(i, j \geq 0\) we have \[\begin{align*} \Ext^q_{X \times Y}(\mathcal{E}_i \boxtimes F(\mathcal{G}_i), \mathcal{E}_j \boxtimes F(\mathcal{G}_j)) & = \bigoplus\nolimits_p \Ext^{q + p}_X(\mathcal{E}_i, \mathcal{E}_j) \otimes_k \Ext^{-p}_Y(F(\mathcal{G}_i), F(\mathcal{G}_j)) \\ & = \bigoplus\nolimits_p \Ext^{q + p}_X(\mathcal{E}_i, \mathcal{E}_j) \otimes_k \Ext^{-p}_X(\mathcal{G}_i, \mathcal{G}_j) \end{align*}\] The second equality holds because \(F\) is fully faithful and the first by Derived Categories of Schemes, Lemma 0FXZ. We find these \(\Ext^q\) are zero for \(q < 0\). Hence by Derived Categories, Lemma 0D83 we can build an infinite Postnikov system \(K_0, K_1, K_2, \ldots\) in \(D_{perf}(\mathcal{O}_{X \times Y})\) for the complex (0G0A). Parallel to what happens with \(M_0, M_1, M_2, \ldots\) this means we obtain morphisms \(K_0 \to K_1[1] \to K_2[2] \to \ldots\) and \(K_n \to \mathcal{E}_n \boxtimes F(\mathcal{G}_n)\) and \(\mathcal{E}_n \boxtimes F(\mathcal{G}_n) \to K_{n - 1}\) in \(D_{perf}(\mathcal{O}_{X \times Y})\) satisfying certain conditions documented in Derived Categories, Definition 0D7Z.
Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module whose support has a finite number of points, i.e., with \(\dim(\text{Supp}(\mathcal{F})) = 0\). Consider the exact functor of triangulated categories \[D_{perf}(\mathcal{O}_{X \times Y}) \longrightarrow D_{perf}(\mathcal{O}_Y),\quad N \longmapsto R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} N)\] It follows that the objects \(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} K_i)\) form a Postnikov system for the complex in \(D_{perf}(\mathcal{O}_Y)\) with terms \[R\text{pr}_{2, *}( (\mathcal{F} \otimes \mathcal{E}_i) \boxtimes F(\mathcal{G}_i)) = \Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k F(\mathcal{G}_i) = F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k \mathcal{G}_i)\] Here we have used that \(\mathcal{F} \otimes \mathcal{E}_i\) has vanishing higher cohomology as its support has dimension \(0\). On the other hand, applying the exact functor \[D_{perf}(\mathcal{O}_{X \times X}) \longrightarrow D_{perf}(\mathcal{O}_Y),\quad N \longmapsto F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} N))\] we find that the objects \(F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n))\) form a second infinite Postnikov system for the complex in \(D_{perf}(\mathcal{O}_Y)\) with terms \[F(R\text{pr}_{2, *}( (\mathcal{F} \otimes \mathcal{E}_i) \boxtimes \mathcal{G}_i)) = F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k \mathcal{G}_i)\] This is the same as before! By uniqueness of Postnikov systems (Derived Categories, Lemma 0D83) which applies because \[\Ext^q_Y( F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k \mathcal{G}_i), F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_j) \otimes_k \mathcal{G}_j)) = 0, \quad q < 0\] as \(F\) is fully faithful, we find a system of isomorphisms \[F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n[n])) \cong R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} K_n[n])\] in \(D_{perf}(\mathcal{O}_Y)\) compatible with the morphisms in \(D_{perf}(\mathcal{O}_Y)\) induced by the morphisms \[M_{n - 1}[n - 1] \to M_n[n] \quad\text{and}\quad K_{n - 1}[n - 1] \to K_n[n]\] \[M_n \to \mathcal{E}_n \boxtimes \mathcal{G}_n \quad\text{and}\quad K_n \to \mathcal{E}_n \boxtimes F(\mathcal{G}_n)\] \[\mathcal{E}_n \boxtimes \mathcal{G}_n \to M_{n - 1} \quad\text{and}\quad \mathcal{E}_n \boxtimes F(\mathcal{G}_n) \to K_{n - 1}\] which are part of the structure of Postnikov systems. For \(n\) sufficiently large we obtain a direct sum decomposition \[F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n[n])) = F(\mathcal{F}) \oplus F(R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \mathcal{F}_n ))[n]\] corresponding to the direct sum decomposition of \(M_n\) constructed above (we are using the flatness of \(\mathcal{F}_n\) over \(X\) via \(\text{pr}_1\) to write a usual tensor product in the formula above, but this isn’t essential for the argument). By Lemma 0FZ8 we find there exists an integer \(m \geq 0\) such that the first summand in this direct sum decomposition has nonzero cohomology sheaves only in the interval \([-m, m]\) and the second summand in this direct sum decomposition has nonzero cohomology sheaves only in the interval \([-m - n, m + \dim(X) - n]\). We conclude the system \(K_0 \to K_1[1] \to K_2[2] \to \ldots\) in \(D_{perf}(\mathcal{O}_{X \times Y})\) satisfies the assumptions of Lemma 0FZ9 after possibly replacing \(m\) by a larger integer. We conclude we can write \[K_n[n] = K \oplus C_n\] for \(n \gg 0\) compatible with transition maps and with \(C_n\) having nonzero cohomology sheaves only in the range \([-m - n, m - n]\). Denote \(G\) the Fourier-Mukai functor corresponding to \(K\). Putting everything together we find \[\begin{matrix} G(\mathcal{F}) \oplus R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}}^\mathbf{L} C_n) \cong \\ R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} K_n[n]) \cong \\ F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n[n])) \cong \\ F(\mathcal{F}) \oplus F(R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \mathcal{F}_n ))[n] \end{matrix}\] Looking at the degrees that objects live in we conclude that for \(n \gg m\) we obtain an isomorphism \[F(\mathcal{F}) \cong G(\mathcal{F})\] Moreover, recall that this holds for every coherent \(\mathcal{F}\) on \(X\) whose support has dimension \(0\).
Lemma
Let \(k\) be a field. Let \(X\) and \(Y\) be smooth proper schemes over \(k\). Given a \(k\)-linear, exact, fully faithful functor \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) there exists a Fourier-Mukai functor \(F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) whose kernel is in \(D_{perf}(\mathcal{O}_{X \times Y})\) which is a sibling to \(F\).
Proof
Apply Lemma 0G06 to \(F\) and the functor \(G\) constructed above.
The following theorem is also true without assuming \(X\) is projective, see [Noah].
Theorem
Let \(k\) be a field. Let \(X\) and \(Y\) be smooth proper schemes over \(k\) with \(X\) projective over \(k\). Any \(k\)-linear fully faithful exact functor \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) is a Fourier-Mukai functor for some kernel in \(D_{perf}(\mathcal{O}_{X \times Y})\).
Proof
Let \(F'\) be the Fourier-Mukai functor which is a sibling of \(F\) as in Lemma 0G0B. By Proposition 0FZW we have \(F \cong F'\) provided we can show that \(\textit{Coh}(\mathcal{O}_X)\) has enough negative objects. However, if \(X = \Spec(k)\) for example, then this isn’t true. Thus we first decompose \(X = \coprod X_i\) into its connected (and irreducible) components and we argue that it suffices to prove the result for each of the (fully faithful) composition functors \[F_i : D_{perf}(\mathcal{O}_{X_i}) \to D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\] Details omitted. Thus we may assume \(X\) is irreducible.
The case \(\dim(X) = 0\). Here \(X\) is the spectrum of a finite (separable) extension \(k'/k\) and hence \(D_{perf}(\mathcal{O}_X)\) is equivalent to the category of graded \(k'\)-vector spaces such that \(\mathcal{O}_X\) corresponds to the trivial \(1\)-dimensional vector space in degree \(0\). It is straightforward to see that any two siblings \(F, F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) are isomorphic. Namely, we are given an isomorphism \(F(\mathcal{O}_X) \cong F'(\mathcal{O}_X)\) compatible the action of the \(k\)-algebra \(k' = \text{End}_{D_{perf}(\mathcal{O}_X)}(\mathcal{O}_X)\) which extends canonically to an isomorphism on any graded \(k'\)-vector space.
The case \(\dim(X) > 0\). Here \(X\) is a projective smooth variety of dimension \(> 1\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. We have to show there exists a coherent module \(\mathcal{N}\) such that
there is a surjection \(\mathcal{N} \to \mathcal{F}\) and
\(\Hom(\mathcal{F}, \mathcal{N}) = 0\).
Choose an ample invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\). We claim that \(\mathcal{N} = (\mathcal{L}^{\otimes n})^{\oplus r}\) will work for \(n \ll 0\) and \(r\) large enough. Condition (1) follows from Properties, Proposition 01Q3. Finally, we have \[\Hom(\mathcal{F}, \mathcal{L}^{\otimes n}) = H^0(X, \SheafHom(\mathcal{F}, \mathcal{L}^{\otimes n})) = H^0(X, \SheafHom(\mathcal{F}, \mathcal{O}_X) \otimes \mathcal{L}^{\otimes n})\] Since the dual \(\SheafHom(\mathcal{F}, \mathcal{O}_X)\) is torsion free, this vanishes for \(n \ll 0\) by Varieties, Lemma 0FD7. This finishes the proof.
Proposition
Let \(k\) be a field. Let \(X\) and \(Y\) be smooth proper schemes over \(k\). If \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) is a \(k\)-linear exact equivalence of triangulated categories then there exists a Fourier-Mukai functor \(F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) whose kernel is in \(D_{perf}(\mathcal{O}_{X \times Y})\) which is an equivalence and a sibling of \(F\).
Proof
The functor \(F'\) of Lemma 0G0B is an equivalence by Lemma 0FZV.
Lemma
Let \(k\) be a field. Let \(X\) be a smooth proper scheme over \(k\). Let \(K \in D_{perf}(\mathcal{O}_{X \times X})\). If the Fourier-Mukai functor \(\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)\) is isomorphic to the identity functor, then \(K \cong \Delta_*\mathcal{O}_X\) in \(_{perf}(\mathcal{O}_{X \times X})\).
Proof
Let \(i\) be the minimal integer such that the cohomology sheaf \(H^i(K)\) is nonzero. Let \(\mathcal{E}\) and \(\mathcal{G}\) be finite locally free \(\mathcal{O}_X\)-modules. Then \[\begin{align*} H^i(X \times X, K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} (\mathcal{E} \boxtimes \mathcal{G})) & = H^i(X, R\text{pr}_{2, *}(K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} (\mathcal{E} \boxtimes \mathcal{G}))) \\ & = H^i(X, \Phi_K(\mathcal{E}) \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}) \\ & \cong H^i(X, \mathcal{E} \otimes \mathcal{G}) \end{align*}\] which is zero if \(i < 0\). On the other hand, we can choose \(\mathcal{E}\) and \(\mathcal{G}\) such that there is a surjection \(\mathcal{E}^\vee \boxtimes \mathcal{G}^\vee \to H^i(K)\) by Lemma 0FZ0. In this case the left hand side of the equalities is nonzero. Hence we conclude that \(H^i(K) = 0\) for \(i < 0\).
Let \(i\) be the maximal integer such that \(H^i(K)\) is nonzero. The same argument with \(\mathcal{E}\) and \(\mathcal{G}\) support of dimension \(0\) shows that \(i \leq 0\). Hence we conclude that \(K\) is given by a single coherent \(\mathcal{O}_{X \times X}\)-module \(\mathcal{K}\) sitting in degree \(0\).
Since \(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})\) is \(\mathcal{F}\), by taking \(\mathcal{F}\) supported at closed points we see that the support of \(\mathcal{K}\) is finite over \(X\) via \(\text{pr}_2\). Since \(R\text{pr}_{2, *}(\mathcal{K}) \cong \mathcal{O}_X\) we conclude by Functors and Morphisms, Lemma 0FZP that \(\mathcal{K} = s_*\mathcal{O}_X\) for some section \(s : X \to X \times X\) of the second projection. Then \(\Phi_K(M) = f^*M\) where \(f = \text{pr}_1 \circ s\) and this can happen only if \(s\) is the diagonal morphism as desired.
A category of Fourier-Mukai kernels
Let \(S\) be a scheme. We claim there is a category with
Objects are proper smooth schemes over \(S\).
Morphisms from \(X\) to \(Y\) are isomorphism classes of objects of \(D_{perf}(\mathcal{O}_{X \times_S Y})\).
Composition of the isomorphism class of \(K \in D_{perf}(\mathcal{O}_{X \times_S Y})\) and the isomorphism class of \(K'\) in \(D_{perf}(\mathcal{O}_{Y \times_S Z})\) is the isomorphism class of \[R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K')\] which is in \(D_{perf}(\mathcal{O}_{X \times_S Z})\) by Derived Categories of Schemes, Lemma 0B91.
The identity morphism from \(X\) to \(X\) is the isomorphism class of \(\Delta_{X/S, *}\mathcal{O}_X\) which is in \(D_{perf}(\mathcal{O}_{X \times_S X})\) by More on Morphisms, Lemma 0G2E and the fact that \(\Delta_{X/S}\) is a perfect morphism by Divisors, Lemma 067U and More on Morphisms, Lemma 068C.
Let us check that associativity of composition of morphisms holds; we omit verifying that the identity morphisms are indeed identities. To see this suppose we have \(X, Y, Z, W\) and \(c \in D_{perf}(\mathcal{O}_{X \times_S Y})\), \(c' \in D_{perf}(\mathcal{O}_{Y \times_S Z})\), and \(c'' \in D_{perf}(\mathcal{O}_{Z \times_S W})\). Then we have \[\begin{align*} c'' \circ (c' \circ c) & \cong \text{pr}^{134}_{14, *}( \text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *} \text{pr}^{1234}_{134, *} ((\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c'') \\ & \cong \text{pr}^{1234}_{14, *}( (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c'') \end{align*}\] Here we use the notation \[p^{1234}_{134} : X \times_S Y \times_S Z \times_S W \to X \times_S Z \times_S W \quad\text{and}\quad p^{134}_{14} : X \times_S Z \times_S W \to X \times_S W\] the projections and similarly for other indices. We also write \(\text{pr}_*\) instead of \(R\text{pr}_*\) and \(\text{pr}^*\) instead of \(L\text{pr}^*\) and we drop all super and sub scripts on \(\otimes\). The first equality is the definition of the composition. The second equality holds because \(\text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *} = \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}\) by base change (Derived Categories of Schemes, Lemma 08IB). The third equality holds because pullbacks compose correctly and pass through tensor products, see Cohomology, Lemmas 0D5S and 079U. The fourth equality follows from the “projection formula” for \(p^{1234}_{134}\), see Derived Categories of Schemes, Lemma 08EU. The fifth equality is that proper pushforward is compatible with composition, see Cohomology, Lemma 0D5T. Since tensor product is associative this concludes the proof of associativity of composition.
Lemma
Let \(S' \to S\) be a morphism of schemes. The rule which sends
a smooth proper scheme \(X\) over \(S\) to \(X' = S' \times_S X\), and
the isomorphism class of an object \(K\) of \(D_{perf}(\mathcal{O}_{X \times_S Y})\) to the isomorphism class of \(L(X' \times_{S'} Y' \to X \times_S Y)^*K\) in \(D_{perf}(\mathcal{O}_{X' \times_{S'} Y'})\)
is a functor from the category defined for \(S\) to the category defined for \(S'\).
Proof
To see this suppose we have \(X, Y, Z\) and \(K \in D_{perf}(\mathcal{O}_{X \times_S Y})\) and \(M \in D_{perf}(\mathcal{O}_{Y \times_S Z})\). Denote \(K' \in D_{perf}(\mathcal{O}_{X' \times_{S'} Y'})\) and \(M' \in D_{perf}(\mathcal{O}_{Y' \times_{S'} Z'})\) their pullbacks as in the statement of the lemma. The diagram \[\xymatrix{ X' \times_{S'} Y' \times_{S'} Z' \ar[r] \ar[d]_{\text{pr}'_{13}} & X \times_S Y \times_S Z \ar[d]^{\text{pr}_{13}} \\ X' \times_{S'} Z' \ar[r] & X \times_S Z }\] is cartesian and \(\text{pr}_{13}\) is proper and smooth. By Derived Categories of Schemes, Lemma 0B91 we see that the derived pullback by the lower horizontal arrow of the composition \[R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*M)\] indeed is (canonically) isomorphic to \[R\text{pr}'_{13, *}( L(\text{pr}'_{12})^*K' \otimes_{\mathcal{O}_{X' \times_{S'} Y' \times_{S'} Z'}}^\mathbf{L} L(\text{pr}'_{23})^*M')\] as desired. Some details omitted.
Relative equivalences
In this section we prove some lemmas about the following concept.
Definition
Let \(S\) be a scheme. Let \(X \to S\) and \(Y \to S\) be smooth proper morphisms. An object \(K \in D_{perf}(\mathcal{O}_{X \times_S Y})\) is said to be the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y\) over \(S\) if there exist an object \(K' \in D_{perf}(\mathcal{O}_{X \times_S Y})\) such that \[\Delta_{X/S, *}\mathcal{O}_X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S X}}^\mathbf{L} L\text{pr}_{23}^*K')\] in \(D(\mathcal{O}_{X \times_S X})\) and \[\Delta_{Y/S, *}\mathcal{O}_Y \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K' \otimes_{\mathcal{O}_{Y \times_S X \times_S Y}}^\mathbf{L} L\text{pr}_{23}^*K)\] in \(D(\mathcal{O}_{Y \times_S Y})\). In other words, the isomorphism class of \(K\) defines an invertible arrow in the category defined in Section 0G0F.
The language is intentionally cumbersome.
Lemma
With notation as in Definition 0G0I let \(K\) be the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y\) over \(S\). Then the corresponding Fourier-Mukai functors \(\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) (Lemma 0FYR) and \(\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) (Lemma 0FYT) are equivalences.
Proof
Lemma
With notation as in Definition 0G0I let \(K\) be the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y\) over \(S\). Let \(S_1 \to S\) be a morphism of schemes. Let \(X_1 = S_1 \times_S X\) and \(Y_1 = S_1 \times_S Y\). Then the pullback \(K_1 = L(X_1 \times_{S_1} Y_1 \to X \times_S Y)^*K\) is the Fourier-Mukai kernel of a relative equivalence from \(X_1\) to \(Y_1\) over \(S_1\).
Proof
Let \(K' \in D_{perf}(\mathcal{O}_{Y \times_S X})\) be the object assumed to exist in Definition 0G0I. Denote \(K'_1\) the pullback of \(K'\) by \(Y_1 \times_{S_1} X_1 \to Y \times_S X\). Then it suffices to prove that we have \[\Delta_{X_1/S_1, *}\mathcal{O}_X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K_1 \otimes_{\mathcal{O}_{X_1 \times_{S_1} Y_1 \times_{S_1} X_1}}^\mathbf{L} L\text{pr}_{23}^*K_1')\] in \(D(\mathcal{O}_{X_1 \times_{S_1} X_1})\) and similarly for the other condition. Since \[\xymatrix{ X_1 \times_{S_1} Y_1 \times_{S_1} X_1 \ar[r] \ar[d]_{\text{pr}_{13}} & X \times_S Y \times_S X \ar[d]^{\text{pr}_{13}} \\ X_1 \times_{S_1} X_1 \ar[r] & X \times_S X }\] is cartesian it suffices by Derived Categories of Schemes, Lemma 0B91 to prove that \[\Delta_{X_1/S_1, *}\mathcal{O}_{X_1} \cong L(X_1 \times_{S_1} X_1 \to X \times_S X)^*\Delta_{X/S, *}\mathcal{O}_X\] This in turn will be true if \(X\) and \(X_1 \times_{S_1} X_1\) are tor independent over \(X \times_S X\), see Derived Categories of Schemes, Lemma 08IB. This tor independence can be seen directly but also follows from the more general More on Morphisms, Lemma 0CTA applied to the square with corners \(X, X, X, S\) and its base change by \(S_1 \to S\).
Lemma
Let \(S = \lim_{i \in I} S_i\) be a limit of a directed system of schemes with affine transition morphisms \(g_{i'i} : S_{i'} \to S_i\). We assume that \(S_i\) is quasi-compact and quasi-separated for all \(i \in I\). Let \(0 \in I\). Let \(X_0 \to S_0\) and \(Y_0 \to S_0\) be smooth proper morphisms. We set \(X_i = S_i \times_{S_0} X_0\) for \(i \geq 0\) and \(X = S \times_{S_0} X_0\) and similarly for \(Y_0\). If \(K\) is the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y\) over \(S\) then for some \(i \geq 0\) there exists a Fourier-Mukai kernel of a relative equivalence from \(X_i\) to \(Y_i\) over \(S_i\).
Proof
Let \(K' \in D_{perf}(\mathcal{O}_{Y \times_S X})\) be the object assumed to exist in Definition 0G0I. Since \(X \times_S Y = \lim X_i \times_{S_i} Y_i\) there exists an \(i\) and objects \(K_i\) and \(K'_i\) in \(D_{perf}(\mathcal{O}_{Y_i \times_{S_i} X_i})\) whose pullbacks to \(Y \times_S X\) give \(K\) and \(K'\). See Derived Categories of Schemes, Lemma 09RF. By Derived Categories of Schemes, Lemma 0B91 the object \[R\text{pr}_{13, *}(L\text{pr}_{12}^*K_i \otimes_{\mathcal{O}_{X_i \times_{S_i} Y_i \times_{S_i} X_i}}^\mathbf{L} L\text{pr}_{23}^*K_i')\] is perfect and its pullback to \(X \times_S X\) is equal to \[R\text{pr}_{13, *}(L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S X}}^\mathbf{L} L\text{pr}_{23}^*K') \cong \Delta_{X/S, *}\mathcal{O}_X\] See proof of Lemma 0G0K. On the other hand, since \(X_i \to S\) is smooth and separated the object \[\Delta_{i, *}\mathcal{O}_{X_i}\] of \(D(\mathcal{O}_{X_i \times_{S_i} X_i})\) is also perfect (by More on Morphisms, Lemmas 0FDP and 0B6G) and its pullback to \(X \times_S X\) is equal to \[\Delta_{X/S, *}\mathcal{O}_X\] See proof of Lemma 0G0K. Thus by Derived Categories of Schemes, Lemma 09RF after increasing \(i\) we may assume that \[\Delta_{i, *}\mathcal{O}_{X_i} \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K_i \otimes_{\mathcal{O}_{X_i \times_{S_i} Y_i \times_{S_i} X_i}}^\mathbf{L} L\text{pr}_{23}^*K_i')\] as desired. The same works for the roles of \(K\) and \(K'\) reversed.
No deformations
The title of this section refers to Lemma 0G0R
Lemma
Let \((R, \mathfrak m, \kappa) \to (A, \mathfrak n, \lambda)\) be a flat local ring homorphism of local rings which is essentially of finite presentation. Let \(\overline{f}_1, \ldots, \overline{f}_r \in \mathfrak n/\mathfrak m A \subset A/\mathfrak m A\) be a regular sequence. Let \(K \in D(A)\). Assume
\(K\) is perfect,
\(K \otimes_A^\mathbf{L} A/\mathfrak m A\) is isomorphic in \(D(A/\mathfrak m A)\) to the Koszul complex on \(\overline{f}_1, \ldots, \overline{f}_r\).
Then \(K\) is isomorphic in \(D(A)\) to a Koszul complex on a regular sequence \(f_1, \ldots, f_r \in A\) lifting the given elements \(\overline{f}_1, \ldots, \overline{f}_r\). Moreover, \(A/(f_1, \ldots, f_r)\) is flat over \(R\).
Proof
Let us use chain complexes in the proof of this lemma. The Koszul complex \(K_\bullet(\overline{f}_1, \ldots, \overline{f}_r)\) is defined in More on Algebra, Definition 0623. By More on Algebra, Lemma 0BCB we can represent \(K\) by a complex \[K_\bullet : A \to A^{\oplus r} \to \ldots \to A^{\oplus r} \to A\] whose tensor product with \(A/\mathfrak mA\) is equal (!) to \(K_\bullet(\overline{f}_1, \ldots, \overline{f}_r)\). Denote \(f_1, \ldots, f_r \in A\) the components of the arrow \(A^{\oplus r} \to A\). These \(f_i\) are lifts of the \(\overline{f}_i\). By Algebra, Lemma 0470 \(f_1, \ldots, f_r\) form a regular sequence in \(A\) and \(A/(f_1, \ldots, f_r)\) is flat over \(R\). Let \(J = (f_1, \ldots, f_r) \subset A\). Consider the diagram \[\xymatrix{ K_\bullet \ar[rd] \ar@{..>}[rr]_{\varphi_\bullet} & & K_\bullet(f_1, \ldots, f_r) \ar[ld] \\ & A/J }\] Since \(f_1, \ldots, f_r\) is a regular sequence the south-west arrow is a quasi-isomorphism (see More on Algebra, Lemma 062F). Hence we can find the dotted arrow making the diagram commute for example by Algebra, Lemma 00LS. Reducing modulo \(\mathfrak m\) we obtain a commutative diagram \[\xymatrix{ K_\bullet(\overline{f}_1, \ldots, \overline{f}_r) \ar[rd] \ar[rr]_{\overline{\varphi}_\bullet} & & K_\bullet(\overline{f}_1, \ldots, \overline{f}_r) \ar[ld] \\ & (A/\mathfrak m A)/(\overline{f}_1, \ldots, \overline{f}_r) }\] by our choice of \(K_\bullet\). Thus \(\overline{\varphi}\) is an isomorphism in the derived category \(D(A/\mathfrak m A)\). It follows that \(\overline{\varphi} \otimes_{A/\mathfrak m A}^\mathbf{L} \lambda\) is an isomorphism. Since \(\overline{f}_i \in \mathfrak n / \mathfrak m A\) we see that \[\text{Tor}_i^{A/\mathfrak m A}( K_\bullet(\overline{f}_1, \ldots, \overline{f}_r), \lambda) = K_i(\overline{f}_1, \ldots, \overline{f}_r) \otimes_{A/\mathfrak m A} \lambda\] Hence \(\varphi_i \bmod \mathfrak n\) is invertible. Since \(A\) is local this means that \(\varphi_i\) is an isomorphism and the proof is complete.
Lemma
Let \(R \to S\) be a finite type flat ring map of Noetherian rings. Let \(\mathfrak q \subset S\) be a prime ideal lying over \(\mathfrak p \subset R\). Let \(K \in D(S)\) be perfect. Let \(f_1, \ldots, f_r \in \mathfrak q S_\mathfrak q\) be a regular sequence such that \(S_\mathfrak q/(f_1, \ldots, f_r)\) is flat over \(R\) and such that \(K \otimes_S^\mathbf{L} S_\mathfrak q\) is isomorphic to the Koszul complex on \(f_1, \ldots, f_r\). Then there exists a \(g \in S\), \(g \not \in \mathfrak q\) such that
\(f_1, \ldots, f_r\) are the images of \(f'_1, \ldots, f'_r \in S_g\),
\(f'_1, \ldots, f'_r\) form a regular sequence in \(S_g\),
\(S_g/(f'_1, \ldots, f'_r)\) is flat over \(R\),
\(K \otimes_S^\mathbf{L} S_g\) is isomorphic to the Koszul complex on \(f_1, \ldots, f_r\).
Proof
We can find \(g \in S\), \(g \not \in \mathfrak q\) with property (1) by the definition of localizations. After replacing \(g\) by \(gg'\) for some \(g' \in S\), \(g' \not \in \mathfrak q\) we may assume (2) holds, see Algebra, Lemma 061L. By Algebra, Theorem 00RC we find that \(S_g/(f'_1, \ldots, f'_r)\) is flat over \(R\) in an open neighbourhood of \(\mathfrak q\). Hence after once more replacing \(g\) by \(gg'\) for some \(g' \in S\), \(g' \not \in \mathfrak q\) we may assume (3) holds as well. Finally, we get (4) for a further replacement by More on Algebra, Lemma 0BC7.
For a generalization of the following lemma, please see More on Morphisms of Spaces, Lemma 05XD.
Lemma
Let \(S\) be a Noetherian scheme. Let \(s \in S\). Let \(p : X \to Y\) be a morphism of schemes over \(S\). Assume
\(Y \to S\) and \(X \to S\) proper,
\(X\) is flat over \(S\),
\(X_s \to Y_s\) an isomorphism.
Then there exists an open neighbourhood \(U \subset S\) of \(s\) such that the base change \(X_U \to Y_U\) is an isomorphism.
Proof
The morphism \(p\) is proper by Morphisms, Lemma 01W5. By Cohomology of Schemes, Lemma 02OH there is an open \(Y_s \subset V \subset Y\) such that \(p|_{p^{-1}(V)} : p^{-1}(V) \to V\) is finite. By More on Morphisms, Theorem 039B there is an open \(X_s \subset U \subset X\) such that \(p|_U : U \to Y\) is flat. After removing the images of \(X \setminus U\) and \(Y \setminus V\) (which are closed subsets not containing \(s\)) we may assume \(p\) is flat and finite. Then \(p\) is open (Morphisms, Lemma 01UA) and \(Y_s \subset p(X) \subset Y\) hence after shrinking \(S\) we may assume \(p\) is surjective. As \(p_s : X_s \to Y_s\) is an isomorphism, the map \[p^\sharp : \mathcal{O}_Y \longrightarrow p_*\mathcal{O}_X\] of coherent \(\mathcal{O}_Y\)-modules (\(p\) is finite) becomes an isomorphism after pullback by \(i : Y_s \to Y\) (by Cohomology of Schemes, Lemma 02KG for example). By Nakayama’s lemma, this implies that \(\mathcal{O}_{Y, y} \to (p_*\mathcal{O}_X)_y\) is surjective for all \(y \in Y_s\). Hence there is an open \(Y_s \subset V \subset Y\) such that \(p^\sharp|_V\) is surjective (Modules, Lemma 01B8). Hence after shrinking \(S\) once more we may assume \(p^\sharp\) is surjective which means that \(p\) is a closed immersion (as \(p\) is already finite). Thus now \(p\) is a surjective flat closed immersion of Noetherian schemes and hence an isomorphism, see Morphisms, Section 04PV.
Lemma
Let \(k\) be a field. Let \(S\) be a finite type scheme over \(k\) with \(k\)-rational point \(s\). Let \(Y \to S\) be a smooth proper morphism. Let \(X = Y_s \times S \to S\) be the constant family with fibre \(Y_s\). Let \(K\) be the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y\) over \(S\). Assume the restriction \[L(Y_s \times_S Y_s \to X \times_S Y)^*K \cong \Delta_{Y_s/k, *} \mathcal{O}_{Y_s}\] in \(D(\mathcal{O}_{Y_s \times Y_s})\). Then there is an open neighbourhood \(s \in U \subset S\) such that \(Y|_U\) is isomorphic to \(Y_s \times U\) over \(U\).
Proof
Denote \(i : Y_s \times Y_s = X_s \times Y_s \to X \times_S Y\) the natural closed immersion. (We will write \(Y_s\) and not \(X_s\) for the fibre of \(X\) over \(s\) from now on.) Let \(z \in Y_s \times Y_s = (X \times_S Y)_s \subset X \times_S Y\) be a closed point. As indicated we think of \(z\) both as a closed point of \(Y_s \times Y_s\) as well as a closed point of \(X \times_S Y\).
Case I: \(z \not \in \Delta_{Y_s/k}(Y_s)\). Denote \(\mathcal{O}_z\) the coherent \(\mathcal{O}_{Y_s \times Y_s}\)-module supported at \(z\) whose value is \(\kappa(z)\). Then \(i_*\mathcal{O}_z\) is the coherent \(\mathcal{O}_{X \times_S Y}\)-module supported at \(z\) whose value is \(\kappa(z)\). Our assumption means that \[K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} i_*\mathcal{O}_z = Li^*K \otimes_{\mathcal{O}_{Y_s \times Y_s}}^\mathbf{L} \mathcal{O}_z = 0\] Hence by Lemma 0G03 we find an open neighbourhood \(U(z) \subset X \times_S Y\) of \(z\) such that \(K|_{U(z)} = 0\). In this case we set \(Z(z) = \emptyset\) as closed subscheme of \(U(z)\).
Case II: \(z \in \Delta_{Y_s/k}(Y_s)\). Since \(Y_s\) is smooth over \(k\) we know that \(\Delta_{Y_s/k} : Y_s \to Y_s \times Y_s\) is a regular immersion, see More on Morphisms, Lemma 0FDP. Choose a regular sequence \(\overline{f}_1, \ldots, \overline{f}_r \in \mathcal{O}_{Y_s \times Y_s, z}\) cutting out the ideal sheaf of \(\Delta_{Y_s/k}(Y_s)\). Since a regular sequence is Koszul-regular (More on Algebra, Lemma 062F) our assumption means that \[K_z \otimes_{\mathcal{O}_{X \times_S Y, z}}^\mathbf{L} \mathcal{O}_{Y_s \times Y_s, z} \in D(\mathcal{O}_{Y_s \times Y_s, z})\] is represented by the Koszul complex on \(\overline{f}_1, \ldots, \overline{f}_r\) over \(\mathcal{O}_{Y_s \times Y_s, z}\). By Lemma 0G0N applied to \(\mathcal{O}_{S, s} \to \mathcal{O}_{X \times_S Y, z}\) we conclude that \(K_z \in D(\mathcal{O}_{X \times_S Y, z})\) is represented by the Koszul complex on a regular sequence \(f_1, \ldots, f_r \in \mathcal{O}_{X \times_S Y, z}\) lifting the regular sequence \(\overline{f}_1, \ldots, \overline{f}_r\) such that moreover \(\mathcal{O}_{X \times_S Y}/(f_1, \ldots, f_r)\) is flat over \(\mathcal{O}_{S, s}\). By some limit arguments (Lemma 0G0P) we conclude that there exists an affine open neighbourhood \(U(z) \subset X \times_S Y\) of \(z\) and a closed subscheme \(Z(z) \subset U(z)\) such that
\(Z(z) \to U(z)\) is a regular closed immersion,
\(K|_{U(z)}\) is quasi-isomorphic to \(\mathcal{O}_{Z(z)}\),
\(Z(z) \to S\) is flat,
\(Z(z)_s = \Delta_{Y_s/k}(Y_s) \cap U(z)_s\) as closed subschemes of \(U(z)_s\).
By property (2), for \(z, z' \in Y_s \times Y_s\), we find that \(Z(z) \cap U(z') = Z(z') \cap U(z)\) as closed subschemes. Hence we obtain an open neighbourhood \[U = \bigcup\nolimits_{z \in Y_s \times Y_s\text{ closed}} U(z)\] of \(Y_s \times Y_s\) in \(X \times_S Y\) and a closed subscheme \(Z \subset U\) such that (1) \(Z \to U\) is a regular closed immersion, (2) \(Z \to S\) is flat, and (3) \(Z_s = \Delta_{Y_s/k}(Y_s)\). Since \(X \times_S Y \to S\) is proper, after replacing \(S\) by an open neighbourhood of \(s\) we may assume \(U = X \times_S Y\). Since the projections \(Z_s \to Y_s\) and \(Z_s \to X_s\) are isomorphisms, we conclude that after shrinking \(S\) we may assume \(Z \to Y\) and \(Z \to X\) are isomorphisms, see Lemma 0G0Q. This finishes the proof.
Lemma
Let \(k\) be an algebraically closed field. Let \(X\) be a smooth proper scheme over \(k\). Let \(f : Y \to S\) be a smooth proper morphism with \(S\) of finite type over \(k\). Let \(K\) be the Fourier-Mukai kernel of a relative equivalence from \(X \times S\) to \(Y\) over \(S\). Then \(S\) can be covered by open subschemes \(U\) such that there is a \(U\)-isomorphism \(f^{-1}(U) \cong Y_0 \times U\) for some \(Y_0\) proper and smooth over \(k\).
Proof
Choose a closed point \(s \in S\). Since \(k\) is algebraically closed this is a \(k\)-rational point. Set \(Y_0 = Y_s\). The restriction \(K_0\) of \(K\) to \(X \times Y_0\) is the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y_0\) over \(\Spec(k)\) by Lemma 0G0K. Let \(K'_0\) in \(D_{perf}(\mathcal{O}_{Y_0 \times X})\) be the object assumed to exist in Definition 0G0I. Then \(K'_0\) is the Fourier-Mukai kernel of a relative equivalence from \(Y_0\) to \(X\) over \(\Spec(k)\) by the symmetry inherent in Definition 0G0I. Hence by Lemma 0G0K we see that the pullback \[M = (Y_0 \times X \times S \to Y_0 \times X)^*K'_0\] on \((Y_0 \times S) \times_S (X \times S) = Y_0 \times X \times S\) is the Fourier-Mukai kernel of a relative equivalence from \(Y_0 \times S\) to \(X \times S\) over \(S\). Now consider the kernel \[K_{new} = R\text{pr}_{13, *}(L\text{pr}_{12}^*M \otimes_{\mathcal{O}_{(Y_0 \times S) \times_S (X \times S) \times_S Y}}^\mathbf{L} L\text{pr}_{23}^*K)\] on \((Y_0 \times S) \times_S Y\). This is the Fourier-Mukai kernel of a relative equivalence from \(Y_0 \times S\) to \(Y\) over \(S\) since it is the composition of two invertible arrows in the category constructed in Section 0G0F. Moreover, this composition passes through base change (Lemma 0G0G). Hence we see that the pullback of \(K_{new}\) to \(((Y_0 \times S) \times_S Y)_s = Y_0 \times Y_0\) is equal to the composition of \(K_0\) and \(K'_0\) and hence equal to the identity in this category. In other words, we have \[L(Y_0 \times Y_0 \to (Y_0 \times S) \times_S Y)^*K_{new} \cong \Delta_{Y_0/k, *}\mathcal{O}_{Y_0}\] Thus by Lemma 0G0R we conclude that \(Y \to S\) is isomorphic to \(Y_0 \times S\) in an open neighbourhood of \(s\). This finishes the proof.
Countability
In this section we prove some elementary lemmas about countability of certain sets. Let \(\mathcal{C}\) be a category. In this section we will say that \(\mathcal{C}\) is countable if
for any \(X, Y \in \Ob(\mathcal{C})\) the set \(\Mor_\mathcal{C}(X, Y)\) is countable, and
the set of isomorphism classes of objects of \(\mathcal{C}\) is countable.
Lemma
Let \(R\) be a countable Noetherian ring. Then the category of schemes of finite type over \(R\) is countable.
Proof
Omitted.
Lemma
Let \(\mathcal{A}\) be a countable abelian category. Then \(D^b(\mathcal{A})\) is countable.
Proof
It suffices to prove the statement for \(D(\mathcal{A})\) as the others are full subcategories of this one. Since every object in \(D(\mathcal{A})\) is a complex of objects of \(\mathcal{A}\) it is immediate that the set of isomorphism classes of objects of \(D^b(\mathcal{A})\) is countable. Moreover, for bounded complexes \(A^\bullet\) and \(B^\bullet\) of \(\mathcal{A}\) it is clear that \(\Hom_{K^b(\mathcal{A})}(A^\bullet, B^\bullet)\) is countable. We have \[\Hom_{D^b(\mathcal{A})}(A^\bullet, B^\bullet) = \colim_{s : (A')^\bullet \to A^\bullet \text{ qis and }(A')^\bullet\text{ bounded}} \Hom_{K^b(\mathcal{A})}((A')^\bullet, B^\bullet)\] by Derived Categories, Lemma 05RW. Thus this is a countable set as a countable colimit of
Lemma
Let \(X\) be a scheme of finite type over a countable Noetherian ring. Then the categories \(D_{perf}(\mathcal{O}_X)\) and \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) are countable.
Proof
Observe that \(X\) is Noetherian by Morphisms, Lemma 01T6. Hence \(D_{perf}(\mathcal{O}_X)\) is a full subcategory of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) by Derived Categories of Schemes, Lemma 0FXU. Thus it suffices to prove the result for \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). Recall that \(D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X))\) by Derived Categories of Schemes, Proposition 0FDB. Hence by Lemma 0G0V it suffices to prove that \(\textit{Coh}(\mathcal{O}_X)\) is countable. This we omit.
Lemma
Let \(K\) be an algebraically closed field. Let \(S\) be a finite type scheme over \(K\). Let \(X \to S\) and \(Y \to S\) be finite type morphisms. There exists a countable set \(I\) and for \(i \in I\) a pair \((S_i \to S, h_i)\) with the following properties
\(S_i \to S\) is a morphism of finite type, set \(X_i = X \times_S S_i\) and \(Y_i = Y \times_S S_i\),
\(h_i : X_i \to Y_i\) is an isomorphism over \(S_i\), and
for any closed point \(s \in S(K)\) if \(X_s \cong Y_s\) over \(K = \kappa(s)\) then \(s\) is in the image of \(S_i \to S\) for some \(i\).
Proof
The field \(K\) is the filtered union of its countable subfields. Dually, \(\Spec(K)\) is the cofiltered limit of the spectra of the countable subfields of \(K\). Hence Limits, Lemma 01ZM guarantees that we can find a countable subfield \(k\) and morphisms \(X_0 \to S_0\) and \(Y_0 \to S_0\) of schemes of finite type over \(k\) such that \(X \to S\) and \(Y \to S\) are the base changes of these.
By Lemma 0G0U there is a countable set \(I\) and pairs \((S_{0, i} \to S_0, h_{0, i})\) such that
\(S_{0, i} \to S_0\) is a morphism of finite type, set \(X_{0, i} = X_0 \times_{S_0} S_{0, i}\) and \(Y_{0, i} = Y_0 \times_{S_0} S_{0, i}\),
\(h_{0, i} : X_{0, i} \to Y_{0, i}\) is an isomorphism over \(S_{0, i}\).
such that every pair \((T \to S_0, h_T)\) with \(T \to S_0\) of finite type and \(h_T : X_0 \times_{S_0} T \to Y_0 \times_{S_0} T\) an isomorphism is isomorphic to one of these. Denote \((S_i \to S, h_i)\) the base change of \((S_{0, i} \to S_0, h_{0, i})\) by \(\Spec(K) \to \Spec(k)\). We claim this works.
Let \(s \in S(K)\) and let \(h_s : X_s \to Y_s\) be an isomorphism over \(K = \kappa(s)\). We can write \(K\) as the filtered union of its finitely generated \(k\)-subalgebras. Hence by Limits, Proposition 01ZC and Lemma 01ZM we can find such a finitely generated \(k\)-subalgebra \(K \supset A \supset k\) such that
there is a commutative diagram \[\xymatrix{ \Spec(K) \ar[d]_s \ar[r] & \Spec(A) \ar[d]^{s'} \\ S \ar[r] & S_0}\] for some morphism \(s' : \Spec(A) \to S_0\) over \(k\),
\(h_s\) is the base change of an isomorphism \(h_{s'} : X_0 \times_{S_0, s'} \Spec(A) \to X_0 \times_{S_0, s'} \Spec(A)\) over \(A\).
Of course, then \((s' : \Spec(A) \to S_0, h_{s'})\) is isomorphic to the pair \((S_{0, i} \to S_0, h_{0, i})\) for some \(i \in I\). This concludes the proof because the commutative diagram in (1) shows that \(s\) is in the image of the base change of \(s'\) to \(\Spec(K)\).
Lemma
Let \(K\) be an algebraically closed field. There exists a countable set \(I\) and for \(i \in I\) a pair \((S_i/K, X_i \to S_i, Y_i \to S_i, M_i)\) with the following properties
\(S_i\) is a scheme of finite type over \(K\),
\(X_i \to S_i\) and \(Y_i \to S_i\) are proper smooth morphisms of schemes,
\(M_i \in D_{perf}(\mathcal{O}_{X_i \times_{S_i} Y_i})\) is the Fourier-Mukai kernel of a relative equivalence from \(X_i\) to \(Y_i\) over \(S_i\), and
for any smooth proper schemes \(X\) and \(Y\) over \(K\) such that there is a \(K\)-linear exact equivalence \(D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\) there exists an \(i \in I\) and a \(s \in S_i(K)\) such that \(X \cong (X_i)_s\) and \(Y \cong (Y_i)_s\).
Proof
Choose a countable subfield \(k \subset K\) for example the prime field. By Lemmas 0G0U and 0G0W there exists a countable set of isomorphism classes of systems over \(k\) satisfying parts (1), (2), (3) of the lemma. Thus we can choose a countable set \(I\) and for each \(i \in I\) such a system \[(S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i})\] over \(k\) such that each isomorphism class occurs at least once. Denote \((S_i/K, X_i \to S_i, Y_i \to S_i, M_i)\) the base change of the displayed system to \(K\). This system has properties (1), (2), (3), see Lemma 0G0K. Let us prove property (4).
Consider smooth proper schemes \(X\) and \(Y\) over \(K\) such that there is a \(K\)-linear exact equivalence \(F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\). By Proposition 0G0D we may assume that there exists an object \(M \in D_{perf}(\mathcal{O}_{X \times Y})\) such that \(F = \Phi_M\) is the corresponding Fourier-Mukai functor. By Lemma 0FYY there is an \(M'\) in \(D_{perf}(\mathcal{O}_{Y \times X})\) such that \(\Phi_{M'}\) is the right adjoint to \(\Phi_M\). Since \(\Phi_M\) is an equivalence, this means that \(\Phi_{M'}\) is the quasi-inverse to \(\Phi_M\). By Lemma 0FYY we see that the Fourier-Mukai functors defined by the objects \[A = R\text{pr}_{13, *}( L\text{pr}_{12}^*M \otimes_{\mathcal{O}_{X \times Y \times X}}^\mathbf{L} L\text{pr}_{23}^*M')\] in \(D_{perf}(\mathcal{O}_{X \times X})\) and \[B = R\text{pr}_{13, *}( L\text{pr}_{12}^*M' \otimes_{\mathcal{O}_{Y \times X \times Y}}^\mathbf{L} L\text{pr}_{23}^*M)\] in \(D_{perf}(\mathcal{O}_{Y \times Y})\) are isomorphic to \(\text{id} : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)\) and \(\text{id} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_Y)\) Hence \(A \cong \Delta_{X/K, *}\mathcal{O}_X\) and \(B \cong \Delta_{Y/K, *}\mathcal{O}_Y\) by Lemma 0G0E. Hence we see that \(M\) is the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y\) over \(K\) by definition.
We can write \(K\) as the filtered colimit of its finite type \(k\)-subalgebras \(A \subset K\). By Limits, Lemma 01ZM we can find \(X_0, Y_0\) of finite type over \(A\) whose base changes to \(K\) produces \(X\) and \(Y\). By Limits, Lemmas 081F and 0C0C after enlarging \(A\) we may assume \(X_0\) and \(Y_0\) are smooth and proper over \(A\). By Lemma 0G0L after enlarging \(A\) we may assume \(M\) is the pullback of some \(M_0 \in D_{perf}(\mathcal{O}_{X_0 \times_{\Spec(A)} Y_0})\) which is the Fourier-Mukai kernel of a relative equivalence from \(X_0\) to \(Y_0\) over \(\Spec(A)\). Thus we see that \((S_0/k, X_0 \to S_0, Y_0 \to S_0, M_0)\) is isomorphic to \((S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i})\) for some \(i \in I\). Since \(S_i = S_{0, i} \times_{\Spec(k)} \Spec(K)\) we conclude that (4) is true with \(s : \Spec(K) \to S_i\) induced by the morphism \(\Spec(K) \to \Spec(A) \cong S_{0, i}\) we get from \(A \subset K\).
Countability of derived equivalent varieties
In this section we prove a result of Anel and Toën, see [AT].
Definition
Let \(k\) be a field. Let \(X\) and \(Y\) be smooth projective schemes over \(k\). We say \(X\) and \(Y\) are derived equivalent if there exists a \(k\)-linear exact equivalence \(D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)\).
Here is the result
Theorem
Let \(K\) be an algebraically closed field. Let \(\mathbf{X}\) be a smooth proper scheme over \(K\). There are at most countably many isomorphism classes of smooth proper schemes \(\mathbf{Y}\) over \(K\) which are derived equivalent to \(\mathbf{X}\).
Proof
Choose a countable set \(I\) and for \(i \in I\) systems \((S_i/K, X_i \to S_i, Y_i \to S_i, M_i)\) satisfying properties (1), (2), (3), and (4) of Lemma 0G0Y. Pick \(i \in I\) and set \(S = S_i\), \(X = X_i\), \(Y = Y_i\), and \(M = M_i\). Clearly it suffice to show that the set of isomorphism classes of fibres \(Y_s\) for \(s \in S(K)\) such that \(X_s \cong \mathbf{X}\) is countable. This we prove in the next paragraph.
Let \(S\) be a finite type scheme over \(K\), let \(X \to S\) and \(Y \to S\) be proper smooth morphisms, and let \(M \in D_{perf}(\mathcal{O}_{X \times_S Y})\) be the Fourier-Mukai kernel of a relative equivalence from \(X\) to \(Y\) over \(S\). We will show the set of isomorphism classes of fibres \(Y_s\) for \(s \in S(K)\) such that \(X_s \cong \mathbf{X}\) is countable. By Lemma 0G0X applied to the families \(\mathbf{X} \times S \to S\) and \(X \to S\) there exists a countable set \(I\) and for \(i \in I\) a pair \((S_i \to S, h_i)\) with the following properties
\(S_i \to S\) is a morphism of finite type, set \(X_i = X \times_S S_i\),
\(h_i : \mathbf{X} \times S_i \to X_i\) is an isomorphism over \(S_i\), and
for any closed point \(s \in S(K)\) if \(\mathbf{X} \cong X_s\) over \(K = \kappa(s)\) then \(s\) is in the image of \(S_i \to S\) for some \(i\).
Set \(Y_i = Y \times_S S_i\). Denote \(M_i \in D_{perf}(\mathcal{O}_{X_i \times_{S_i} Y_i})\) the pullback of \(M\). By Lemma 0G0K \(M_i\) is the Fourier-Mukai kernel of a relative equivalence from \(X_i\) to \(Y_i\) over \(S_i\). Since \(I\) is countable, by property (3) it suffices to prove that the set of isomorphism classes of fibres \(Y_{i, s}\) for \(s \in S_i(K)\) is countable. In fact, this number is finite by Lemma 0G0S and the proof is complete.