Derived Categories of Spaces

Introduction

In this chapter we discuss derived categories of modules on algebraic spaces. There do not seem to be good introductory references addressing this topic; it is covered in the literature by referring to papers dealing with derived categories of modules on algebraic stacks, for example see [olsson_sheaves].

Conventions

If \(\mathcal{A}\) is an abelian category and \(M\) is an object of \(\mathcal{A}\) then we also denote \(M\) the object of \(K(\mathcal{A})\) and/or \(D(\mathcal{A})\) corresponding to the complex which has \(M\) in degree \(0\) and is zero in all other degrees.

If we have a ring \(A\), then \(K(A)\) denotes the homotopy category of complexes of \(A\)-modules and \(D(A)\) the associated derived category. Similarly, if we have a ringed space \((X, \mathcal{O}_X)\) the symbol \(K(\mathcal{O}_X)\) denotes the homotopy category of complexes of \(\mathcal{O}_X\)-modules and \(D(\mathcal{O}_X)\) the associated derived category.

Generalities

In this section we put some general results on cohomology of unbounded complexes of modules on algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Given an étale morphism \(V \to Y\), set \(U = V \times_Y X\) and denote \(g : U \to V\) the projection morphism. Then \((Rf_*E)|_V = Rg_*(E|_U)\) for \(E\) in \(D(\mathcal{O}_X)\).

Proof

Represent \(E\) by a K-injective complex \(\mathcal{I}^\bullet\) of \(\mathcal{O}_X\)-modules. Then \(Rf_*(E) = f_*\mathcal{I}^\bullet\) and \(Rg_*(E|_U) = g_*(\mathcal{I}^\bullet|_U)\) by Cohomology on Sites, Lemma 08FI. Hence the result follows from Properties of Spaces, Lemma 03LX.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(T \subset |X|\) be a closed subset. We say \(E\) is supported on \(T\) if the cohomology sheaves \(H^i(E)\) are supported on \(T\).

Derived category of quasi-coherent modules on the small étale site

Let \(X\) be a scheme. In this section we show that \(D_\QCoh(\mathcal{O}_X)\) can be defined in terms of the small étale site \(X_\etale\) of \(X\). Denote \(\mathcal{O}_\etale\) the structure sheaf on \(X_\etale\). Consider the morphism of ringed sites [08H7]\[\begin{equation} \epsilon : (X_\etale, \mathcal{O}_\etale) \longrightarrow (X_{Zar}, \mathcal{O}_X). \end{equation}\] denoted \(\text{id}_{small, \etale, Zar}\) in Descent, Lemma 070S.

Lemma

The morphism \(\epsilon\) of (08H7) is a flat morphism of ringed sites. In particular the functor \(\epsilon^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_\etale)\) is exact. Moreover, if \(\epsilon^*\mathcal{F} = 0\), then \(\mathcal{F} = 0\).

Proof

The flatness of the morphism \(\epsilon\) is Descent, Lemma 0GNA. Here is another proof. We have to show that \(\mathcal{O}_\etale\) is a flat \(\epsilon^{-1}\mathcal{O}_X\)-module. To do this it suffices to check \(\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}\) is flat for any geometric point \(\overline{x}\) of \(X\), see Modules on Sites, Lemma 05VC, Sites, Lemma 05V1, and Étale Cohomology, Remarks 040S. By Étale Cohomology, Lemma 04HX we see that \(\mathcal{O}_{\etale, \overline{x}}\) is the strict henselization of \(\mathcal{O}_{X, x}\). Thus \(\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}\) is faithfully flat by More on Algebra, Lemma 07QM.

The exactness of \(\epsilon^*\) follows from the flatness of \(\epsilon\) by Modules on Sites, Lemma 04JC.

Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. If \(\epsilon^*\mathcal{F} = 0\), then with notation as above \[0 = \epsilon^*\mathcal{F}_{\overline{x}} = \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{\etale, \overline{x}}\] (Modules on Sites, Lemma 05V5) for all geometric points \(\overline{x}\). By faithful flatness of \(\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}\) we conclude \(\mathcal{F}_x = 0\) for all \(x \in X\).

Let \(X\) be a scheme. Notation as in (08H7). Recall that \(\epsilon^* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_\etale)\) is an equivalence by Descent, Proposition 03DX and Remark 03FH. Moreover, \(\QCoh(\mathcal{O}_\etale)\) forms a Serre subcategory of \(\textit{Mod}(\mathcal{O}_\etale)\) by Descent, Lemma 06VE. Hence we can let \(D_\QCoh(\mathcal{O}_\etale)\) be the triangulated subcategory of \(D(\mathcal{O}_\etale)\) whose objects are the complexes with quasi-coherent cohomology sheaves, see Derived Categories, Section 06UP. The functor \(\epsilon^*\) is exact (Lemma 08H8) hence induces \(\epsilon^* : D(\mathcal{O}_X) \to D(\mathcal{O}_\etale)\) and since pullbacks of quasi-coherent modules are quasi-coherent also \(\epsilon^* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_\etale)\).

Lemma

Let \(X\) be a scheme. The functor \(\epsilon^* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_\etale)\) defined above is an equivalence.

Proof

We will prove this by showing the functor \(R\epsilon_* : D(\mathcal{O}_\etale) \to D(\mathcal{O}_X)\) induces a quasi-inverse. We will use freely that \(\epsilon_*\) is given by restriction to \(X_{Zar} \subset X_\etale\) and the description of \(\epsilon^* = \text{id}_{small, \etale, Zar}^*\) in Descent, Lemma 070S.

For a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the adjunction map \(\mathcal{F} \to \epsilon_*\epsilon^*\mathcal{F}\) is an isomorphism by the fact that \(\mathcal{F}^a\) (Descent, Definition 03DU) is a sheaf as proved in Descent, Lemma 03DT. Conversely, every quasi-coherent \(\mathcal{O}_\etale\)-module \(\mathcal{H}\) is of the form \(\epsilon^*\mathcal{F}\) for some quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), see Descent, Proposition 03DX. Then \(\mathcal{F} = \epsilon_*\mathcal{H}\) by what we just said and we conclude that the adjunction map \(\epsilon^*\epsilon_*\mathcal{H} \to \mathcal{H}\) is an isomorphism for all quasi-coherent \(\mathcal{O}_\etale\)-modules \(\mathcal{H}\).

Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_\etale)\) and denote \(\mathcal{H}^q = H^q(E)\) its \(q\)th cohomology sheaf. Let \(\mathcal{B}\) be the set of affine objects of \(X_\etale\). Then \(H^p(U, \mathcal{H}^q) = 0\) for all \(p > 0\), all \(q \in \mathbf{Z}\), and all \(U \in \mathcal{B}\), see Descent, Proposition 03DW and Cohomology of Schemes, Lemma 01XB. By Cohomology on Sites, Lemma 0BKZ this means that \[H^q(U, E) = H^0(U, \mathcal{H}^q)\] for all \(U \in \mathcal{B}\). In particular, we find that this holds for affine opens \(U \subset X\). It follows that the \(q\)th cohomology of \(R\epsilon_*E\) over \(U\) is the value of the sheaf \(\epsilon_*\mathcal{H}^q\) over \(U\). Applying sheafification we obtain \[H^q(R\epsilon_*E) = \epsilon_*\mathcal{H}^q\] which in particular shows that \(R\epsilon_*\) induces a functor \(D_\QCoh(\mathcal{O}_\etale) \to D_\QCoh(\mathcal{O}_X)\). Since \(\epsilon^*\) is exact we then obtain \(H^q(\epsilon^*R\epsilon_*E) = \epsilon^*\epsilon_*\mathcal{H}^q = \mathcal{H}^q\) (by discussion above). Thus the adjunction map \(\epsilon^*R\epsilon_*E \to E\) is an isomorphism.

Conversely, for \(F \in D_\QCoh(\mathcal{O}_X)\) the adjunction map \(F \to R\epsilon_*\epsilon^*F\) is an isomorphism for the same reason, i.e., because the cohomology sheaves of \(R\epsilon_*\epsilon^*F\) are isomorphic to \(\epsilon_*H^m(\epsilon^*F) = \epsilon_*\epsilon^*H^m(F) = H^m(F)\).

Derived category of quasi-coherent modules

Let \(S\) be a scheme. Lemma 071Q shows that the category \(D_\QCoh(\mathcal{O}_S)\) can be defined in terms of complexes of \(\mathcal{O}_S\)-modules on the scheme \(S\) or by complexes of \(\mathcal{O}\)-modules on the small étale site of \(S\). Hence the following definition is compatible with the definition in the case of schemes.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). The derived category of \(\mathcal{O}_X\)-modules with quasi-coherent cohomology sheaves is denoted \(D_\QCoh(\mathcal{O}_X)\).

This makes sense by Properties of Spaces, Lemma 03M1 and Derived Categories, Section 06UP. Thus we obtain a canonical functor [08F1]\[\begin{equation} D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X) \end{equation}\] see Derived Categories, Equation (06UR).

Observe that a flat morphism \(f : Y \to X\) of algebraic spaces induces an exact functor \(f^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y)\), see Morphisms of Spaces, Lemma 073C and Modules on Sites, Lemma 04JC. In particular \(Lf^* : D(\mathcal{O}_X) \to D(\mathcal{O}_Y)\) is computed on any representative complex (Derived Categories, Lemma 015F). We will write \(Lf^* = f^*\) when \(f\) is flat and we have \(H^i(f^*E) = f^*H^i(E)\) for \(E\) in \(D(\mathcal{O}_X)\) in this case. We will use this often when \(f\) is étale. Of course in the étale case the pullback functor is just the restriction to \(Y_\etale\), see Properties of Spaces, Equation (03LW).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E\) be an object of \(D(\mathcal{O}_X)\). The following are equivalent

1. \(E\) is in \(D_\QCoh(\mathcal{O}_X)\),

2. for every étale morphism \(\varphi : U \to X\) where \(U\) is an affine scheme \(\varphi^*E\) is an object of \(D_\QCoh(\mathcal{O}_U)\),

3. for every étale morphism \(\varphi : U \to X\) where \(U\) is a scheme \(\varphi^*E\) is an object of \(D_\QCoh(\mathcal{O}_U)\),

4. there exists a surjective étale morphism \(\varphi : U \to X\) where \(U\) is a scheme such that \(\varphi^*E\) is an object of \(D_\QCoh(\mathcal{O}_U)\), and

5. there exists a surjective étale morphism of algebraic spaces \(f : Y \to X\) such that \(Lf^*E\) is an object of \(D_\QCoh(\mathcal{O}_Y)\).

Proof

This follows immediately from the discussion preceding the lemma and Properties of Spaces, Lemma 03M0.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Then \(D_\QCoh(\mathcal{O}_X)\) has direct sums.

Proof

By Injectives, Lemma 07D9 the derived category \(D(\mathcal{O}_X)\) has direct sums and they are computed by taking termwise direct sums of any representatives. Thus it is clear that the cohomology sheaf of a direct sum is the direct sum of the cohomology sheaves as taking direct sums is an exact functor (in any Grothendieck abelian category). The lemma follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see Properties of Spaces, Lemma 03M1.

We will need some information on derived limits. We warn the reader that in the lemma below the derived limit will typically not be an object of \(D_\QCoh\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \((K_n)\) be an inverse system of \(D_\QCoh(\mathcal{O}_X)\) with derived limit \(K = R\lim K_n\) in \(D(\mathcal{O}_X)\). Assume \(H^q(K_{n + 1}) \to H^q(K_n)\) is surjective for all \(q \in \mathbf{Z}\) and \(n \geq 1\). Then

1. \(H^q(K) = \lim H^q(K_n)\),

2. \(R\lim H^q(K_n) = \lim H^q(K_n)\), and

3. for every affine open \(U \subset X\) we have \(H^p(U, \lim H^q(K_n)) = 0\) for \(p > 0\).

Proof

Let \(\mathcal{B} \subset \Ob(X_\etale)\) be the set of affine objects. Since \(H^q(K_n)\) is quasi-coherent we have \(H^p(U, H^q(K_n)) = 0\) for \(U \in \mathcal{B}\) by the discussion in Cohomology of Spaces, Section 071Y and Cohomology of Schemes, Lemma 01XB. Moreover, the maps \(H^0(U, H^q(K_{n + 1})) \to H^0(U, H^q(K_n))\) are surjective for \(U \in \mathcal{B}\) by similar reasoning. Part (1) follows from Cohomology on Sites, Lemma 0A09 whose conditions we have just verified. Parts (2) and (3) follow from Cohomology on Sites, Lemma 0BKY.

Lemma

Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). The functor \(Lf^*\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\).

Proof

Choose a diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] where \(U\) and \(V\) are schemes, the vertical arrows are étale, and \(a\) is surjective. Since \(a^* \circ Lf^* = Lh^* \circ b^*\) the result follows from Lemma 08F2 and the case of schemes which is Derived Categories of Schemes, Lemma 08DW.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). For objects \(K, L\) of \(D_\QCoh(\mathcal{O}_X)\) the derived tensor product \(K \otimes^\mathbf{L} L\) is in \(D_\QCoh(\mathcal{O}_X)\).

Proof

Let \(\varphi : U \to X\) be a surjective étale morphism from a scheme \(U\). Since \(\varphi^*(K \otimes_{\mathcal{O}_X}^\mathbf{L} L) = \varphi^*K \otimes_{\mathcal{O}_U}^\mathbf{L} \varphi^*L\) we see from Lemma 08F2 that this follows from the case of schemes which is Derived Categories of Schemes, Lemma 08DX.

The following lemma will help us to “compute” a right derived functor on an object of \(D_\QCoh(\mathcal{O}_X)\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Then the map \(E \to R\lim \tau_{\geq -n}E\) of Derived Categories, Remark 0H72 is an isomorphism¹.

Proof

Denote \(\mathcal{H}^i = H^i(E)\) the \(i\)th cohomology sheaf of \(E\). Let \(\mathcal{B}\) be the set of affine objects of \(X_\etale\). Then \(H^p(U, \mathcal{H}^i) = 0\) for all \(p > 0\), all \(i \in \mathbf{Z}\), and all \(U \in \mathcal{B}\) as \(U\) is an affine scheme. See discussion in Cohomology of Spaces, Section 071Y and Cohomology of Schemes, Lemma 01XB. Thus the lemma follows from Cohomology on Sites, Lemma 0D6P with \(d = 0\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(F : \textit{Mod}(\mathcal{O}_X) \to \textit{Ab}\) be a functor and \(N \geq 0\) an integer. Assume that

1. \(F\) is left exact,

2. \(F\) commutes with countable direct products,

3. \(R^pF(\mathcal{F}) = 0\) for all \(p \geq N\) and \(\mathcal{F}\) quasi-coherent.

Then for \(E \in D_\QCoh(\mathcal{O}_X)\)

1. \(H^i(RF(\tau_{\leq a}E)) \to H^i(RF(E))\) is an isomorphism for \(i \leq a\),

2. \(H^i(RF(E)) \to H^i(RF(\tau_{\geq b - N + 1}E))\) is an isomorphism for \(i \geq b\),

3. if \(H^i(E) = 0\) for \(i \not \in [a, b]\) for some \(-\infty \leq a \leq b \leq \infty\), then \(H^i(RF(E)) = 0\) for \(i \not \in [a, b + N - 1]\).

Proof

Statement (1) is Derived Categories, Lemma 05TC.

Proof of statement (2). Write \(E_n = \tau_{\geq -n}E\). We have \(E = R\lim E_n\), see Lemma 08F6. Thus \(RF(E) = R\lim RF(E_n)\) in \(D(\textit{Ab})\) by Injectives, Lemma 08U1. Thus for every \(i \in \mathbf{Z}\) we have a short exact sequence \[0 \to R^1\lim H^{i - 1}(RF(E_n)) \to H^i(RF(E)) \to \lim H^i(RF(E_n)) \to 0\] see More on Algebra, Remark 08U5. To prove (2) we will show that the term on the left is zero and that the term on the right equals \(H^i(RF(E_{-b + N - 1})\) for any \(b\) with \(i \geq b\).

For every \(n\) we have a distinguished triangle \[H^{-n}(E)[n] \to E_n \to E_{n - 1} \to H^{-n}(E)[n + 1]\] (Derived Categories, Remark 08J5) in \(D(\mathcal{O}_X)\). Since \(H^{-n}(E)\) is quasi-coherent we have \[H^i(RF(H^{-n}(E)[n])) = R^{i + n}F(H^{-n}(E)) = 0\] for \(i + n \geq N\) and \[H^i(RF(H^{-n}(E)[n + 1])) = R^{i + n + 1}F(H^{-n}(E)) = 0\] for \(i + n + 1 \geq N\). We conclude that \[H^i(RF(E_n)) \to H^i(RF(E_{n - 1}))\] is an isomorphism for \(n \geq N - i\). Thus the systems \(H^i(RF(E_n))\) all satisfy the ML condition and the \(R^1\lim\) term in our short exact sequence is zero (see discussion in More on Algebra, Section 07KV). Moreover, the system \(H^i(RF(E_n))\) is constant starting with \(n = N - i - 1\) as desired.

Proof of (3). Under the assumption on \(E\) we have \(\tau_{\leq a - 1}E = 0\) and we get the vanishing of \(H^i(RF(E))\) for \(i \leq a - 1\) from (1). Similarly, we have \(\tau_{\geq b + 1}E = 0\) and hence we get the vanishing of \(H^i(RF(E))\) for \(i \geq b + N\) from part (2).

Total direct image

The following lemma is the analogue of Cohomology of Spaces, Lemma 073G.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-separated and quasi-compact morphism of algebraic spaces over \(S\).

1. The functor \(Rf_*\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\).

2. If \(Y\) is quasi-compact, there exists an integer \(N = N(X, Y, f)\) such that for an object \(E\) of \(D_\QCoh(\mathcal{O}_X)\) with \(H^m(E) = 0\) for \(m > 0\) we have \(H^m(Rf_*E) = 0\) for \(m \geq N\).

3. In fact, if \(Y\) is quasi-compact we can find \(N = N(X, Y, f)\) such that for every morphism of algebraic spaces \(Y' \to Y\) the same conclusion holds for the functor \(R(f')_*\) where \(f' : X' \to Y'\) is the base change of \(f\).

Proof

Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). To prove (1) we have to show that \(Rf_*E\) has quasi-coherent cohomology sheaves. This question is local on \(Y\), hence we may assume \(Y\) is quasi-compact. Pick \(N = N(X, Y, f)\) as in Cohomology of Spaces, Lemma 073G. Thus \(R^pf_*\mathcal{F} = 0\) for all quasi-coherent \(\mathcal{O}_X\)-modules \(\mathcal{F}\) and all \(p \geq N\). Moreover \(R^pf_*\mathcal{F}\) is quasi-coherent for all \(p\) by Cohomology of Spaces, Lemma 0720. These statements remain true after base change.

First, assume \(E\) is bounded below. We will show (1) and (2) and (3) hold for such \(E\) with our choice of \(N\). In this case we can for example use the spectral sequence \[R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E\] (Derived Categories, Lemma 015J), the quasi-coherence of \(R^pf_*H^q(E)\), and the vanishing of \(R^pf_*H^q(E)\) for \(p \geq N\) to see that (1), (2), and (3) hold in this case.

Next we prove (2) and (3). Say \(H^m(E) = 0\) for \(m > 0\). Let \(V\) be an affine object of \(Y_\etale\). We have \(H^p(V \times_Y X, \mathcal{F}) = 0\) for \(p \geq N\), see Cohomology of Spaces, Lemma 08EX. Hence we may apply Lemma 08F7 to the functor \(\Gamma(V \times_Y X, -)\) to see that \[R\Gamma(V, Rf_*E) = R\Gamma(V \times_Y X, E)\] has vanishing cohomology in degrees \(\geq N\). Since this holds for all \(V\) affine in \(Y_\etale\) we conclude that \(H^m(Rf_*E) = 0\) for \(m \geq N\).

Next, we prove (1) in the general case. Recall that there is a distinguished triangle \[\tau_{\leq -n - 1}E \to E \to \tau_{\geq -n}E \to (\tau_{\leq -n - 1}E)[1]\] in \(D(\mathcal{O}_X)\), see Derived Categories, Remark 08J5. By (2) we see that \(Rf_*\tau_{\leq -n - 1}E\) has vanishing cohomology sheaves in degrees \(\geq -n + N\). Thus, given an integer \(q\) we see that \(R^qf_*E\) is equal to \(R^qf_*\tau_{\geq -n}E\) for some \(n\) and the result above applies.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-separated and quasi-compact morphism of algebraic spaces over \(S\). Then \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) commutes with direct sums.

Proof

Let \(E_i\) be a family of objects of \(D_\QCoh(\mathcal{O}_X)\) and set \(E = \bigoplus E_i\). We want to show that the map \[\bigoplus Rf_*E_i \longrightarrow Rf_*E\] is an isomorphism. We will show it induces an isomorphism on cohomology sheaves in degree \(0\) which will imply the lemma. Choose an integer \(N\) as in Lemma 08FA. Then \(R^0f_*E = R^0f_*\tau_{\geq -N}E\) and \(R^0f_*E_i = R^0f_*\tau_{\geq -N}E_i\) by the lemma cited. Observe that \(\tau_{\geq -N}E = \bigoplus \tau_{\geq -N}E_i\). Thus we may assume all of the \(E_i\) have vanishing cohomology sheaves in degrees \(< -N\). Next we use the spectral sequences \[R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E \quad\text{and}\quad R^pf_*H^q(E_i) \Rightarrow R^{p + q}f_*E_i\] (Derived Categories, Lemma 015J) to reduce to the case of a direct sum of quasi-coherent sheaves. This case is handled by Cohomology of Spaces, Lemma 07U6.

Remark

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of representable algebraic spaces \(X\) and \(Y\) over \(S\). Let \(f_0 : X_0 \to Y_0\) be a morphism of schemes representing \(f\) (awkward but temporary notation). Then the diagram \[\xymatrix{ D_\QCoh(\mathcal{O}_{X_0}) \ar@{=}[rrrrrr]_{\text{Lemma \href{spaces-perfect.html#spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}{071Q}}} & & & & & & D_\QCoh(\mathcal{O}_X) \\ D_\QCoh(\mathcal{O}_{Y_0}) \ar[u]^{Lf^*_0} \ar@{=}[rrrrrr]^{\text{Lemma \href{spaces-perfect.html#spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}{071Q}}} & & & & & & D_\QCoh(\mathcal{O}_Y) \ar[u]_{Lf^*} }\] (Lemma 08F4 and Derived Categories of Schemes, Lemma 08DW) is commutative. This follows as the equivalences \(D_\QCoh(\mathcal{O}_{X_0}) \to D_\QCoh(\mathcal{O}_X)\) and \(D_\QCoh(\mathcal{O}_{Y_0}) \to D_\QCoh(\mathcal{O}_Y)\) of Lemma 071Q come from pulling back by the (flat) morphisms of ringed sites \(\epsilon : X_\etale \to X_{0, Zar}\) and \(\epsilon : Y_\etale \to Y_{0, Zar}\) and the diagram of ringed sites \[\xymatrix{ X_{0, Zar} \ar[d]_{f_0} & X_\etale \ar[l]^\epsilon \ar[d]^f \\ Y_{0, Zar} & Y_\etale \ar[l]_\epsilon }\] is commutative (details omitted). If \(f\) is quasi-compact and quasi-separated, equivalently if \(f_0\) is quasi-compact and quasi-separated, then we claim \[\xymatrix{ D_\QCoh(\mathcal{O}_{X_0}) \ar[d]_{Rf_{0, *}} \ar@{=}[rrrrrr]_{\text{Lemma \href{spaces-perfect.html#spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}{071Q}}} & & & & & & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D_\QCoh(\mathcal{O}_{Y_0}) \ar@{=}[rrrrrr]^{\text{Lemma \href{spaces-perfect.html#spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}{071Q}}} & & & & & & D_\QCoh(\mathcal{O}_Y) }\] (Lemma 08FA and Derived Categories of Schemes, Lemma 08D5) is commutative as well. This also follows from the commutative diagram of sites displayed above as the proof of Lemma 071Q shows that the functor \(R\epsilon_*\) gives the equivalences \(D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_{X_0})\) and \(D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_{Y_0})\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). Then \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) reflects isomorphisms.

Proof

The statement means that a morphism \(\alpha : E \to F\) of \(D_\QCoh(\mathcal{O}_X)\) is an isomorphism if \(Rf_*\alpha\) is an isomorphism. We may check this on cohomology sheaves. In particular, the question is étale local on \(Y\). Hence we may assume \(Y\) and therefore \(X\) is affine. In this case the problem reduces to the case of schemes (Derived Categories of Schemes, Lemma 08I8) via Lemma 071Q and Remark 08GH.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). For \(E\) in \(D_\QCoh(\mathcal{O}_Y)\) we have \(Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_Y} f_*\mathcal{O}_X\).

Proof

Since \(f\) is affine the map \(f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X\) is an isomorphism (Cohomology of Spaces, Lemma 073H). There is a canonical map \(E \otimes^\mathbf{L} f_*\mathcal{O}_X = E \otimes^\mathbf{L} Rf_*\mathcal{O}_X \to Rf_* Lf^* E\) adjoint to the map \[Lf^*(E \otimes^\mathbf{L} Rf_*\mathcal{O}_X) = Lf^*E \otimes^\mathbf{L} Lf^*Rf_*\mathcal{O}_X \longrightarrow Lf^* E \otimes^\mathbf{L} \mathcal{O}_X = Lf^* E\] coming from \(1 : Lf^*E \to Lf^*E\) and the canonical map \(Lf^*Rf_*\mathcal{O}_X \to \mathcal{O}_X\). To check the map so constructed is an isomorphism we may work locally on \(Y\). Hence we may assume \(Y\) and therefore \(X\) is affine. In this case the problem reduces to the case of schemes (Derived Categories of Schemes, Lemma 08I9) via Lemma 071Q and Remark 08GH.

Being proper over a base

This section is the analogue of Cohomology of Schemes, Section 0CYK. As usual with material having to do with topology on the sets of points, we have to be careful translating the material to algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(T \subset |X|\) be a closed subset. The following are equivalent

1. the morphism \(Z \to Y\) is proper if \(Z\) is the reduced induced algebraic space structure on \(T\) (Properties of Spaces, Definition 047X),

2. for some closed subspace \(Z \subset X\) with \(|Z| = T\) the morphism \(Z \to Y\) is proper, and

3. for any closed subspace \(Z \subset X\) with \(|Z| = T\) the morphism \(Z \to Y\) is proper.

Proof

The implications (3) \(\Rightarrow\) (1) and (1) \(\Rightarrow\) (2) are immediate. Thus it suffices to prove that (2) implies (3). We urge the reader to find their own proof of this fact. Let \(Z'\) and \(Z''\) be closed subspaces with \(T = |Z'| = |Z''|\) such that \(Z' \to Y\) is a proper morphism of algebraic spaces. We have to show that \(Z'' \to Y\) is proper too. Let \(Z''' = Z' \cup Z''\) be the scheme theoretic union, see Morphisms of Spaces, Definition 0CYZ. Then \(Z'''\) is another closed subspace with \(|Z'''| = T\). This follows for example from the description of scheme theoretic unions in Morphisms of Spaces, Lemma 0CZ1. Since \(Z'' \to Z'''\) is a closed immersion it suffices to prove that \(Z''' \to Y\) is proper (see Morphisms of Spaces, Lemmas 04XZ and 04XY). The morphism \(Z' \to Z'''\) is a bijective closed immersion and in particular surjective and universally closed. Then the fact that \(Z' \to Y\) is separated implies that \(Z''' \to Y\) is separated, see Morphisms of Spaces, Lemma 05Z2. Moreover \(Z''' \to Y\) is locally of finite type as \(X \to Y\) is locally of finite type (Morphisms of Spaces, Lemmas 06ED and 03XG). Since \(Z' \to Y\) is quasi-compact and \(Z' \to Z'''\) is a universal homeomorphism we see that \(Z''' \to Y\) is quasi-compact. Finally, since \(Z' \to Y\) is universally closed, we see that the same thing is true for \(Z''' \to Y\) by Morphisms of Spaces, Lemma 08AJ. This finishes the proof.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(T \subset |X|\) be a closed subset. We say \(T\) is proper over \(Y\) if the equivalent conditions of Lemma 0CZC are satisfied.

The lemma used in the definition above is false if the morphism \(f : X \to Y\) is not locally of finite type. Therefore we urge the reader not to use this terminology if \(f\) is not locally of finite type.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(T' \subset T \subset |X|\) be closed subsets. If \(T\) is proper over \(Y\), then the same is true for \(T'\).

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Consider a cartesian diagram of algebraic spaces over \(S\) \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] with \(f\) locally of finite type. If \(T\) is a closed subset of \(|X|\) proper over \(Y\), then \(|g'|^{-1}(T)\) is a closed subset of \(|X'|\) proper over \(Y'\).

Proof

Observe that the statement makes sense as \(f'\) is locally of finite type by Morphisms of Spaces, Lemma 03XH. Let \(Z \subset X\) be the reduced induced closed subspace structure on \(T\). Denote \(Z' = (g')^{-1}(Z)\) the scheme theoretic inverse image. Then \(Z' = X' \times_X Z = (Y' \times_Y X) \times_X Z = Y' \times_Y Z\) is proper over \(Y'\) as a base change of \(Z\) over \(Y\) (Morphisms of Spaces, Lemma 04WP). On the other hand, we have \(T' = |Z'|\). Hence the lemma holds.

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(f : X \to Y\) be a morphism of algebraic spaces which are locally of finite type over \(B\).

1. If \(Y\) is separated over \(B\) and \(T \subset |X|\) is a closed subset proper over \(B\), then \(|f|(T)\) is a closed subset of \(|Y|\) proper over \(B\).

2. If \(f\) is universally closed and \(T \subset |X|\) is a closed subset proper over \(B\), then \(|f|(T)\) is a closed subset of \(Y\) proper over \(B\).

3. If \(f\) is proper and \(T \subset |Y|\) is a closed subset proper over \(B\), then \(|f|^{-1}(T)\) is a closed subset of \(|X|\) proper over \(B\).

Proof

Proof of (1). Assume \(Y\) is separated over \(B\) and \(T \subset |X|\) is a closed subset proper over \(B\). Let \(Z\) be the reduced induced closed subspace structure on \(T\) and apply Morphisms of Spaces, Lemma 0AGD to \(Z \to Y\) over \(B\) to conclude.

Proof of (2). Assume \(f\) is universally closed and \(T \subset |X|\) is a closed subset proper over \(B\). Let \(Z\) be the reduced induced closed subspace structure on \(T\) and let \(Z'\) be the reduced induced closed subspace structure on \(|f|(T)\). We obtain an induced morphism \(Z \to Z'\). Denote \(Z'' = f^{-1}(Z')\) the scheme theoretic inverse image. Then \(Z'' \to Z'\) is universally closed as a base change of \(f\) (Morphisms of Spaces, Lemma 04WP). Hence \(Z \to Z'\) is universally closed as a composition of the closed immersion \(Z \to Z''\) and \(Z'' \to Z'\) (Morphisms of Spaces, Lemmas 04XZ and 04XY). We conclude that \(Z' \to B\) is separated by Morphisms of Spaces, Lemma 05Z2. Since \(Z \to B\) is quasi-compact and \(Z \to Z'\) is surjective we see that \(Z' \to B\) is quasi-compact. Since \(Z' \to B\) is the composition of \(Z' \to Y\) and \(Y \to B\) we see that \(Z' \to B\) is locally of finite type (Morphisms of Spaces, Lemmas 06ED and 03XG). Finally, since \(Z \to B\) is universally closed, we see that the same thing is true for \(Z' \to B\) by Morphisms of Spaces, Lemma 08AJ. This finishes the proof.

Proof of (3). Assume \(f\) is proper and \(T \subset |Y|\) is a closed subset proper over \(B\). Let \(Z\) be the reduced induced closed subspace structure on \(T\). Denote \(Z' = f^{-1}(Z)\) the scheme theoretic inverse image. Then \(Z' \to Z\) is proper as a base change of \(f\) (Morphisms of Spaces, Lemma 04WP). Whence \(Z' \to B\) is proper as the composition of \(Z' \to Z\) and \(Z \to B\) (Morphisms of Spaces, Lemma 04XY). This finishes the proof.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(T_i \subset |X|\), \(i = 1, \ldots, n\) be closed subsets. If \(T_i\), \(i = 1, \ldots, n\) are proper over \(Y\), then the same is true for \(T_1 \cup \ldots \cup T_n\).

Proof

Let \(Z_i\) be the reduced induced closed subscheme structure on \(T_i\). The morphism \[Z_1 \amalg \ldots \amalg Z_n \longrightarrow X\] is finite by Morphisms of Spaces, Lemmas 081Y and 0CZ2. As finite morphisms are universally closed (Morphisms of Spaces, Lemma 04NZ) and since \(Z_1 \amalg \ldots \amalg Z_n\) is proper over \(S\) we conclude by Lemma 0CZG part (2) that the image \(Z_1 \cup \ldots \cup Z_n\) is proper over \(S\).

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. Then the support \(\text{Supp}(\mathcal{F})\) of \(\mathcal{F}\) is a closed subset of \(|X|\), see Morphisms of Spaces, Lemma 07TZ. Hence it makes sense to say “the support of \(\mathcal{F}\) is proper over \(Y\)”.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. The following are equivalent

1. the support of \(\mathcal{F}\) is proper over \(Y\),

2. the scheme theoretic support of \(\mathcal{F}\) (Morphisms of Spaces, Definition 07U1) is proper over \(Y\), and

3. there exists a closed subspace \(Z \subset X\) and a finite type, quasi-coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\) such that (a) \(Z \to Y\) is proper, and (b) \((Z \to X)_*\mathcal{G} = \mathcal{F}\).

Proof

The support \(\text{Supp}(\mathcal{F})\) of \(\mathcal{F}\) is a closed subset of \(|X|\), see Morphisms of Spaces, Lemma 07TZ. Hence we can apply Definition 0CZD. Since the scheme theoretic support of \(\mathcal{F}\) is a closed subspace whose underlying closed subset is \(\text{Supp}(\mathcal{F})\) we see that (1) and (2) are equivalent by Definition 0CZD. It is clear that (2) implies (3). Conversely, if (3) is true, then \(\text{Supp}(\mathcal{F}) \subset |Z|\) and hence \(\text{Supp}(\mathcal{F})\) is proper over \(Y\) for example by Lemma 0CZE.

Lemma

Let \(S\) be a scheme. Consider a cartesian diagram of algebraic spaces over \(S\) \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] with \(f\) locally of finite type. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. If the support of \(\mathcal{F}\) is proper over \(Y\), then the support of \((g')^*\mathcal{F}\) is proper over \(Y'\).

Proof

Observe that the statement makes sense because \((g')*\mathcal{F}\) is of finite type by Modules on Sites, Lemma 03DO. We have \(\text{Supp}((g')^*\mathcal{F}) = |g'|^{-1}(\text{Supp}(\mathcal{F}))\) by Morphisms of Spaces, Lemma 07TZ. Thus the lemma follows from Lemma 0CZF.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is locally of finite type. Let \(\mathcal{F}\), \(\mathcal{G}\) be finite type, quasi-coherent \(\mathcal{O}_X\)-module.

1. If the supports of \(\mathcal{F}\), \(\mathcal{G}\) are proper over \(Y\), then the same is true for \(\mathcal{F} \oplus \mathcal{G}\), for any extension of \(\mathcal{G}\) by \(\mathcal{F}\), for \(\Im(u)\) and \(\Coker(u)\) given any \(\mathcal{O}_X\)-module map \(u : \mathcal{F} \to \mathcal{G}\), and for any quasi-coherent quotient of \(\mathcal{F}\) or \(\mathcal{G}\).

2. If \(Y\) is locally Noetherian, then the category of coherent \(\mathcal{O}_X\)-modules with support proper over \(Y\) is a Serre subcategory (Homology, Definition 02MO) of the abelian category of coherent \(\mathcal{O}_X\)-modules.

Proof

Proof of (1). Let \(T\), \(T'\) be the support of \(\mathcal{F}\) and \(\mathcal{G}\). Then all the sheaves mentioned in (1) have support contained in \(T \cup T'\). Thus the assertion itself is clear from Lemmas 0CZE and 0CZH provided we check that these sheaves are finite type and quasi-coherent. For quasi-coherence we refer the reader to Properties of Spaces, Section 03G5. For “finite type” we refer the reader to Properties of Spaces, Section 05VR.

Proof of (2). The proof is the same as the proof of (1). Note that the assertions make sense as \(X\) is locally Noetherian by Morphisms of Spaces, Lemma 04ZK and by the description of the category of coherent modules in Cohomology of Spaces, Section 07U9.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is locally of finite type and \(Y\) locally Noetherian. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module with support proper over \(Y\). Then \(R^pf_*\mathcal{F}\) is a coherent \(\mathcal{O}_Y\)-module for all \(p \geq 0\).

Proof

By Lemma 0CZI there exists a closed immersion \(i : Z \to X\) with \(g = f \circ i : Z \to Y\) proper and \(\mathcal{F} = i_*\mathcal{G}\) for some coherent module \(\mathcal{G}\) on \(Z\). We see that \(R^pg_*\mathcal{G}\) is coherent on \(S\) by Cohomology of Spaces, Lemma 08AR. On the other hand, \(R^qi_*\mathcal{G} = 0\) for \(q > 0\) (Cohomology of Spaces, Lemma 07UH). By Cohomology on Sites, Lemma 0734 we get \(R^pf_*\mathcal{F} = R^pg_*\mathcal{G}\) and the lemma follows.

Derived category of coherent modules

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). In this case the category \(\textit{Coh}(\mathcal{O}_X) \subset \textit{Mod}(\mathcal{O}_X)\) of coherent \(\mathcal{O}_X\)-modules is a weak Serre subcategory, see Homology, Section 02MN and Cohomology of Spaces, Lemma 07UC. Denote \[D_{\textit{Coh}}(\mathcal{O}_X) \subset D(\mathcal{O}_X)\] the subcategory of complexes whose cohomology sheaves are coherent, see Derived Categories, Section 06UP. Thus we obtain a canonical functor [08GJ]\[\begin{equation} D(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_X) \end{equation}\] see Derived Categories, Equation (06UR).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is locally of finite type and \(Y\) is Noetherian. Let \(E\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) such that the support of \(H^i(E)\) is proper over \(Y\) for all \(i\). Then \(Rf_*E\) is an object of \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\).

Proof

Consider the spectral sequence \[R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E\] see Derived Categories, Lemma 015J. By assumption and Lemma 08GC the sheaves \(R^pf_*H^q(E)\) are coherent. Hence \(R^{p + q}f_*E\) is coherent, i.e., \(E \in D_{\textit{Coh}}(\mathcal{O}_Y)\). Boundedness from below is trivial. Boundedness from above follows from Cohomology of Spaces, Lemma 073G or from Lemma 08FA.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is locally of finite type and \(Y\) is Noetherian. Let \(E\) be an object of \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) such that the support of \(H^i(E)\) is proper over \(S\) for all \(i\). Then \(Rf_*E\) is an object of \(D^+_{\textit{Coh}}(\mathcal{O}_Y)\).

Proof

The proof is the same as the proof of Lemma 08GK. You can also deduce it from Lemma 08GK by considering what the exact functor \(Rf_*\) does to the distinguished triangles \(\tau_{\leq a}E \to E \to \tau_{\geq a + 1}E \to \tau_{\leq a}E[1]\).

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). If \(L\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) and \(K\) in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\), then \(R\SheafHom(K, L)\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

We can check whether an object of \(D(\mathcal{O}_X)\) is in \(D_{\textit{Coh}}(\mathcal{O}_X)\) étale locally on \(X\), see Cohomology of Spaces, Lemma 07UB. Hence this lemma follows from the case of schemes, see Derived Categories of Schemes, Lemma 0D0C.

Lemma

Let \(A\) be a Noetherian ring. Let \(X\) be a proper algebraic space over \(A\). For \(L\) in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) and \(K\) in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\), the \(A\)-modules \(\Ext_{\mathcal{O}_X}^n(K, L)\) are finite.

Proof

Recall that \[\Ext_{\mathcal{O}_X}^n(K, L) = H^n(X, R\SheafHom_{\mathcal{O}_X}(K, L)) = H^n(\Spec(A), Rf_*R\SheafHom_{\mathcal{O}_X}(K, L))\] see Cohomology on Sites, Lemma 08JA and Cohomology on Sites, Section 072X. Thus the result follows from Lemmas 0D0S and 0D0R.

Induction principle

In this section we discuss an induction principle for algebraic spaces analogous to what is Cohomology of Schemes, Lemma 08DR for schemes. To formulate it we introduce the notion of an elementary distinguished square; this terminology is borrowed from [MV]. The principle as formulated here is implicit in the paper [GruRay] by Raynaud and Gruson. A related principle for algebraic stacks is [rydh_etale_devissage, Theorem D] by David Rydh.

Definition

Let \(S\) be a scheme. A commutative diagram \[\xymatrix{ U \times_W V \ar[r] \ar[d] & V \ar[d]^f \\ U \ar[r]^j & W }\] of algebraic spaces over \(S\) is called an elementary distinguished square if

1. \(U\) is an open subspace of \(W\) and \(j\) is the inclusion morphism,

2. \(f\) is étale, and

3. setting \(T = W \setminus U\) (with reduced induced subspace structure) the morphism \(f^{-1}(T) \to T\) is an isomorphism.

We will indicate this by saying: “Let \((U \subset W, f : V \to W)\) be an elementary distinguished square.”

Note that if \((U \subset W, f : V \to W)\) is an elementary distinguished square, then we have \(W = U \cup f(V)\). Thus \(\{U \to W, V \to W\}\) is an étale covering of \(W\). It turns out that these étale coverings have nice properties and that in some sense there are “enough” of them.

Lemma

Let \(S\) be a scheme. Let \((U \subset W, f : V \to W)\) be an elementary distinguished square of algebraic spaces over \(S\).

1. If \(V' \subset V\) and \(U \subset U' \subset W\) are open subspaces and \(W' = U' \cup f(V')\) then \((U' \subset W', f|_{V'} : V' \to W')\) is an elementary distinguished square.

2. If \(p : W' \to W\) is a morphism of algebraic spaces, then \((p^{-1}(U) \subset W', V \times_W W' \to W')\) is an elementary distinguished square.

3. If \(S' \to S\) is a morphism of schemes, then \((S' \times_S U \subset S' \times_S W, S' \times_S V \to S' \times_S W)\) is an elementary distinguished square.

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(P\) be a property of the quasi-compact and quasi-separated objects of \(X_{spaces, \etale}\). Assume that

1. \(P\) holds for every affine object of \(X_{spaces, \etale}\),

2. for every elementary distinguished square \((U \subset W, f : V \to W)\) such that

   1. \(W\) is a quasi-compact and quasi-separated object of \(X_{spaces, \etale}\),

   2. \(U\) is quasi-compact,

   3. \(V\) is affine, and

   4. \(P\) holds for \(U\), \(V\), and \(U \times_W V\),

   then \(P\) holds for \(W\).

Then \(P\) holds for every quasi-compact and quasi-separated object of \(X_{spaces, \etale}\) and in particular for \(X\).

Proof

We first claim that \(P\) holds for every representable quasi-compact and quasi-separated object of \(X_{spaces, \etale}\). Namely, suppose that \(U \to X\) is étale and \(U\) is a quasi-compact and quasi-separated scheme. By assumption (1) property \(P\) holds for every affine open of \(U\). Moreover, if \(W, V \subset U\) are quasi-compact open with \(V\) affine and \(P\) holds for \(W\), \(V\), and \(W \cap V\), then \(P\) holds for \(W \cup V\) by (2) (as the pair \((W \subset W \cup V, V \to W \cup V)\) is an elementary distinguished square). Thus \(P\) holds for \(U\) by the induction principle for schemes, see Cohomology of Schemes, Lemma 08DR.

To finish the proof it suffices to prove \(P\) holds for \(X\) (because we can simply replace \(X\) by any quasi-compact and quasi-separated object of \(X_{spaces, \etale}\) we want to prove the result for). We will use the filtration \[\emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X\] and the morphisms \(f_p : V_p \to U_p\) of Decent Spaces, Lemma 07ST. We will prove that \(P\) holds for \(U_p\) by descending induction on \(p\). Note that \(P\) holds for \(U_{n + 1}\) by (1) as an empty algebraic space is affine. Assume \(P\) holds for \(U_{p + 1}\). Note that \((U_{p + 1} \subset U_p, f_p : V_p \to U_p)\) is an elementary distinguished square, but (2) may not apply as \(V_p\) may not be affine. However, as \(V_p\) is a quasi-compact scheme we may choose a finite affine open covering \(V_p = V_{p, 1} \cup \ldots \cup V_{p, m}\). Set \(W_{p, 0} = U_{p + 1}\) and \[W_{p, i} = U_{p + 1} \cup f_p(V_{p, 1} \cup \ldots \cup V_{p, i})\] for \(i = 1, \ldots, m\). These are quasi-compact open subspaces of \(X\). Then we have \[U_{p + 1} = W_{p, 0} \subset W_{p, 1} \subset \ldots \subset W_{p, m} = U_p\] and the pairs \[(W_{p, 0} \subset W_{p, 1}, f_p|_{V_{p, 1}}), (W_{p, 1} \subset W_{p, 2}, f_p|_{V_{p, 2}}),\ldots, (W_{p, m - 1} \subset W_{p, m}, f_p|_{V_{p, m}})\] are elementary distinguished squares by Lemma 08GN. Note that \(P\) holds for each \(V_{p, i}\) (as affine schemes) and for \(W_{p, i} \times_{W_{p, i + 1}} V_{p, i + 1}\) as this is a quasi-compact open of \(V_{p, i + 1}\) and hence \(P\) holds for it by the first paragraph of this proof. Thus (2) applies to each of these and we inductively conclude \(P\) holds for \(W_{p, 1}, \ldots, W_{p, m} = U_p\).

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\mathcal{B} \subset \Ob(X_{spaces, \etale})\). Let \(P\) be a property of the elements of \(\mathcal{B}\). Assume that

1. every \(W \in \mathcal{B}\) is quasi-compact and quasi-separated,

2. if \(W \in \mathcal{B}\) and \(U \subset W\) is quasi-compact open, then \(U \in \mathcal{B}\),

3. if \(V \in \Ob(X_{spaces, \etale})\) is affine, then (a) \(V \in \mathcal{B}\) and (b) \(P\) holds for \(V\),

4. for every elementary distinguished square \((U \subset W, f : V \to W)\) such that

   1. \(W \in \mathcal{B}\),

   2. \(U\) is quasi-compact,

   3. \(V\) is affine, and

   4. \(P\) holds for \(U\), \(V\), and \(U \times_W V\),

   then \(P\) holds for \(W\).

Then \(P\) holds for every \(W \in \mathcal{B}\).

Proof

This is proved in exactly the same manner as the proof of Lemma 08GP. (We remark that (4)(d) makes sense as \(U \times_W V\) is a quasi-compact open of \(V\) hence an element of \(\mathcal{B}\) by conditions (2) and (3).)

Remark

How to choose the collection \(\mathcal{B}\) in Lemma 08GQ? Here are some examples:

1. If \(X\) is quasi-compact and separated, then we can choose \(\mathcal{B}\) to be the set of quasi-compact and separated objects of \(X_{spaces, \etale}\). Then \(X \in \mathcal{B}\) and \(\mathcal{B}\) satisfies (1), (2), and (3)(a). With this choice of \(\mathcal{B}\) Lemma 08GQ reproduces Lemma 08GP.

2. If \(X\) is quasi-compact with affine diagonal over \(\mathbf{Z}\) (as in Properties of Spaces, Definition 03BS), then we can choose \(\mathcal{B}\) to be the set of objects of \(X_{spaces, \etale}\) which are quasi-compact and have affine diagonal over \(\mathbf{Z}\). Again \(X \in \mathcal{B}\) and \(\mathcal{B}\) satisfies (1), (2), and (3)(a).

3. If \(X\) is quasi-compact and quasi-separated, then the smallest subset \(\mathcal{B}\) which contains \(X\) and satisfies (1), (2), and (3)(a) is given by the rule \(W \in \mathcal{B}\) if and only if either \(W\) is a quasi-compact open subspace of \(X\), or \(W\) is a quasi-compact open of an affine object of \(X_{spaces, \etale}\).

Here is a variant where we extend the truth from an open to larger opens.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(W \subset X\) be a quasi-compact open subspace. Let \(P\) be a property of quasi-compact open subspaces of \(X\). Assume that

1. \(P\) holds for \(W\), and

2. for every elementary distinguished square \((W_1 \subset W_2, f : V \to W_2)\) such that

   1. \(W_1\), \(W_2\) are quasi-compact open subspaces of \(X\),

   2. \(W \subset W_1\),

   3. \(V\) is affine, and

   4. \(P\) holds for \(W_1\),

   then \(P\) holds for \(W_2\).

Then \(P\) holds for \(X\).

Proof

We can deduce this from Lemma 08GQ, but instead we will give a direct argument by explicitly redoing the proof of Lemma 08GP. We will use the filtration \[\emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X\] and the morphisms \(f_p : V_p \to U_p\) of Decent Spaces, Lemma 07ST. We will prove that \(P\) holds for \(W_p = W \cup U_p\) by descending induction on \(p\). This will finish the proof as \(W_1 = X\). Note that \(P\) holds for \(W_{n + 1} = W \cup U_{n + 1} = W\) by (1). Assume \(P\) holds for \(W_{p + 1}\). Observe that \(W_p \setminus W_{p + 1}\) (with reduced induced subspace structure) is a closed subspace of \(U_p \setminus U_{p + 1}\). Since \((U_{p + 1} \subset U_p, f_p : V_p \to U_p)\) is an elementary distinguished square, the same is true for \((W_{p + 1} \subset W_p, f_p : V_p \to W_p)\). However (2) may not apply as \(V_p\) may not be affine. However, as \(V_p\) is a quasi-compact scheme we may choose a finite affine open covering \(V_p = V_{p, 1} \cup \ldots \cup V_{p, m}\). Set \(W_{p, 0} = W_{p + 1}\) and \[W_{p, i} = W_{p + 1} \cup f_p(V_{p, 1} \cup \ldots \cup V_{p, i})\] for \(i = 1, \ldots, m\). These are quasi-compact open subspaces of \(X\) containing \(W\). Then we have \[W_{p + 1} = W_{p, 0} \subset W_{p, 1} \subset \ldots \subset W_{p, m} = W_p\] and the pairs \[(W_{p, 0} \subset W_{p, 1}, f_p|_{V_{p, 1}}), (W_{p, 1} \subset W_{p, 2}, f_p|_{V_{p, 2}}),\ldots, (W_{p, m - 1} \subset W_{p, m}, f_p|_{V_{p, m}})\] are elementary distinguished squares by Lemma 08GN. Now (2) applies to each of these and we inductively conclude \(P\) holds for \(W_{p, 1}, \ldots, W_{p, m} = W_p\).

Mayer-Vietoris

In this section we prove that an elementary distinguished square gives rise to various Mayer-Vietoris sequences.

Let \(S\) be a scheme. Let \(U \to X\) be an étale morphism of algebraic spaces over \(S\). In Properties of Spaces, Section 04LX it was shown that \(U_{spaces, \etale} = X_{spaces, \etale}/U\) compatible with structure sheaves. Hence in this situation we often think of the morphism \(j_U : U \to X\) as a localization morphism (see Modules on Sites, Definition 04IX). In particular we think of pullback \(j_U^*\) as restriction to \(U\) and we often denote it by \({}|_U\); this is compatible with Properties of Spaces, Equation (03LW). In particular we see that [08GT]\[\begin{equation} (\mathcal{F}|_U)_{\overline{u}} = \mathcal{F}_{\overline{x}} \end{equation}\] if \(\overline{u}\) is a geometric point of \(U\) and \(\overline{x}\) the image of \(\overline{u}\) in \(X\). Moreover, restriction has an exact left adjoint \(j_{U!}\), see Modules on Sites, Lemmas 03DI and 03DJ. Finally, recall that if \(\mathcal{G}\) is an \(\mathcal{O}_X\)-module, then [08GU]\[\begin{equation} (j_{U!}\mathcal{G})_{\overline{x}} = \bigoplus\nolimits_{\overline{u}} \mathcal{G}_{\overline{u}} \end{equation}\] for any geometric point \(\overline{x} : \Spec(k) \to X\) where the direct sum is over those morphisms \(\overline{u} : \Spec(k) \to U\) such that \(j_U \circ \overline{u} = \overline{x}\), see Modules on Sites, Lemma 0710 and Properties of Spaces, Lemma 04K6.

Lemma

Let \(S\) be a scheme. Let \((U \subset X, V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\).

1. For a sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) we have a short exact sequence \[0 \to j_{U \times_X V!}\mathcal{F}|_{U \times_X V} \to j_{U!}\mathcal{F}|_U \oplus j_{V!}\mathcal{F}|_V \to \mathcal{F} \to 0\]

2. For an object \(E\) of \(D(\mathcal{O}_X)\) we have a distinguished triangle \[j_{U \times_X V!}E|_{U \times_X V} \to j_{U!}E|_U \oplus j_{V!}E|_V \to E \to j_{U \times_X V!}E|_{U \times_X V}[1]\] in \(D(\mathcal{O}_X)\).

Proof

To show the sequence of (1) is exact we may check on stalks at geometric points by Properties of Spaces, Theorem 04K5. Let \(\overline{x}\) be a geometric point of \(X\). By Equations (08GT) and (08GU) taking stalks at \(\overline{x}\) we obtain the sequence \[0 \to \bigoplus\nolimits_{(\overline{u}, \overline{v})} \mathcal{F}_{\overline{x}} \to \bigoplus\nolimits_{\overline{u}} \mathcal{F}_{\overline{x}} \oplus \bigoplus\nolimits_{\overline{v}} \mathcal{F}_{\overline{x}} \to \mathcal{F}_{\overline{x}} \to 0\] This sequence is exact because for every \(\overline{x}\) there either is exactly one \(\overline{u}\) mapping to \(\overline{x}\), or there is no \(\overline{u}\) and exactly one \(\overline{v}\) mapping to \(\overline{x}\).

Proof of (2). We have seen in Cohomology on Sites, Section 08FH that the restriction functors and the extension by zero functors on derived categories are computed by just applying the functor to any complex. Let \(\mathcal{E}^\bullet\) be a complex of \(\mathcal{O}_X\)-modules representing \(E\). The distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 014Z and especially Lemma 0152) to the short exact sequence of complexes of \(\mathcal{O}_X\)-modules \[0 \to j_{U \times_X V!}\mathcal{E}^\bullet|_{U \times_X V} \to j_{U!}\mathcal{E}^\bullet|_U \oplus j_{V!}\mathcal{E}^\bullet|_V \to \mathcal{E}^\bullet \to 0\] which is short exact by (1).

Lemma

Let \(S\) be a scheme. Let \((U \subset X, V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\).

1. For every sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) we have a short exact sequence \[0 \to \mathcal{F} \to j_{U, *}\mathcal{F}|_U \oplus j_{V, *}\mathcal{F}|_V \to j_{U \times_X V, *}\mathcal{F}|_{U \times_X V} \to 0\]

2. For any object \(E\) of \(D(\mathcal{O}_X)\) we have a distinguished triangle \[E \to Rj_{U, *}E|_U \oplus Rj_{V, *}E|_V \to Rj_{U \times_X V, *}E|_{U \times_X V} \to E[1]\] in \(D(\mathcal{O}_X)\).

Proof

Let \(W\) be an object of \(X_\etale\). We claim the sequence \[0 \to \mathcal{F}(W) \to \mathcal{F}(W \times_X U) \oplus \mathcal{F}(W \times_X V) \to \mathcal{F}(W \times_X U \times_X V)\] is exact and that an element of the last group can locally on \(W\) be lifted to the middle one. By Lemma 08GN the pair \((W \times_X U \subset W, V \times_X W \to W)\) is an elementary distinguished square. Thus we may assume \(W = X\) and it suffices to prove the same thing for \[0 \to \mathcal{F}(X) \to \mathcal{F}(U) \oplus \mathcal{F}(V) \to \mathcal{F}(U \times_X V)\] We have seen that \[0 \to j_{U \times_X V!}\mathcal{O}_{U \times_X V} \to j_{U!}\mathcal{O}_U \oplus j_{V!}\mathcal{O}_V \to \mathcal{O}_X \to 0\] is a exact sequence of \(\mathcal{O}_X\)-modules in Lemma 08GV and applying the left exact functor \(\Hom_{\mathcal{O}_X}(- , \mathcal{F})\) gives the sequence above. This also means that the obstruction to lifting \(s \in \mathcal{F}(U \times_X V)\) to an element of \(\mathcal{F}(U) \oplus \mathcal{F}(V)\) lies in \(\Ext^1_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{F}) = H^1(X, \mathcal{F})\). By locality of cohomology (Cohomology on Sites, Lemma 01FW) this obstruction vanishes étale locally on \(X\) and the proof of (1) is complete.

Proof of (2). Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(E\) whose terms \(\mathcal{I}^n\) are injective objects of \(\textit{Mod}(\mathcal{O}_X)\), see Injectives, Theorem 079P. Then \(\mathcal{I}^\bullet|U\) is a K-injective complex (Cohomology on Sites, Lemma 08FI). Hence \(Rj_{U, *}E|_U\) is represented by \(j_{U, *}\mathcal{I}^\bullet|_U\). Similarly for \(V\) and \(U \times_X V\). Hence the distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 014Z and especially Lemma 0152) to the short exact sequence of complexes \[0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet|_U \oplus j_{V, *}\mathcal{I}^\bullet|_V \to j_{U \times_X V, *}\mathcal{I}^\bullet|_{U \times_X V} \to 0.\] This sequence is exact by (1).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \((U \subset X, V \to X)\) be an elementary distinguished square. Denote \(a = f|_U : U \to Y\), \(b = f|_V : V \to Y\), and \(c = f|_{U \times_X V} : U \times_X V \to Y\) the restrictions. For every object \(E\) of \(D(\mathcal{O}_X)\) there exists a distinguished triangle \[Rf_*E \to Ra_*(E|_U) \oplus Rb_*(E|_V) \to Rc_*(E|_{U \times_X V}) \to Rf_*E[1]\] in \(D(\mathcal{O}_Y)\). This triangle is functorial in \(E\).

Proof

Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(E\). We may assume \(\mathcal{I}^n\) is an injective object of \(\textit{Mod}(\mathcal{O}_X)\) for all \(n\), see Injectives, Theorem 079P. Then \(Rf_*E\) is computed by \(f_*\mathcal{I}^\bullet\). Similarly for \(U\), \(V\), and \(U \times_X V\) by Cohomology on Sites, Lemma 08FI. Hence the distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 014Z and especially Lemma 0152) to the short exact sequence of complexes \[0 \to f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet|_U \oplus b_*\mathcal{I}^\bullet|_V \to c_*\mathcal{I}^\bullet|_{U \times_X V} \to 0.\] To see this is a short exact sequence of complexes we argue as follows. Pick an injective object \(\mathcal{I}\) of \(\textit{Mod}(\mathcal{O}_X)\). Apply \(f_*\) to the short exact sequence \[0 \to \mathcal{I} \to j_{U, *}\mathcal{I}|_U \oplus j_{V, *}\mathcal{I}|_V \to j_{U \times_X V, *}\mathcal{I}|_{U \times_X V} \to 0\] of Lemma 08GW and use that \(R^1f_*\mathcal{I} = 0\) to get a short exact sequence \[0 \to f_*\mathcal{I} \to f_*j_{U, *}\mathcal{I}|_U \oplus f_*j_{V, *}\mathcal{I}|_V \to f_*j_{U \times_X V, *}\mathcal{I}|_{U \times_X V} \to 0\] The proof is finished by observing that \(a_* = f_*j_{U, *}\) and similarly for \(b_*\) and \(c_*\).

Lemma

Let \(S\) be a scheme. Let \((U \subset X, V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\). For objects \(E\), \(F\) of \(D(\mathcal{O}_X)\) we have a Mayer-Vietoris sequence \[\xymatrix{ & \ldots \ar[r] & \Ext^{-1}(E_{U \times_X V}, F_{U \times_X V}) \ar[lld] \\ \Hom(E, F) \ar[r] & \Hom(E_U, F_U) \oplus \Hom(E_V, F_V) \ar[r] & \Hom(E_{U \times_X V}, F_{U \times_X V}) }\] where the subscripts denote restrictions to the relevant opens and the \(\Hom\)’s are taken in the relevant derived categories.

Proof

Use the distinguished triangle of Lemma 08GV to obtain a long exact sequence of \(\Hom\)’s (from Derived Categories, Lemma 0149) and use that \(\Hom(j_{U!}E|_U, F) = \Hom(E|_U, F|_U)\) by Cohomology on Sites, Lemma 08FJ.

Lemma

Let \(S\) be a scheme. Let \((U \subset X, V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\). For an object \(E\) of \(D(\mathcal{O}_X)\) we have a distinguished triangle \[R\Gamma(X, E) \to R\Gamma(U, E) \oplus R\Gamma(V, E) \to R\Gamma(U \times_X V, E) \to R\Gamma(X, E)[1]\] and in particular a long exact cohomology sequence \[\ldots \to H^n(X, E) \to H^n(U, E) \oplus H^n(V, E) \to H^n(U \times_X V, E) \to H^{n + 1}(X, E) \to \ldots\] The construction of the distinguished triangle and the long exact sequence is functorial in \(E\).

Proof

Choose a K-injective complex \(\mathcal{I}^\bullet\) representing \(E\) whose terms \(\mathcal{I}^n\) are injective objects of \(\textit{Mod}(\mathcal{O}_X)\), see Injectives, Theorem 079P. In the proof of Lemma 08GW we found a short exact sequence of complexes \[0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet|_U \oplus j_{V, *}\mathcal{I}^\bullet|_V \to j_{U \times_X V, *}\mathcal{I}^\bullet|_{U \times_X V} \to 0\] Since \(H^1(X, \mathcal{I}^n) = 0\), we see that taking global sections gives an exact sequence of complexes \[0 \to \Gamma(X, \mathcal{I}^\bullet) \to \Gamma(U, \mathcal{I}^\bullet) \oplus \Gamma(V, \mathcal{I}^\bullet) \to \Gamma(U \times_X V, \mathcal{I}^\bullet) \to 0\] Since these complexes represent \(R\Gamma(X, E)\), \(R\Gamma(U, E)\), \(R\Gamma(V, E)\), and \(R\Gamma(U \times_X V, E)\) we get a distinguished triangle by Derived Categories, Section 014Z and especially Lemma 0152.

Lemma

Let \(S\) be a scheme. Let \(j : U \to X\) be a étale morphism of algebraic spaces over \(S\). Given an étale morphism \(V \to X\), set \(W = V \times_X U\) and denote \(j_W : W \to V\) the projection morphism. Then \((j_!E)|_V = j_{W!}(E|_W)\) for \(E\) in \(D(\mathcal{O}_U)\).

Proof

This is true because \((j_!\mathcal{F})|_V = j_{W!}(\mathcal{F}|_W)\) for an \(\mathcal{O}_X\)-module \(\mathcal{F}\) as follows immediately from the construction of the functors \(j_!\) and \(j_{W!}\), see Modules on Sites, Lemma 03DI.

Lemma

Let \(S\) be a scheme. Let \((U \subset X, j : V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\). Set \(T = |X| \setminus |U|\).

1. If \(E\) is an object of \(D(\mathcal{O}_X)\) supported on \(T\), then (a) \(E \to Rj_*(E|_V)\) and (b) \(j_!(E|_V) \to E\) are isomorphisms.

2. If \(F\) is an object of \(D(\mathcal{O}_V)\) supported on \(j^{-1}T\), then (a) \(F \to (j_!F)|_V\), (b) \((Rj_*F)|_V \to F\), and (c) \(j_!F \to Rj_*F\) are isomorphisms.

Proof

Let \(E\) be an object of \(D(\mathcal{O}_X)\) whose cohomology sheaves are supported on \(T\). Then we see that \(E|_U = 0\) and \(E|_{U \times_X V} = 0\) as \(T\) doesn’t meet \(U\) and \(j^{-1}T\) doesn’t meet \(U \times_X V\). Thus (1)(a) follows from Lemma 08GW. In exactly the same way (1)(b) follows from Lemma 08GV.

Let \(F\) be an object of \(D(\mathcal{O}_V)\) whose cohomology sheaves are supported on \(j^{-1}T\). By Lemma 08GE we have \((Rj_*F)|_U = Rj_{W, *}(F|_W) = 0\) because \(F|_W = 0\) by our assumption. Similarly \((j_!F)|_U = j_{W!}(F|_W) = 0\) by Lemma 08HA. Thus \(j_!F\) and \(Rj_*F\) are supported on \(T\) and \((j_!F)|_V\) and \((Rj_*F)|_V\) are supported on \(j^{-1}(T)\). To check that the maps (2)(a), (b), (c) are isomorphisms in the derived category, it suffices to check that these map induce isomorphisms on stalks of cohomology sheaves at geometric points of \(T\) and \(j^{-1}(T)\) by Properties of Spaces, Theorem 04K5. This we may do after replacing \(X\) by \(V\), \(U\) by \(U \times_X V\), \(V\) by \(V \times_X V\) and \(F\) by \(F|_{V \times_X V}\) (restriction via first projection), see Lemmas 08GE, 08HA, and 08GN. Since \(V \times_X V \to V\) has a section this reduces (2) to the case that \(j : V \to X\) has a section.

Assume \(j\) has a section \(\sigma : X \to V\). Set \(V' = \sigma(X)\). This is an open subspace of \(V\). Set \(U' = j^{-1}(U)\). This is another open subspace of \(V\). Then \((U' \subset V, V' \to V)\) is an elementary distinguished square. Observe that \(F|_{U'} = 0\) and \(F|_{V' \cap U'} = 0\) because \(F\) is supported on \(j^{-1}(T)\). Denote \(j' : V' \to V\) the open immersion and \(j_{V'} : V' \to X\) the composition \(V' \to V \to X\) which is the inverse of \(\sigma\). Set \(F' = \sigma^*F\). The distinguished triangles of Lemmas 08GV and 08GW show that \(F = j'_!(F|_{V'})\) and \(F = Rj'_*(F|_{V'})\). It follows that \(j_!F = j_!j'_!(F|_{V'}) = j_{V'!}F = F'\) because \(j_{V'} : V' \to X\) is an isomorphism and the inverse of \(\sigma\). Similarly, \(Rj_*F = Rj_*Rj'_*F = Rj_{V', *}F = F'\). This proves (2)(c). To prove (2)(a) and (2)(b) it suffices to show that \(F = F'|_V\). This is clear because both \(F\) and \(F'|_V\) restrict to zero on \(U'\) and \(U' \cap V'\) and the same object on \(V'\).

We can glue complexes!

Lemma

Let \(S\) be a scheme. Let \((U \subset X, V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\). Suppose given

1. an object \(A\) of \(D(\mathcal{O}_U)\),

2. an object \(B\) of \(D(\mathcal{O}_V)\), and

3. an isomorphism \(c : A|_{U \times_X V} \to B|_{U \times_X V}\).

Then there exists an object \(F\) of \(D(\mathcal{O}_X)\) and isomorphisms \(f : F|_U \to A\), \(g : F|_V \to B\) such that \(c = g|_{U \times_X V} \circ f^{-1}|_{U \times_X V}\). Moreover, given

1. an object \(E\) of \(D(\mathcal{O}_X)\),

2. a morphism \(a : A \to E|_U\) of \(D(\mathcal{O}_U)\),

3. a morphism \(b : B \to E|_V\) of \(D(\mathcal{O}_V)\),

such that \[a|_{U \times_X V} = b|_{U \times_X V} \circ c.\] Then there exists a morphism \(F \to E\) in \(D(\mathcal{O}_X)\) whose restriction to \(U\) is \(a \circ f\) and whose restriction to \(V\) is \(b \circ g\).

Proof

Denote \(j_U\), \(j_V\), \(j_{U \times_X V}\) the corresponding morphisms towards \(X\). Choose a distinguished triangle \[F \to Rj_{U, *}A \oplus Rj_{V, *}B \to Rj_{U \times_X V, *}(B|_{U \times_X V}) \to F[1]\] Here the map \(Rj_{V, *}B \to Rj_{U \times_X V, *}(B|_{U \times_X V})\) is the obvious one. The map \(Rj_{U, *}A \to Rj_{U \times_X V, *}(B|_{U \times_X V})\) is the composition of \(Rj_{U, *}A \to Rj_{U \times_X V, *}(A|_{U \times_X V})\) with \(Rj_{U \times_X V, *}c\). Restricting to \(U\) we obtain \[F|_U \to A \oplus (Rj_{V, *}B)|_U \to (Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U \to F|_U[1]\] Denote \(j : U \times_X V \to U\). Compatibility of restriction and total direct image (Lemma 08GE) shows that both \((Rj_{V, *}B)|_U\) and \((Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U\) are canonically isomorphic to \(Rj_*(B|_{U \times_X V})\). Hence the second arrow of the last displayed equation has a section, and we conclude that the morphism \(F|_U \to A\) is an isomorphism.

To see that the morphism \(F|_V \to B\) is an isomorphism we will use a trick. Namely, choose a distinguished triangle \[F|_V \to B \to B' \to F[1]|_V\] in \(D(\mathcal{O}_V)\). Since \(F|_U \to A\) is an isomorphism, and since we have the isomorphism \(c : A|_{U \times_X V} \to B|_{U \times_X V}\) the restriction of \(F|_V \to B\) is an isomorphism over \(U \times_X V\). Thus \(B'\) is supported on \(j_V^{-1}(T)\) where \(T = |X| \setminus |U|\). On the other hand, there is a morphism of distinguished triangles \[\xymatrix{ F \ar[r] \ar[d] & Rj_{U, *}F|_U \oplus Rj_{V, *}F|_V \ar[r] \ar[d] & Rj_{U \times_X V, *}F|_{U \times_X V} \ar[r] \ar[d] & F[1] \ar[d] \\ F \ar[r] & Rj_{U, *}A \oplus Rj_{V, *}B \ar[r] & Rj_{U \times_X V, *}(B|_{U \times_X V}) \ar[r] & F[1] }\] The all of the vertical maps in this diagram are isomorphisms, except for the map \(Rj_{V, *}F|_V \to Rj_{V, *}B\), hence that is an isomorphism too (Derived Categories, Lemma 014A). This implies that \(Rj_{V, *}B' = 0\). Hence \(B' = 0\) by Lemma 08GG.

The existence of the morphism \(F \to E\) follows from the Mayer-Vietoris sequence for \(\Hom\), see Lemma 08H9.

The coherator

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). The coherator is a functor \[Q_X : \textit{Mod}(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_X)\] which is right adjoint to the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\). It exists for any algebraic space \(X\) and moreover the adjunction mapping \(Q_X(\mathcal{F}) \to \mathcal{F}\) is an isomorphism for every quasi-coherent module \(\mathcal{F}\), see Properties of Spaces, Proposition 077V. Since \(Q_X\) is left exact (as a right adjoint) we can consider its right derived extension \[RQ_X : D(\mathcal{O}_X) \longrightarrow D(\QCoh(\mathcal{O}_X)).\] Since \(Q_X\) is right adjoint to the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) we see that \(RQ_X\) is right adjoint to the canonical functor \(D(\QCoh(\mathcal{O}_X)) \to D(\mathcal{O}_X)\) by Derived Categories, Lemma 09T5.

In this section we will study the functor \(RQ_X\). In Section 0CR3 we will study the (closely related) right adjoint to the inclusion functor \(D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_X)\) (when it exists).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). Then \(f_*\) defines a derived functor \(f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\). This functor has the property that \[\xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{f_*} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) }\] commutes.

Proof

The functor \(f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)\) is exact, see Cohomology of Spaces, Lemma 073H. Hence \(f_*\) defines a derived functor \(f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\) by simply applying \(f_*\) to any representative complex, see Derived Categories, Lemma 015F. For any complex of \(\mathcal{O}_X\)-modules \(\mathcal{F}^\bullet\) there is a canonical map \(f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet\). To finish the proof we show this is a quasi-isomorphism when \(\mathcal{F}^\bullet\) is a complex with each \(\mathcal{F}^n\) quasi-coherent. The statement is étale local on \(Y\) hence we may assume \(Y\) affine. As an affine morphism is representable we reduce to the case of schemes by the compatibility of Remark 08GH. The case of schemes is Derived Categories of Schemes, Lemma 08D7.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is quasi-compact, quasi-separated, and flat. Then, denoting \[\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\] the right derived functor of \(f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)\) we have \(RQ_Y \circ Rf_* = \Phi \circ RQ_X\).

Proof

We will prove this by showing that \(RQ_Y \circ Rf_*\) and \(\Phi \circ RQ_X\) are right adjoint to the same functor \(D(\QCoh(\mathcal{O}_Y)) \to D(\mathcal{O}_X)\).

Since \(f\) is quasi-compact and quasi-separated, we see that \(f_*\) preserves quasi-coherence, see Morphisms of Spaces, Lemma 03M9. Recall that \(\QCoh(\mathcal{O}_X)\) is a Grothendieck abelian category (Properties of Spaces, Proposition 077V). Hence any \(K\) in \(D(\QCoh(\mathcal{O}_X))\) can be represented by a K-injective complex \(\mathcal{I}^\bullet\) of \(\QCoh(\mathcal{O}_X)\), see Injectives, Theorem 079P. Then we can define \(\Phi(K) = f_*\mathcal{I}^\bullet\).

Since \(f\) is flat, the functor \(f^*\) is exact. Hence \(f^*\) defines \(f^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)\) and also \(f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))\). The functor \(f^* = Lf^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)\) is left adjoint to \(Rf_* : D(\mathcal{O}_X) \to D(\mathcal{O}_Y)\), see Cohomology on Sites, Lemma 07A6. Similarly, the functor \(f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))\) is left adjoint to \(\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\) by Derived Categories, Lemma 09T5.

Let \(A\) be an object of \(D(\QCoh(\mathcal{O}_Y))\) and \(E\) an object of \(D(\mathcal{O}_X)\). Then \[\begin{align*} \Hom_{D(\QCoh(\mathcal{O}_Y))}(A, RQ_Y(Rf_*E)) & = \Hom_{D(\mathcal{O}_Y)}(A, Rf_*E) \\ & = \Hom_{D(\mathcal{O}_X)}(f^*A, E) \\ & = \Hom_{D(\QCoh(\mathcal{O}_X))}(f^*A, RQ_X(E)) \\ & = \Hom_{D(\QCoh(\mathcal{O}_Y))}(A, \Phi(RQ_X(E))) \end{align*}\] This implies what we want.

Lemma

Let \(S\) be a scheme. Let \(X\) be an affine algebraic space over \(S\). Set \(A = \Gamma(X, \mathcal{O}_X)\). Then

1. \(Q_X : \textit{Mod}(\mathcal{O}_X) \to \QCoh(\mathcal{O}_X)\) is the functor which sends \(\mathcal{F}\) to the quasi-coherent \(\mathcal{O}_X\)-module associated to the \(A\)-module \(\Gamma(X, \mathcal{F})\),

2. \(RQ_X : D(\mathcal{O}_X) \to D(\QCoh(\mathcal{O}_X))\) is the functor which sends \(E\) to the complex of quasi-coherent \(\mathcal{O}_X\)-modules associated to the object \(R\Gamma(X, E)\) of \(D(A)\),

3. restricted to \(D_\QCoh(\mathcal{O}_X)\) the functor \(RQ_X\) defines a quasi-inverse to (08F1).

Proof

Let \(X_0 = \Spec(A)\) be the affine scheme representing \(X\). Recall that there is a morphism of ringed sites \(\epsilon : X_\etale \to X_{0, Zar}\) which induces equivalences \[\xymatrix{ \QCoh(\mathcal{O}_X) \ar@<1ex>[r]^{{\epsilon_*}} & \QCoh(\mathcal{O}_{X_0}) \ar@<1ex>[l]^{{\epsilon^*}} }\] see Lemma 071Q. Hence we see that \(Q_X = \epsilon^* \circ Q_{X_0} \circ \epsilon_*\) by uniqueness of adjoint functors. Hence (1) follows from the description of \(Q_{X_0}\) in Derived Categories of Schemes, Lemma 08D9 and the fact that \(\Gamma(X_0, \epsilon_*\mathcal{F}) = \Gamma(X, \mathcal{F})\). Part (2) follows from (1) and the fact that the functor from \(A\)-modules to quasi-coherent \(\mathcal{O}_X\)-modules is exact. The third assertion now follows from the result for schemes (Derived Categories of Schemes, Lemma 08D9) and Lemma 071Q.

Next, we prove a criterion for when the functor \(D(\QCoh(\mathcal{O}_X)) \to D_\QCoh(\mathcal{O}_X)\) is an equivalence.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Suppose that for every étale morphism \(j : V \to W\) with \(W \subset X\) quasi-compact open and \(V\) affine the right derived functor \[\Phi : D(\QCoh(\mathcal{O}_V)) \to D(\QCoh(\mathcal{O}_W))\] of the left exact functor \(j_* : \QCoh(\mathcal{O}_V) \to \QCoh(\mathcal{O}_W)\) fits into a commutative diagram \[\xymatrix{ D(\QCoh(\mathcal{O}_V)) \ar[d]_\Phi \ar[r]_{i_V} & D_\QCoh(\mathcal{O}_V) \ar[d]^{Rj_*} \\ D(\QCoh(\mathcal{O}_W)) \ar[r]^{i_W} & D_\QCoh(\mathcal{O}_W) }\] Then the functor (08F1) \[D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X)\] is an equivalence with quasi-inverse given by \(RQ_X\).

Proof

We first use the induction principle to prove \(i_X\) is fully faithful. More precisely, we will use Lemma 09IT. Let \((U \subset W, V \to W)\) be an elementary distinguished square with \(V\) affine and \(U, W\) quasi-compact open in \(X\). Assume that \(i_U\) is fully faithful. We have to show that \(i_W\) is fully faithful. We may replace \(X\) by \(W\), i.e., we may assume \(W = X\) (we do this just to simplify the notation – observe that the condition in the statement of the lemma is preserved under this operation).

Suppose that \(A, B\) are objects of \(D(\QCoh(\mathcal{O}_X))\). We want to show that \[\Hom_{D(\QCoh(\mathcal{O}_X))}(A, B) \longrightarrow \Hom_{D(\mathcal{O}_X)}(i_X(A), i_X(B))\] is bijective. Let \(T = |X| \setminus |U|\).

Assume first \(i_X(B)\) is supported on \(T\). In this case the map \[i_X(B) \to Rj_{V, *}(i_X(B)|_V) = Rj_{V, *}(i_V(B|_V))\] is a quasi-isomorphism (Lemma 08GG). By assumption we have an isomorphism \(i_X(\Phi(B|_V)) \to Rj_{V, *}(i_V(B|_V))\) in \(D(\mathcal{O}_X)\). Moreover, \(\Phi\) and \({-}|_V\) are adjoint functors on the derived categories of quasi-coherent modules (by Derived Categories, Lemma 09T5). The adjunction map \(B \to \Phi(B|_V)\) becomes an isomorphism after applying \(i_X\), whence is an isomorphism in \(D(\QCoh(\mathcal{O}_X))\). Hence \[\begin{align*} \Mor_{D(\QCoh(\mathcal{O}_X))}(A, B) & = \Mor_{D(\QCoh(\mathcal{O}_X))}(A, \Phi(B|_V)) \\ & = \Mor_{D(\QCoh(\mathcal{O}_V))}(A|_V, B|_V) \\ & = \Mor_{D(\mathcal{O}_V)}(i_V(A|_V), i_V(B|_V)) \\ & = \Mor_{D(\mathcal{O}_X)}(i_X(A), Rj_{V, *}(i_V(B|_V))) \\ & = \Mor_{D(\mathcal{O}_X)}(i_X(A), i_X(B)) \end{align*}\] as desired. Here we have used that \(i_V\) is fully faithful (Lemma 08H0).

In general, choose any complex \(\mathcal{B}^\bullet\) of quasi-coherent \(\mathcal{O}_X\)-modules representing \(B\). Next, choose any quasi-isomorphism \(s : \mathcal{B}^\bullet|_U \to \mathcal{C}^\bullet\) of complexes of quasi-coherent modules on \(U\). As \(j_U : U \to X\) is quasi-compact and quasi-separated the functor \(j_{U, *}\) transforms quasi-coherent modules into quasi-coherent modules (Morphisms of Spaces, Lemma 03M9). Thus there is a canonical map \(\mathcal{B}^\bullet \to j_{U, *}(\mathcal{B}^\bullet|_U) \to j_{U, *}\mathcal{C}^\bullet\) of complexes of quasi-coherent modules on \(X\). Set \(B'' = j_{U, *}\mathcal{C}^\bullet\) in \(D(\QCoh(\mathcal{O}_X))\) and choose a distinguished triangle \[B \to B'' \to B' \to B[1]\] in \(D(\QCoh(\mathcal{O}_X))\). Since the first arrow of the triangle restricts to an isomorphism over \(U\) we see that \(B'\) is supported on \(T\). Hence in the diagram \[\xymatrix{ \Hom_{D(\QCoh(\mathcal{O}_X))}(A, B'[-1]) \ar[r] \ar[d] & \Hom_{D(\mathcal{O}_X)}(i_X(A), i_X(B')[-1]) \ar[d] \\ \Hom_{D(\QCoh(\mathcal{O}_X))}(A, B) \ar[r] \ar[d] & \Hom_{D(\mathcal{O}_X)}(i_X(A), i_X(B)) \ar[d] \\ \Hom_{D(\QCoh(\mathcal{O}_X))}(A, B'') \ar[r] \ar[d] & \Hom_{D(\mathcal{O}_X)}(i_X(A), i_X(B'')) \ar[d] \\ \Hom_{D(\QCoh(\mathcal{O}_X))}(A, B') \ar[r] & \Hom_{D(\mathcal{O}_X)}(i_X(A), i_X(B')) }\] we have exact columns and the top and bottom horizontal arrows are bijective. Finally, choose a complex \(\mathcal{A}^\bullet\) of quasi-coherent modules representing \(A\).

Let \(\alpha : i_X(A) \to i_X(B)\) be a morphism between in \(D(\mathcal{O}_X)\). The restriction \(\alpha|_U\) comes from a morphism in \(D(\QCoh(\mathcal{O}_U))\) as \(i_U\) is fully faithful. Hence there exists a choice of \(s : \mathcal{B}^\bullet|_U \to \mathcal{C}^\bullet\) as above such that \(\alpha|_U\) is represented by an actual map of complexes \(\mathcal{A}^\bullet|_U \to \mathcal{C}^\bullet\). This corresponds to a map of complexes \(\mathcal{A} \to j_{U, *}\mathcal{C}^\bullet\). In other words, the image of \(\alpha\) in \(\Hom_{D(\mathcal{O}_X)}(i_X(A), i_X(B''))\) comes from an element of \(\Hom_{D(\QCoh(\mathcal{O}_X))}(A, B'')\). A diagram chase then shows that \(\alpha\) comes from a morphism \(A \to B\) in \(D(\QCoh(\mathcal{O}_X))\). Finally, suppose that \(a : A \to B\) is a morphism of \(D(\QCoh(\mathcal{O}_X))\) which becomes zero in \(D(\mathcal{O}_X)\). After choosing \(\mathcal{B}^\bullet\) suitably, we may assume \(a\) is represented by a morphism of complexes \(a^\bullet : \mathcal{A}^\bullet \to \mathcal{B}^\bullet\). Since \(i_U\) is fully faithul the restriction \(a^\bullet|_U\) is zero in \(D(\QCoh(\mathcal{O}_U))\). Thus we can choose \(s\) such that \(s \circ a^\bullet|_U : \mathcal{A}^\bullet|_U \to \mathcal{C}^\bullet\) is homotopic to zero. Applying the functor \(j_{U, *}\) we conclude that \(\mathcal{A}^\bullet \to j_{U, *}\mathcal{C}^\bullet\) is homotopic to zero. Thus \(a\) maps to zero in \(\Hom_{D(\QCoh(\mathcal{O}_X))}(A, B'')\). Thus we may assume that \(a\) is the image of an element of \(b \in \Hom_{D(\QCoh(\mathcal{O}_X))}(A, B'[-1])\). The image of \(b\) in \(\Hom_{D(\mathcal{O}_X)}(i_X(A), i_X(B')[-1])\) comes from a \(\gamma \in \Hom_{D(\mathcal{O}_X)}(A, B''[-1])\) (as \(a\) maps to zero in the group on the right). Since we’ve seen above the horizontal arrows are surjective, we see that \(\gamma\) comes from a \(c\) in \(\Hom_{D(\QCoh(\mathcal{O}_X))}(A, B''[-1])\) which implies \(a = 0\) as desired.

At this point we know that \(i_X\) is fully faithful for our original \(X\). Since \(RQ_X\) is its right adjoint, we see that \(RQ_X \circ i_X = \text{id}\) (Categories, Lemma 07RB). To finish the proof we show that for any \(E\) in \(D_\QCoh(\mathcal{O}_X)\) the map \(i_X(RQ_X(E)) \to E\) is an isomorphism. Choose a distinguished triangle \[i_X(RQ_X(E)) \to E \to E' \to i_X(RQ_X(E))[1]\] in \(D_\QCoh(\mathcal{O}_X)\). A formal argument using the above shows that \(i_X(RQ_X(E')) = 0\). Thus it suffices to prove that for \(E \in D_\QCoh(\mathcal{O}_X)\) the condition \(i_X(RQ_X(E)) = 0\) implies that \(E = 0\). Consider an étale morphism \(j : V \to X\) with \(V\) affine. By Lemmas 08H0 and 08GZ and our assumption we have \[Rj_*(E|_V) = Rj_*(i_V(RQ_V(E|_V))) = i_X(\Phi(RQ_V(E|_V))) = i_X(RQ_X(Rj_*(E|_V)))\] Choose a distinguished triangle \[E \to Rj_*(E|_V) \to E' \to E[1]\] Apply \(RQ_X\) to get a distinguished triangle \[0 \to RQ_X(Rj_*(E|_V)) \to RQ_X(E') \to 0[1]\] in other words the map in the middle is an isomorphism. Combined with the string of equalities above we find that our first distinguished triangle becomes a distinguished triangle \[E \to i_X(RQ_X(E')) \to E' \to E[1]\] where the middle morphism is the adjunction map. However, the composition \(E \to E'\) is zero, hence \(E \to i_X(RQ_X(E'))\) is zero by adjunction! Since this morphism is isomorphic to the morphism \(E \to Rj_*(E|_V)\) adjoint to \(\text{id} : E|_V \to E|_V\) we conclude that \(E|_V\) is zero. Since this holds for all affine \(V\) étale over \(X\) we conclude \(E\) is zero as desired.

Proposition

Let \(S\) be a scheme. Let \(X\) be a quasi-compact algebraic space over \(S\) with affine diagonal over \(\mathbf{Z}\) (as in Properties of Spaces, Definition 03BS). Then the functor (08F1) \[D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X)\] is an equivalence with quasi-inverse given by \(RQ_X\).

Proof

Let \(V \to W\) be an étale morphism with \(V\) affine and \(W\) a quasi-compact open subspace of \(X\). Then the morphism \(V \to W\) is affine as \(W\) has affine diagonal over \(\mathbf{Z}\) and \(V\) is affine (Morphisms of Spaces, Lemma 08GB). Lemma 08GY then guarantees that the assumption of Lemma 09TG holds. Hence we conclude.

Lemma

Let \(S\) be a scheme and let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(X\) and \(Y\) are quasi-compact and have affine diagonal over \(\mathbf{Z}\) (as in Properties of Spaces, Definition 03BS). Then, denoting \[\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\] the right derived functor of \(f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)\) the diagram \[\xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_\Phi \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) }\] is commutative.

Proof

Observe that the horizontal arrows in the diagram are equivalences of categories by Proposition 08H1. Hence we can identify these categories (and similarly for other quasi-compact algebraic spaces with affine diagonal) and then the statement of the lemma is that the canonical map \(\Phi(K) \to Rf_*(K)\) is an isomorphism for all \(K\) in \(D(\QCoh(\mathcal{O}_X))\). Note that if \(K_1 \to K_2 \to K_3 \to K_1[1]\) is a distinguished triangle in \(D(\QCoh(\mathcal{O}_X))\) and the statement is true for two-out-of-three, then it is true for the third.

Let \(\mathcal{B} \subset \Ob(X_{spaces, \etale})\) be the set of objects which are quasi-compact and have affine diagonal. For \(U \in \mathcal{B}\) and any morphism \(g : U \to Z\) where \(Z\) is a quasi-compact algebraic space over \(S\) with affine diagonal, denote \[\Phi_g : D(\QCoh(\mathcal{O}_U)) \to D(\QCoh(\mathcal{O}_Z))\] the derived extension of \(g_*\). Let \(P(U) =\) “for any \(K\) in \(D(\QCoh(\mathcal{O}_U))\) and any \(g : U \to Z\) as above the map \(\Phi_g(K) \to Rg_*K\) is an isomorphism”. By Remark 08GR conditions (1), (2), and (3)(a) of Lemma 08GQ hold and we are left with proving (3)(b) and (4).

Checking condition (3)(b). Let \(U\) be an affine scheme étale over \(X\). Let \(g : U \to Z\) be as above. Since the diagonal of \(Z\) is affine the morphism \(g : U \to Z\) is affine (Morphisms of Spaces, Lemma 08GB). Hence \(P(U)\) holds by Lemma 08GY.

Checking condition (4). Let \((U \subset W, V \to W)\) be an elementary distinguished square in \(X_{spaces, \etale}\) with \(U, W, V\) in \(\mathcal{B}\) and \(V\) affine. Assume that \(P\) holds for \(U\), \(V\), and \(U \times_W V\). We have to show that \(P\) holds for \(W\). Let \(g : W \to Z\) be a morphism to a quasi-compact algebraic space with affine diagonal. Let \(K\) be an object of \(D(\QCoh(\mathcal{O}_W))\). Consider the distinguished triangle \[K \to Rj_{U, *}K|_U \oplus Rj_{V, *}K|_V \to Rj_{U \times_W V, *}K|_{U \times_W V} \to K[1]\] in \(D(\mathcal{O}_W)\). By the two-out-of-three property mentioned above, it suffices to show that \(\Phi_g(Rj_{U, *}K|_U) \to Rg_*(Rj_{U, *}K|_U)\) is an isomorphism and similarly for \(V\) and \(U \times_W V\). This is discussed in the next paragraph.

Let \(j : U \to W\) be a morphism \(X_{spaces, \etale}\) with \(U, W\) in \(\mathcal{B}\) and \(P\) holds for \(U\). Let \(g : W \to Z\) be a morphism to a quasi-compact algebraic space with affine diagonal. To finish the proof we have to show that \(\Phi_g(Rj_*K) \to Rg_*(Rj_*K)\) is an isomorphism for any \(K\) in \(D(\QCoh(\mathcal{O}_U))\). Let \(\mathcal{I}^\bullet\) be a K-injective complex in \(\QCoh(\mathcal{O}_U)\) representing \(K\). From \(P(U)\) applied to \(j\) we see that \(j_*\mathcal{I}^\bullet\) represents \(Rj_*K\). Since \(j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_W)\) has an exact left adjoint \(j^* : \QCoh(\mathcal{O}_W) \to \QCoh(\mathcal{O}_U)\) we see that \(j_*\mathcal{I}^\bullet\) is a K-injective complex in \(\QCoh(\mathcal{O}_W)\), see Derived Categories, Lemma 08BJ. Hence \(\Phi_g(Rj_*K)\) is represented by \(g_*j_*\mathcal{I}^\bullet = (g \circ j)_*\mathcal{I}^\bullet\). By \(P(U)\) applied to \(g \circ j\) we see that this represents \(R_{g \circ j, *}(K) = Rg_*(Rj_*K)\). This finishes the proof.

The coherator for Noetherian spaces

We need a little bit more about injective modules to treat the case of a Noetherian algebraic space.

Lemma

Let \(S\) be a Noetherian affine scheme. Every injective object of \(\QCoh(\mathcal{O}_S)\) is a filtered colimit \(\colim_i \mathcal{F}_i\) of quasi-coherent sheaves of the form \[\mathcal{F}_i = (Z_i \to S)_*\mathcal{G}_i\] where \(Z_i\) is the spectrum of an Artinian ring and \(\mathcal{G}_i\) is a coherent module on \(Z_i\).

Proof

Let \(S = \Spec(A)\). Let \(\mathcal{J}\) be an injective object of \(\QCoh(\mathcal{O}_S)\). Since \(\QCoh(\mathcal{O}_S)\) is equivalent to the category of \(A\)-modules we see that \(\mathcal{J}\) is equal to \(\widetilde{J}\) for some injective \(A\)-module \(J\). By Dualizing Complexes, Proposition 08YA we can write \(J = \bigoplus E_\alpha\) with \(E_\alpha\) indecomposable and therefore isomorphic to the injective hull of a residue field at a point. Thus (because finite disjoint unions of Artinian schemes are Artinian) we may assume that \(J\) is the injective hull of \(\kappa(\mathfrak p)\) for some prime \(\mathfrak p\) of \(A\). Then \(J = \bigcup J[\mathfrak p^n]\) where \(J[\mathfrak p^n]\) is the injective hull of \(\kappa(\mathfrak p)\) over \(A_\mathfrak p/\mathfrak p^nA_\mathfrak p\), see Dualizing Complexes, Lemma 08Z4. Thus \(\widetilde{J}\) is the colimit of the sheaves \((Z_n \to S)_*\mathcal{G}_n\) where \(Z_n = \Spec(A_\mathfrak p/\mathfrak p^nA_\mathfrak p)\) and \(\mathcal G_n\) the coherent sheaf associated to the finite \(A_\mathfrak p/\mathfrak p^nA_\mathfrak p\)-module \(J[\mathfrak p^n]\). Finiteness follows from Dualizing Complexes, Lemma 08YX.

Lemma

Let \(S\) be an affine scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Every injective object of \(\QCoh(\mathcal{O}_X)\) is a direct summand of a filtered colimit \(\colim_i \mathcal{F}_i\) of quasi-coherent sheaves of the form \[\mathcal{F}_i = (Z_i \to X)_*\mathcal{G}_i\] where \(Z_i\) is the spectrum of an Artinian ring and \(\mathcal{G}_i\) is a coherent module on \(Z_i\).

Proof

Choose an affine scheme \(U\) and a surjective étale morphism \(j : U \to X\) (Properties of Spaces, Lemma 03H6). Then \(U\) is a Noetherian affine scheme. Choose an injective object \(\mathcal{J}'\) of \(\QCoh(\mathcal{O}_U)\) such that there exists an injection \(\mathcal{J}|_U \to \mathcal{J}'\). Then \[\mathcal{J} \to j_*\mathcal{J}'\] is an injective morphism in \(\QCoh(\mathcal{O}_X)\), hence identifies \(\mathcal{J}\) as a direct summand of \(j_*\mathcal{J}'\). Thus the result follows from the corresponding result for \(\mathcal{J}'\) proved in Lemma 09TI.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a flat, quasi-compact, and quasi-separated morphism of algebraic spaces over \(S\). If \(\mathcal{J}\) is an injective object of \(\QCoh(\mathcal{O}_X)\), then \(f_*\mathcal{J}\) is an injective object of \(\QCoh(\mathcal{O}_Y)\).

Proof

Since \(f\) is quasi-compact and quasi-separated, the functor \(f_*\) transforms quasi-coherent sheaves into quasi-coherent sheaves (Morphisms of Spaces, Lemma 03M9). The functor \(f^*\) is a left adjoint to \(f_*\) which transforms injections into injections. Hence the result follows from Homology, Lemma 015Z

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). If \(\mathcal{J}\) is an injective object of \(\QCoh(\mathcal{O}_X)\), then

1. \(H^p(U, \mathcal{J}|_U) = 0\) for \(p > 0\) and for every quasi-compact and quasi-separated algebraic space \(U\) étale over \(X\),

2. for any morphism \(f : X \to Y\) of algebraic spaces over \(S\) with \(Y\) quasi-separated we have \(R^pf_*\mathcal{J} = 0\) for \(p > 0\).

Proof

Proof of (1). Write \(\mathcal{J}\) as a direct summand of \(\colim \mathcal{F}_i\) with \(\mathcal{F}_i = (Z_i \to X)_*\mathcal{G}_i\) as in Lemma 09TJ. It is clear that it suffices to prove the vanishing for \(\colim \mathcal{F}_i\). Since pullback commutes with colimits and since \(U\) is quasi-compact and quasi-separated, it suffices to prove \(H^p(U, \mathcal{F}_i|_U) = 0\) for \(p > 0\), see Cohomology of Spaces, Lemma 073E. Observe that \(Z_i \to X\) is an affine morphism, see Morphisms of Spaces, Lemma 09TF. Thus \[\mathcal{F}_i|_U = (Z_i \times_X U \to U)_*\mathcal{G}'_i = R(Z_i \times_X U \to U)_*\mathcal{G}'_i\] where \(\mathcal{G}'_i\) is the pullback of \(\mathcal{G}_i\) to \(Z_i \times_X U\), see Cohomology of Spaces, Lemma 07U8. Since \(Z_i \times_X U\) is affine we conclude that \(\mathcal{G}'_i\) has no higher cohomology on \(Z_i \times_X U\). By the Leray spectral sequence we conclude the same thing is true for \(\mathcal{F}_i|_U\) (Cohomology on Sites, Lemma 0733).

Proof of (2). Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(V \to Y\) be an étale morphism with \(V\) affine. Then \(V \times_Y X \to X\) is an étale morphism and \(V \times_Y X\) is a quasi-compact and quasi-separated algebraic space étale over \(X\) (details omitted). Hence \(H^p(V \times_Y X, \mathcal{J})\) is zero by part (1). Since \(R^pf_*\mathcal{J}\) is the sheaf associated to the presheaf \(V \mapsto H^p(V \times_Y X, \mathcal{J})\) the result is proved.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of Noetherian algebraic spaces over \(S\). Then \(f_*\) on quasi-coherent sheaves has a right derived extension \(\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\) such that the diagram \[\xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{\Phi} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) }\] commutes.

Proof

Since \(X\) and \(Y\) are Noetherian the morphism is quasi-compact and quasi-separated (see Morphisms of Spaces, Lemma 073B). Thus \(f_*\) preserve quasi-coherence, see Morphisms of Spaces, Lemma 03M9. Next, let \(K\) be an object of \(D(\QCoh(\mathcal{O}_X))\). Since \(\QCoh(\mathcal{O}_X)\) is a Grothendieck abelian category (Properties of Spaces, Proposition 077V), we can represent \(K\) by a K-injective complex \(\mathcal{I}^\bullet\) such that each \(\mathcal{I}^n\) is an injective object of \(\QCoh(\mathcal{O}_X)\), see Injectives, Theorem 079P. Thus we see that the functor \(\Phi\) is defined by setting \[\Phi(K) = f_*\mathcal{I}^\bullet\] where the right hand side is viewed as an object of \(D(\QCoh(\mathcal{O}_Y))\). To finish the proof of the lemma it suffices to show that the canonical map \[f_*\mathcal{I}^\bullet \longrightarrow Rf_*\mathcal{I}^\bullet\] is an isomorphism in \(D(\mathcal{O}_Y)\). To see this it suffices to prove the map induces an isomorphism on cohomology sheaves. Pick any \(m \in \mathbf{Z}\). Let \(N = N(X, Y, f)\) be as in Lemma 08FA. Consider the short exact sequence \[0 \to \sigma_{\geq m - N - 1}\mathcal{I}^\bullet \to \mathcal{I}^\bullet \to \sigma_{\leq m - N - 2}\mathcal{I}^\bullet \to 0\] of complexes of quasi-coherent sheaves on \(X\). By Lemma 08FA we see that the cohomology sheaves of \(Rf_*\sigma_{\leq m - N - 2}\mathcal{I}^\bullet\) are zero in degrees \(\geq m - 1\). Thus we see that \(R^mf_*\mathcal{I}^\bullet\) is isomorphic to \(R^mf_*\sigma_{\geq m - N - 1}\mathcal{I}^\bullet\). In other words, we may assume that \(\mathcal{I}^\bullet\) is a bounded below complex of injective objects of \(\QCoh(\mathcal{O}_X)\). This case follows from Leray’s acyclicity lemma (Derived Categories, Lemma 015E) with required vanishing because of Lemma 09TL.

Proposition

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Then the functor (08F1) \[D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X)\] is an equivalence with quasi-inverse given by \(RQ_X\).

Proof

Follows immediately from Lemmas 09TM and 09TG.

Pseudo-coherent and perfect complexes

In this section we study the general notions defined in Cohomology on Sites, Sections 08FK, 08FS, 08FY, and 08G4 for the étale site of an algebraic space. In particular we match this with what happens for schemes.

First we compare the notion of a pseudo-coherent complex on a scheme and on its associated small étale site.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. The following are equivalent

1. \(\mathcal{F}\) is of finite type as an \(\mathcal{O}_X\)-module, and

2. \(\epsilon^*\mathcal{F}\) is of finite type as an \(\mathcal{O}_\etale\)-module on the small étale site of \(X\).

Here \(\epsilon\) is as in (08H7).

Proof

The implication (1) \(\Rightarrow\) (2) is a general fact, see Modules on Sites, Lemma 03DO. Assume (2). By assumption there exists an étale covering \(\{f_i : X_i \to X\}\) such that \(\epsilon^*\mathcal{F}|_{(X_i)_\etale}\) is generated by finitely many sections. Let \(x \in X\). We will show that \(\mathcal{F}\) is generated by finitely many sections in a neighbourhood of \(x\). Say \(x\) is in the image of \(X_i \to X\) and denote \(X' = X_i\). Let \(s_1, \ldots, s_n \in \Gamma(X', \epsilon^*\mathcal{F}|_{X'_\etale})\) be generating sections. As \(\epsilon^*\mathcal{F} = \epsilon^{-1}\mathcal{F} \otimes_{\epsilon^{-1}\mathcal{O}_X} \mathcal{O}_\etale\) we can find an étale morphism \(X'' \to X'\) such that \(x\) is in the image of \(X'' \to X\) and such that \(s_i|_{X''} = \sum s_{ij} \otimes a_{ij}\) for some sections \(s_{ij} \in \epsilon^{-1}\mathcal{F}(X'')\) and \(a_{ij} \in \mathcal{O}_\etale(X'')\). Denote \(U \subset X\) the image of \(X'' \to X\). This is an open subscheme as \(f'' : X'' \to X\) is étale (Morphisms, Lemma 03WT). After possibly shrinking \(X''\) more we may assume \(s_{ij}\) come from elements \(t_{ij} \in \mathcal{F}(U)\) as follows from the construction of the inverse image functor \(\epsilon^{-1}\). Now we claim that \(t_{ij}\) generate \(\mathcal{F}|_U\) which finishes the proof of the lemma. Namely, the corresponding map \(\mathcal{O}_U^{\oplus N} \to \mathcal{F}|_U\) has the property that its pullback by \(f''\) to \(X''\) is surjective. Since \(f'' : X'' \to U\) is a surjective flat morphism of schemes, this implies that \(\mathcal{O}_U^{\oplus N} \to \mathcal{F}|_U\) is surjective by looking at stalks and using that \(\mathcal{O}_{U, f''(z)} \to \mathcal{O}_{X'', z}\) is faithfully flat for all \(z \in X''\).

In the situation above the morphism of sites \(\epsilon\) is flat hence defines a pullback on complexes of modules.

Lemma

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

1. \(E\) is \(m\)-pseudo-coherent, and

2. \(\epsilon^*E\) is \(m\)-pseudo-coherent on the small étale site of \(X\).

Here \(\epsilon\) is as in (08H7).

Proof

The implication (1) \(\Rightarrow\) (2) is a general fact, see Cohomology on Sites, Lemma 08H4. Assume \(\epsilon^*E\) is \(m\)-pseudo-coherent. We will use without further mention that \(\epsilon^*\) is an exact functor and that therefore \[\epsilon^*H^i(E) = H^i(\epsilon^*E).\] To show that \(E\) is \(m\)-pseudo-coherent we may work locally on \(X\), hence we may assume that \(X\) is quasi-compact (for example affine). Since \(X\) is quasi-compact every étale covering \(\{U_i \to X\}\) has a finite refinement. Thus we see that \(\epsilon^*E\) is an object of \(D^{-}(\mathcal{O}_\etale)\), see comments following Cohomology on Sites, Definition 08FT. By Lemma 08H8 it follows that \(E\) is an object of \(D^-(\mathcal{O}_X)\).

Let \(n \in \mathbf{Z}\) be the largest integer such that \(H^n(E)\) is nonzero; then \(n\) is also the largest integer such that \(H^n(\epsilon^*E)\) is nonzero. We will prove the lemma by induction on \(n - m\). If \(n < m\), then the lemma is clearly true. If \(n \geq m\), then \(H^n(\epsilon^*E)\) is a finite \(\mathcal{O}_\etale\)-module, see Cohomology on Sites, Lemma 08FX. Hence \(H^n(E)\) is a finite \(\mathcal{O}_X\)-module, see Lemma 08HD. After replacing \(X\) by the members of an open covering, we may assume there exists a surjection \(\mathcal{O}_X^{\oplus t} \to H^n(E)\). We may locally on \(X\) lift this to a map of complexes \(\alpha : \mathcal{O}_X^{\oplus t}[-n] \to E\) (details omitted). Choose a distinguished triangle \[\mathcal{O}_X^{\oplus t}[-n] \to E \to C \to \mathcal{O}_X^{\oplus t}[-n + 1]\] Then \(C\) has vanishing cohomology in degrees \(\geq n\). On the other hand, the complex \(\epsilon^*C\) is \(m\)-pseudo-coherent, see Cohomology on Sites, Lemma 08FV. Hence by induction we see that \(C\) is \(m\)-pseudo-coherent. Applying Cohomology on Sites, Lemma 08FV once more we conclude.

Lemma

Let \(X\) be a scheme. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Then

1. \(E\) has tor amplitude in \([a, b]\) if and only if \(\epsilon^*E\) has tor amplitude in \([a, b]\).

2. \(E\) has finite tor dimension if and only if \(\epsilon^*E\) has finite tor dimension.

Here \(\epsilon\) is as in (08H7).

Proof

The easy implication follows from Cohomology on Sites, Lemma 08H5. For the converse, assume that \(\epsilon^*E\) has tor amplitude in \([a, b]\). Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. As \(\epsilon\) is a flat morphism of ringed sites (Lemma 08H8) we have \[\epsilon^*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F}) = \epsilon^*E \otimes^\mathbf{L}_{\mathcal{O}_\etale} \epsilon^*\mathcal{F}\] Thus the (assumed) vanishing of cohomology sheaves on the right hand side implies the desired vanishing of the cohomology sheaves of \(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F}\) via Lemma 08H8.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Then

1. \(E\) as an object of \(D(f^{-1}\mathcal{O}_Y)\) has tor amplitude in \([a, b]\) if and only if \(\epsilon^*E\) has tor amplitude in \([a, b]\) as an object of \(D(f_{small}^{-1}\mathcal{O}_{Y_\etale})\).

2. \(E\) locally has finite tor dimension as an object of \(D(f^{-1}\mathcal{O}_Y)\) if and only if \(\epsilon^*E\) locally has finite tor dimension as an object of \(D(f_{small}^{-1}\mathcal{O}_{Y_\etale})\).

Here \(\epsilon\) is as in (08H7).

Proof

The easy direction in (1) follows from Cohomology on Sites, Lemma 08H5. Let \(x \in X\) be a point and let \(\overline{x}\) be a geometric point lying over \(x\). Let \(y = f(x)\) and denote \(\overline{y}\) the geometric point of \(Y\) coming from \(\overline{x}\). Then \((f^{-1}\mathcal{O}_Y)_x = \mathcal{O}_{Y, y}\) (Sheaves, Lemma 008H) and \[(f_{small}^{-1}\mathcal{O}_{Y_\etale})_{\overline{x}} = \mathcal{O}_{Y_\etale, \overline{y}} = \mathcal{O}_{Y, y}^{sh}\] is the strict henselization (by Étale Cohomology, Lemmas 03Q1 and 04HX). Since the stalk of \(\mathcal{O}_{X_\etale}\) at \(\overline{x}\) is \(\mathcal{O}_{X, x}^{sh}\) we obtain \[(\epsilon^*E)_{\overline{x}} = E_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X, x}^{sh}\] by transitivity of pullbacks. If \(\epsilon^*E\) has tor amplitude in \([a, b]\) as a complex of \(f_{small}^{-1}\mathcal{O}_{Y_\etale}\)-modules, then \((\epsilon^*E)_{\overline{x}}\) has tor amplitude in \([a, b]\) as a complex of \(\mathcal{O}_{Y, y}^{sh}\)-modules (because taking stalks is a pullback and lemma cited above). By More on Flatness, Lemma 0DK0 we find the tor amplitude of \((\epsilon^*E)_{\overline{x}}\) as a complex of \(\mathcal{O}_{Y, y}\)-modules is in \([a, b]\). Since \(\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}\) is faithfully flat (More on Algebra, Lemma 07QM) and since \((\epsilon^*E)_{\overline{x}} = E_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X, x}^{sh}\) we may apply More on Algebra, Lemma 0DJF to conclude the tor amplitude of \(E_x\) as a complex of \(\mathcal{O}_{Y, y}\)-modules is in \([a, b]\). By Cohomology, Lemma 09U9 we conclude that \(E\) as an object of \(D(f^{-1}\mathcal{O}_Y)\) has tor amplitude in \([a, b]\). This gives the reverse implication in (1). Part (2) follows formally from (1).

Lemma

Let \(X\) be a scheme. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Then \(E\) is a perfect object of \(D(\mathcal{O}_X)\) if and only if \(\epsilon^*E\) is a perfect object of \(D(\mathcal{O}_\etale)\). Here \(\epsilon\) is as in (08H7).

Proof

The easy implication follows from the general result contained in Cohomology on Sites, Lemma 08H6. For the converse, we can use the equivalence of Cohomology on Sites, Lemma 08G8 and the corresponding results for pseudo-coherent and complexes of finite tor dimension, namely Lemmas 08HE and 08HF. Some details omitted.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). If \(E\) is an \(m\)-pseudo-coherent object of \(D(\mathcal{O}_X)\), then \(H^i(E)\) is a quasi-coherent \(\mathcal{O}_X\)-module for \(i > m\). If \(E\) is pseudo-coherent, then \(E\) is an object of \(D_\QCoh(\mathcal{O}_X)\).

Proof

Locally \(H^i(E)\) is isomorphic to \(H^i(\mathcal{E}^\bullet)\) with \(\mathcal{E}^\bullet\) strictly perfect. The sheaves \(\mathcal{E}^i\) are direct summands of finite free modules, hence quasi-coherent. The lemma follows.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). For \(m \in \mathbf{Z}\) the following are equivalent

1. \(H^i(E)\) is coherent for \(i \geq m\) and zero for \(i \gg 0\), and

2. \(E\) is \(m\)-pseudo-coherent.

In particular, \(E\) is pseudo-coherent if and only if \(E\) is an object of \(D^-_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

As \(X\) is quasi-compact we can find an affine scheme \(U\) and a surjective étale morphism \(U \to X\) (Properties of Spaces, Lemma 03H6). Observe that \(U\) is Noetherian. Note that \(E\) is \(m\)-pseudo-coherent if and only if \(E|_U\) is \(m\)-pseudo-coherent (follows from the definition or from Cohomology on Sites, Lemma 08FU). Similarly, \(H^i(E)\) is coherent if and only if \(H^i(E)|_U = H^i(E|_U)\) is coherent (see Cohomology of Spaces, Lemma 07UB). Thus we may assume that \(X\) is representable.

If \(X\) is representable by a scheme \(X_0\) then (Lemma 071Q) we can write \(E = \epsilon^*E_0\) where \(E_0\) is an object of \(D_\QCoh(\mathcal{O}_{X_0})\) and \(\epsilon : X_\etale \to (X_0)_{Zar}\) is as in (08H7). In this case \(E\) is \(m\)-pseudo-coherent if and only if \(E_0\) is by Lemma 08HE. Similarly, \(H^i(E_0)\) is of finite type (i.e., coherent) if and only if \(H^i(E)\) is by Lemma 08HD. Finally, \(H^i(E_0) = 0\) if and only if \(H^i(E) = 0\) by Lemma 08H8. Thus we reduce to the case of schemes which is Derived Categories of Schemes, Lemma 08E8.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-separated algebraic space over \(S\). Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Let \(a \leq b\). The following are equivalent

1. \(E\) has tor amplitude in \([a, b]\), and

2. for all \(\mathcal{F}\) in \(\QCoh(\mathcal{O}_X)\) we have \(H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}) = 0\) for \(i \not \in [a, b]\).

Proof

It is clear that (1) implies (2). Assume (2). Let \(j : U \to X\) be an étale morphism with \(U\) affine. As \(X\) is quasi-separated, we see that \(j\) is both quasi-compact and separated. Hence \(j_*\) transforms quasi-coherent modules into quasi-coherent modules (Morphisms of Spaces, Lemma 03M9). Take any quasi-coherent module \(\mathcal{G}\) on \(U\). By assumption we see that \(H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} j_*\mathcal{G})\) vanishes for \(i \not \in [a, b]\). Pulling back by the flat morphism \(j\) we find that \(H^i(E|_U \otimes_{\mathcal{O}_U}^\mathbf{L} j^*j_*\mathcal{G})\) vanishes for \(i \not \in [a, b]\). By Cohomology of Spaces, Lemma 0HAP we see that \(\mathcal{G}\) is a direct summand of \(j^*j_*\mathcal{G}\). It follows that condition (2) implies the vanishing of \(H^i(E|_U \otimes_{\mathcal{O}_U}^\mathbf{L} \mathcal{G})\) for \(i \not \in [a, b]\) for all quasi-coherent \(\mathcal{O}_U\)-modules \(\mathcal{G}\). Since it suffices to prove that \(E|_U\) has tor amplitude in \([a, b]\) we reduce to the case where \(X\) is representable.

If \(X\) is representable by a scheme \(X_0\) then (Lemma 071Q) we can write \(E = \epsilon^*E_0\) where \(E_0\) is an object of \(D_\QCoh(\mathcal{O}_{X_0})\) and \(\epsilon : X_\etale \to (X_0)_{Zar}\) is as in (08H7). For every quasi-coherent module \(\mathcal{F}_0\) on \(X_0\) the module \(\epsilon^*\mathcal{F}_0\) is quasi-coherent on \(X\) and \[H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \epsilon^*\mathcal{F}_0) = \epsilon^*H^i(E_0 \otimes_{\mathcal{O}_{X_0}}^\mathbf{L} \mathcal{F}_0)\] as \(\epsilon\) is flat (Lemma 08H8). Moreover, the vanishing of these sheaves for \(i \not \in [a, b]\) implies the same thing for \(H^i(E_0 \otimes_{\mathcal{O}_{X_0}}^\mathbf{L} \mathcal{F}_0)\) by the same lemma. Thus we’ve reduced the problem to the case of schemes which is treated in Derived Categories of Schemes, Lemma 08EA.

Lemma

Let \(X\) be a scheme. Let \(E, F\) be objects of \(D(\mathcal{O}_X)\). Assume either

1. \(E\) is pseudo-coherent and \(F\) lies in \(D^+(\mathcal{O}_X)\), or

2. \(E\) is perfect and \(F\) arbitrary,

then there is a canonical isomorphism \[\epsilon^*R\SheafHom(E, F) \longrightarrow R\SheafHom(\epsilon^*E, \epsilon^*F)\] Here \(\epsilon\) is as in (08H7).

Proof

Recall that \(\epsilon\) is flat (Lemma 08H8) and hence \(\epsilon^* = L\epsilon^*\). There is a canonical map from left to right by Cohomology on Sites, Remark 08JF. To see this is an isomorphism we can work locally, i.e., we may assume \(X\) is an affine scheme.

In case (1) we can represent \(E\) by a bounded above complex \(\mathcal{E}^\bullet\) of finite free \(\mathcal{O}_X\)-modules, see Derived Categories of Schemes, Lemma 08EE. We may also represent \(F\) by a bounded below complex \(\mathcal{F}^\bullet\) of \(\mathcal{O}_X\)-modules. Applying Cohomology, Lemma 08I5 we see that \(R\SheafHom(E, F)\) is represented by the complex with terms \[\bigoplus\nolimits_{n = - p + q} \SheafHom_{\mathcal{O}_X}(\mathcal{E}^p, \mathcal{F}^q)\] Applying Cohomology on Sites, Lemma 08JI we see that \(R\SheafHom(\epsilon^*E, \epsilon^*F)\) is represented by the complex with terms \[\bigoplus\nolimits_{n = - p + q} \SheafHom_{\mathcal{O}_\etale} (\epsilon^*\mathcal{E}^p, \epsilon^*\mathcal{F}^q)\] Thus the statement of the lemma boils down to the true fact that the canonical map \[\epsilon^*\SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}) \longrightarrow \SheafHom_{\mathcal{O}_\etale} (\epsilon^*\mathcal{E}, \epsilon^*\mathcal{F})\] is an isomorphism for any \(\mathcal{O}_X\)-module \(\mathcal{F}\) and finite free \(\mathcal{O}_X\)-module \(\mathcal{E}\).

In case (2) we can represent \(E\) by a strictly perfect complex \(\mathcal{E}^\bullet\) of \(\mathcal{O}_X\)-modules, use Derived Categories of Schemes, Lemmas 06Z0 and 08EB and the fact that a perfect complex of modules is represented by a finite complex of finite projective modules. Thus we can do the exact same proof as above, replacing the reference to Cohomology, Lemma 08I5 by a reference to Cohomology, Lemma 08DM.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(L, K\) be objects of \(D(\mathcal{O}_X)\). If either

1. \(L\) in \(D^+_\QCoh(\mathcal{O}_X)\) and \(K\) is pseudo-coherent,

2. \(L\) in \(D_\QCoh(\mathcal{O}_X)\) and \(K\) is perfect,

then \(R\SheafHom(K, L)\) is in \(D_\QCoh(\mathcal{O}_X)\).

Proof

This follows from the analogue for schemes (Derived Categories of Schemes, Lemma 0A6H) via the criterion of Lemma 08F2, the criterion of Lemmas 08HE and 08HG, and the result of Lemma 08JP.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(K, L, M\) be objects of \(D_\QCoh(\mathcal{O}_X)\). The map \[K \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, L) \longrightarrow R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] of Cohomology on Sites, Lemma 0BYU is an isomorphism in the following cases

1. \(M\) perfect, or

2. \(K\) is perfect, or

3. \(M\) is pseudo-coherent, \(L \in D^+(\mathcal{O}_X)\), and \(K\) has finite tor dimension.

Proof

Checking whether or not the map is an isomorphism can be done étale locally hence we may assume \(X\) is an affine scheme. Then we can write \(K, L, M\) as \(\epsilon^*K_0, \epsilon^*L_0, \epsilon^*M_0\) for some \(K_0, L_0, M_0\) in \(D_\QCoh(\mathcal{O}_X)\) by Lemma 071Q. Then we see that Lemma 08JP reduces cases (1) and (3) to the case of schemes which is Derived Categories of Schemes, Lemma 0ATN. If \(K\) is perfect but no other assumptions are made, then we do not know that either side of the arrow is in \(D_\QCoh(\mathcal{O}_X)\) but the result is still true because \(K\) will be represented (after localizing further) by a finite complex of finite free modules in which case it is clear.

Approximation by perfect complexes

In this section we continue the discussion started in Derived Categories of Schemes, Section 08EL.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Consider triples \((T, E, m)\) where

1. \(T \subset |X|\) is a closed subset,

2. \(E\) is an object of \(D_\QCoh(\mathcal{O}_X)\), and

3. \(m \in \mathbf{Z}\).

We say approximation holds for the triple \((T, E, m)\) if there exists a perfect object \(P\) of \(D(\mathcal{O}_X)\) supported on \(T\) and a map \(\alpha : P \to E\) which induces isomorphisms \(H^i(P) \to H^i(E)\) for \(i > m\) and a surjection \(H^m(P) \to H^m(E)\).

Approximation cannot hold for every triple. Please read the remarks following Derived Categories of Schemes, Definition 08EM to see why.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). We say approximation by perfect complexes holds on \(X\) if for any closed subset \(T \subset |X|\) such that the morphism \(X \setminus T \to X\) is quasi-compact there exists an integer \(r\) such that for every triple \((T, E, m)\) as in Definition 08HI with

1. \(E\) is \((m - r)\)-pseudo-coherent, and

2. \(H^i(E)\) is supported on \(T\) for \(i \geq m - r\)

approximation holds.

Lemma

Let \(S\) be a scheme. Let \((U \subset X, j : V \to X)\) be an elementary distinguished square of algebraic space over \(S\). Let \(E\) be a perfect object of \(D(\mathcal{O}_V)\) supported on \(j^{-1}(T)\) where \(T = |X| \setminus |U|\). Then \(Rj_*E\) is a perfect object of \(D(\mathcal{O}_X)\).

Proof

Being perfect is local on \(X_\etale\). Thus it suffices to check that \(Rj_*E\) is perfect when restricted to \(U\) and \(V\). We have \(Rj_*E|_V = E\) by Lemma 08GG which is perfect. We have \(Rj_*E|_U = 0\) because \(E|_{V \setminus j^{-1}(T)} = 0\) (use Lemma 08GE).

Lemma

Let \(S\) be a scheme. Let \((U \subset X, j : V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\). Let \(T\) be a closed subset of \(|X| \setminus |U|\) and let \((T, E, m)\) be a triple as in Definition 08HI. If

1. approximation holds for \((j^{-1}T, E|_V, m)\), and

2. the sheaves \(H^i(E)\) for \(i \geq m\) are supported on \(T\),

then approximation holds for \((T, E, m)\).

Proof

Let \(P \to E|_V\) be an approximation of the triple \((j^{-1}T, E|_V, m)\) over \(V\). Then \(Rj_*P\) is a perfect object of \(D(\mathcal{O}_X)\) by Lemma 08HK. On the other hand, \(Rj_*P = j_!P\) by Lemma 08GG. We see that \(j_!P\) is supported on \(T\) for example by (08GU). Hence we obtain an approximation \(Rj_*P = j_!P \to j_!(E|_V) \to E\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\) which is representable by an affine scheme. Then approximation holds for every triple \((T, E, m)\) as in Definition 08HI such that there exists an integer \(r \geq 0\) with

1. \(E\) is \(m\)-pseudo-coherent,

2. \(H^i(E)\) is supported on \(T\) for \(i \geq m - r + 1\),

3. \(X \setminus T\) is the union of \(r\) affine opens.

In particular, approximation by perfect complexes holds for affine schemes.

Proof

Let \(X_0\) be an affine scheme representing \(X\). Let \(T_0 \subset X_0\) by the closed subset corresponding to \(T\). Let \(\epsilon : X_\etale \to X_{0, Zar}\) be the morphism (08H7). We may write \(E = \epsilon^*E_0\) for some object \(E_0\) of \(D_\QCoh(\mathcal{O}_{X_0})\), see Lemma 071Q. Then \(E_0\) is \(m\)-pseudo-coherent, see Lemma 08HE. Comparing stalks of cohomology sheaves (see proof of Lemma 08H8) we see that \(H^i(E_0)\) is supported on \(T_0\) for \(i \geq m - r + 1\). By Derived Categories of Schemes, Lemma 08EQ there exists an approximation \(P_0 \to E_0\) of \((T_0, E_0, m)\). By Lemma 08HG we see that \(P = \epsilon^*P_0\) is a perfect object of \(D(\mathcal{O}_X)\). Pulling back we obtain an approximation \(P = \epsilon^*P_0 \to \epsilon^*E_0 = E\) as desired.

Lemma

Let \(S\) be a scheme. Let \((U \subset X, j : V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\). Assume \(U\) quasi-compact, \(V\) affine, and \(U \times_X V\) quasi-compact. If approximation by perfect complexes holds on \(U\), then approximation by perfect complexes holds on \(X\).

Proof

Let \(T \subset |X|\) be a closed subset with \(X \setminus T \to X\) quasi-compact. Let \(r_U\) be the integer of Definition 08HJ adapted to the pair \((U, T \cap |U|)\). Set \(T' = T \setminus |U|\). Endow \(T'\) with the induced reduced subspace structure. Since \(|T'|\) is contained in \(|X| \setminus |U|\) we see that \(j^{-1}(T') \to T'\) is an isomorphism. Moreover, \(V \setminus j^{-1}(T')\) is quasi-compact as it is the fibre product of \(U \times_X V\) with \(X \setminus T\) over \(X\) and we’ve assumed \(U \times_X V\) quasi-compact and \(X \setminus T \to X\) quasi-compact. Let \(r'\) be the number of affines needed to cover \(V \setminus j^{-1}(T')\). We claim that \(r = \max(r_U, r')\) works for the pair \((X, T)\).

To see this choose a triple \((T, E, m)\) such that \(E\) is \((m - r)\)-pseudo-coherent and \(H^i(E)\) is supported on \(T\) for \(i \geq m - r\). Let \(t\) be the largest integer such that \(H^t(E)|_U\) is nonzero. (Such an integer exists as \(U\) is quasi-compact and \(E|_U\) is \((m - r)\)-pseudo-coherent.) We will prove that \(E\) can be approximated by induction on \(t\).

Base case: \(t \leq m - r'\). This means that \(H^i(E)\) is supported on \(T'\) for \(i \geq m - r'\). Hence Lemma 08HM guarantees the existence of an approximation \(P \to E|_V\) of \((T', E|_V, m)\) on \(V\). Applying Lemma 08HL we see that \((T', E, m)\) can be approximated. Such an approximation is also an approximation of \((T, E, m)\).

Induction step. Choose an approximation \(P \to E|_U\) of \((T \cap |U|, E|_U, m)\). This in particular gives a surjection \(H^t(P) \to H^t(E|_U)\). In the rest of the proof we will use the equivalence of Lemma 071Q (and the compatibilities of Remark 08GH) for the representable algebraic spaces \(V\) and \(U \times_X V\). We will also use the fact that \((m - r)\)-pseudo-coherence, resp. perfectness on the Zariski site and étale site agree, see Lemmas 08HE and 08HG. Thus we can use the results of Derived Categories of Schemes, Section 08EC for the open immersion \(U \times_X V \subset V\). In this way Derived Categories of Schemes, Lemma 08EK implies there exists a perfect object \(Q\) in \(D(\mathcal{O}_V)\) supported on \(j^{-1}(T)\) and an isomorphism \(Q|_{U \times_X V} \to (P \oplus P[1])|_{U \times_X V}\). By Derived Categories of Schemes, Lemma 08EH we can replace \(Q\) by \(Q \otimes^\mathbf{L} I\) and assume that the map \[Q|_{U \times_X V} \longrightarrow (P \oplus P[1])|_{U \times_X V} \longrightarrow P|_{U \times_X V} \longrightarrow E|_{U \times_X V}\] lifts to \(Q \to E|_V\). By Lemma 08HB we find an morphism \(a : R \to E\) of \(D(\mathcal{O}_X)\) such that \(a|_U\) is isomorphic to \(P \oplus P[1] \to E|_U\) and \(a|_V\) isomorphic to \(Q \to E|_V\). Thus \(R\) is perfect and supported on \(T\) and the map \(H^t(R) \to H^t(E)\) is surjective on restriction to \(U\). Choose a distinguished triangle \[R \to E \to E' \to R[1]\] Then \(E'\) is \((m - r)\)-pseudo-coherent (Cohomology on Sites, Lemma 08FV), \(H^i(E')|_U = 0\) for \(i \geq t\), and \(H^i(E')\) is supported on \(T\) for \(i \geq m - r\). By induction we find an approximation \(R' \to E'\) of \((T, E', m)\). Fit the composition \(R' \to E' \to R[1]\) into a distinguished triangle \(R \to R'' \to R' \to R[1]\) and extend the morphisms \(R' \to E'\) and \(R[1] \to R[1]\) into a morphism of distinguished triangles \[\xymatrix{ R \ar[r] \ar[d] & R'' \ar[d] \ar[r] & R' \ar[d] \ar[r] & R[1] \ar[d] \\ R \ar[r] & E \ar[r] & E' \ar[r] & R[1] }\] using TR3. Then \(R''\) is a perfect complex (Cohomology on Sites, Lemma 08G9) supported on \(T\). An easy diagram chase shows that \(R'' \to E\) is the desired approximation.

Theorem

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Then approximation by perfect complexes holds on \(X\).

Proof

This follows from the induction principle of Lemma 08GP and Lemmas 08HN and 08HM.

Generating derived categories

This section is the analogue of Derived Categories of Schemes, Section 09IP. However, we first prove the following lemma which is the analogue of Derived Categories of Schemes, Lemma 09IM.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(W \subset X\) be a quasi-compact open. Let \(T \subset |X|\) be a closed subset such that \(X \setminus T \to X\) is a quasi-compact morphism. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Let \(\alpha : P \to E|_W\) be a map where \(P\) is a perfect object of \(D(\mathcal{O}_W)\) supported on \(T \cap W\). Then there exists a map \(\beta : R \to E\) where \(R\) is a perfect object of \(D(\mathcal{O}_X)\) supported on \(T\) such that \(P\) is a direct summand of \(R|_W\) in \(D(\mathcal{O}_W)\) compatible \(\alpha\) and \(\beta|_W\).

Proof

We will use the induction principle of Lemma 09IT to prove this. Thus we immediately reduce to the case where we have an elementary distinguished square \((W \subset X, f : V \to X)\) with \(V\) affine and \(P \to E|_W\) as in the statement of the lemma. In the rest of the proof we will use Lemma 071Q (and the compatibilities of Remark 08GH) for the representable algebraic spaces \(V\) and \(W \times_X V\). We will also use the fact that perfectness on the Zariski site and étale site agree, see Lemma 08HG.

By Derived Categories of Schemes, Lemma 08EK we can choose a perfect object \(Q\) in \(D(\mathcal{O}_V)\) supported on \(f^{-1}T\) and an isomorphism \(Q|_{W \times_X V} \to (P \oplus P[1])|_{W \times_X V}\). By Derived Categories of Schemes, Lemma 08EH we can replace \(Q\) by \(Q \otimes^\mathbf{L} I\) (still supported on \(f^{-1}T\)) and assume that the map \[Q|_{W \times_X V} \to (P \oplus P[1])|_{W \times_X V} \longrightarrow P|_{W \times_X V} \longrightarrow E|_{W \times_X V}\] lifts to \(Q \to E|_V\). By Lemma 08HB we find an morphism \(a : R \to E\) of \(D(\mathcal{O}_X)\) such that \(a|_W\) is isomorphic to \(P \oplus P[1] \to E|_W\) and \(a|_V\) isomorphic to \(Q \to E|_V\). Thus \(R\) is perfect and supported on \(T\) as desired.

Remark

The proof of Lemma 09IV shows that \[R|_W = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m]\] for some \(m \geq 0\) and \(n_j \geq 0\). Thus the highest degree cohomology sheaf of \(R|_W\) equals that of \(P\). By repeating the construction for the map \(P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] \to R|_W\), taking cones, and using induction we can achieve equality of cohomology sheaves of \(R|_W\) and \(P\) above any given degree.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(W\) be a quasi-compact open subspace of \(X\). Let \(P\) be a perfect object of \(D(\mathcal{O}_W)\). Then \(P\) is a direct summand of the restriction of a perfect object of \(D(\mathcal{O}_X)\).

Proof

Special case of Lemma 09IV.

Theorem

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). The category \(D_\QCoh(\mathcal{O}_X)\) can be generated by a single perfect object. More precisely, there exists a perfect object \(P\) of \(D(\mathcal{O}_X)\) such that for \(E \in D_\QCoh(\mathcal{O}_X)\) the following are equivalent

1. \(E = 0\), and

2. \(\Hom_{D(\mathcal{O}_X)}(P[n], E) = 0\) for all \(n \in \mathbf{Z}\).

Proof

We will prove this using the induction principle of Lemma 08GP.

If \(X\) is affine, then \(\mathcal{O}_X\) is a perfect generator. This follows from Lemma 071Q and Derived Categories of Schemes, Lemma 06Z0.

Assume that \((U \subset X, f : V \to X)\) is an elementary distinguished square with \(U\) quasi-compact such that the theorem holds for \(U\) and \(V\) is an affine scheme. Let \(P\) be a perfect object of \(D(\mathcal{O}_U)\) which is a generator for \(D_\QCoh(\mathcal{O}_U)\). Using Lemma 09IX we may choose a perfect object \(Q\) of \(D(\mathcal{O}_X)\) whose restriction to \(U\) is a direct sum one of whose summands is \(P\). Say \(V = \Spec(A)\). Let \(Z \subset V\) be the reduced closed subscheme which is the inverse image of \(X \setminus U\) and maps isomorphically to it (see Definition 08GM). This is a retrocompact closed subset of \(V\). Choose \(f_1, \ldots, f_r \in A\) such that \(Z = V(f_1, \ldots, f_r)\). Let \(K \in D(\mathcal{O}_V)\) be the perfect object corresponding to the Koszul complex on \(f_1, \ldots, f_r\) over \(A\). Note that since \(K\) is supported on \(Z\), the pushforward \(K' = Rf_*K\) is a perfect object of \(D(\mathcal{O}_X)\) whose restriction to \(V\) is \(K\) (see Lemmas 08HK and 08GG). We claim that \(Q \oplus K'\) is a generator for \(D_\QCoh(\mathcal{O}_X)\).

Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\) such that there are no nontrivial maps from any shift of \(Q \oplus K'\) into \(E\). By Lemma 08GG we have \(K' = f_! K\) and hence \[\Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V)\] Thus by Derived Categories of Schemes, Lemma 09IR (using also Lemma 071Q) the vanishing of these groups implies that \(E|_V\) is isomorphic to \(R(U \times_X V \to V)_*E|_{U \times_X V}\). This implies that \(E = R(U \to X)_*E|_U\) (small detail omitted). If this is the case then \[\Hom_{D(\mathcal{O}_X)}(Q[n], E) = \Hom_{D(\mathcal{O}_U)}(Q|_U[n], E|_U)\] which contains \(\Hom_{D(\mathcal{O}_U)}(P[n], E|_U)\) as a direct summand. Thus by our choice of \(P\) the vanishing of these groups implies that \(E|_U\) is zero. Whence \(E\) is zero.

Remark

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated algebraic spaces over \(S\). Let \(E \in D_\QCoh(\mathcal{O}_Y)\) be a generator (see Theorem 09IY). Then the following are equivalent

1. for \(K \in D_\QCoh(\mathcal{O}_X)\) we have \(Rf_*K = 0\) if and only if \(K = 0\),

2. \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) reflects isomorphisms, and

3. \(Lf^*E\) is a generator for \(D_\QCoh(\mathcal{O}_X)\).

The equivalence between (1) and (2) is a formal consequence of the fact that \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that \(Lf^*\) is the left adjoint to \(Rf_*\). These conditions hold if \(f\) is affine (Lemma 08II) or if \(f\) is an open immersion, or if \(f\) is a composition of such.

The following result is an strengthening of Theorem 09IY proved using exactly the same methods. Let \(T \subset |X|\) be a closed subset where \(X\) is an algebraic space. Let’s denote \(D_T(\mathcal{O}_X)\) the strictly full, saturated, triangulated subcategory consisting of complexes whose cohomology sheaves are supported on \(T\).

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset such that \(|X| \setminus T\) is quasi-compact. With notation as above, the category \(D_{\QCoh, T}(\mathcal{O}_X)\) is generated by a single perfect object.

Proof

We will prove this using the induction principle of Lemma 08GP. The property is true for representable quasi-compact and quasi-separated objects of the site \(X_{spaces, \etale}\) by Derived Categories of Schemes, Lemma 0A9A.

Assume that \((U \subset X, f : V \to X)\) is an elementary distinguished square such that the lemma holds for \(U\) and \(V\) is affine. To finish the proof we have to show that the result holds for \(X\). Let \(P\) be a perfect object of \(D(\mathcal{O}_U)\) supported on \(T \cap U\) which is a generator for \(D_{\QCoh, T \cap U}(\mathcal{O}_U)\). Using Lemma 09IV we may choose a perfect object \(Q\) of \(D(\mathcal{O}_X)\) supported on \(T\) whose restriction to \(U\) is a direct sum one of whose summands is \(P\). Write \(V = \Spec(B)\). Let \(Z = X \setminus U\). Then \(f^{-1}Z\) is a closed subset of \(V\) such that \(V \setminus f^{-1}Z\) is quasi-compact. As \(X\) is quasi-separated, it follows that \(f^{-1}Z \cap f^{-1}T = f^{-1}(Z \cap T)\) is a closed subset of \(V\) such that \(W = V \setminus f^{-1}(Z \cap T)\) is quasi-compact. Thus we can choose \(g_1, \ldots, g_s \in B\) such that \(f^{-1}(Z \cap T) = V(g_1, \ldots, g_s)\). Let \(K \in D(\mathcal{O}_V)\) be the perfect object corresponding to the Koszul complex on \(g_1, \ldots, g_s\) over \(B\). Note that since \(K\) is supported on \(f^{-1}(Z \cap T) \subset V\) closed, the pushforward \(K' = R(V \to X)_*K\) is a perfect object of \(D(\mathcal{O}_X)\) whose restriction to \(V\) is \(K\) (see Lemmas 08HK and 08GG). We claim that \(Q \oplus K'\) is a generator for \(D_{\QCoh, T}(\mathcal{O}_X)\).

Let \(E\) be an object of \(D_{\QCoh, T}(\mathcal{O}_X)\) such that there are no nontrivial maps from any shift of \(Q \oplus K'\) into \(E\). By Lemma 08GG we have \(K' = R(V \to X)_! K\) and hence \[\Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V)\] Thus by Derived Categories of Schemes, Lemma 09IR we have \(E|_V = Rj_*E|_W\) where \(j : W \to V\) is the inclusion. Picture \[\xymatrix{ W \ar[r]_j & V & Z \cap T \ar[l] \ar[d] \\ V \setminus f^{-1}Z \ar[u]^{j'} \ar[ru]_{j''} & & Z \ar[lu] }\] Since \(E\) is supported on \(T\) we see that \(E|_W\) is supported on \(f^{-1}T \cap W = f^{-1}T \cap (V \setminus f^{-1}Z)\) which is closed in \(W\). We conclude that \[E|_V = Rj_*(E|_W) = Rj_*(Rj'_*(E|_{U \cap V})) = Rj''_*(E|_{U \cap V})\] Here the second equality is part (1) of Cohomology, Lemma 08DF which applies because \(V\) is a scheme and \(E\) has quasi-coherent cohomology sheaves hence pushforward along the quasi-compact open immersion \(j'\) agrees with pushforward on the underlying schemes, see Remark 08GH. This implies that \(E = R(U \to X)_*E|_U\) (small detail omitted). If this is the case then \[\Hom_{D(\mathcal{O}_X)}(Q[n], E) = \Hom_{D(\mathcal{O}_U)}(Q|_U[n], E|_U)\] which contains \(\Hom_{D(\mathcal{O}_U)}(P[n], E|_U)\) as a direct summand. Thus by our choice of \(P\) the vanishing of these groups implies that \(E|_U\) is zero. Whence \(E\) is zero.

Compact and perfect objects

This section is the analogue of Derived Categories of Schemes, Section 09M0.

Proposition

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). An object of \(D_\QCoh(\mathcal{O}_X)\) is compact if and only if it is perfect.

Proof

If \(K\) is a perfect object of \(D(\mathcal{O}_X)\) with dual \(K^\vee\) (Cohomology on Sites, Lemma 08JJ) we have \[\Hom_{D(\mathcal{O}_X)}(K, M) = H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} M)\] functorially in \(M\). Since \(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} -\) commutes with direct sums and since \(H^0(X, -)\) commutes with direct sums on \(D_\QCoh(\mathcal{O}_X)\) by Lemma 08FB we conclude that \(K\) is compact in \(D_\QCoh(\mathcal{O}_X)\).

Conversely, let \(K\) be a compact object of \(D_\QCoh(\mathcal{O}_X)\). To show that \(K\) is perfect, it suffices to show that \(K|_U\) is perfect for every affine scheme \(U\) étale over \(X\), see Cohomology on Sites, Lemma 08G6. Observe that \(j : U \to X\) is a quasi-compact and separated morphism. Hence \(Rj_* : D_\QCoh(\mathcal{O}_U) \to D_\QCoh(\mathcal{O}_X)\) commutes with direct sums, see Lemma 08FB. Thus the adjointness of restriction to \(U\) and \(Rj_*\) implies that \(K|_U\) is a perfect object of \(D_\QCoh(\mathcal{O}_U)\). Hence we reduce to the case that \(X\) is affine, in particular a quasi-compact and quasi-separated scheme. Via Lemma 071Q and 08HG we reduce to the case of schemes, i.e., to Derived Categories of Schemes, Proposition 09M1.

Remark

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(G\) be a perfect object of \(D(\mathcal{O}_X)\) which is a generator for \(D_\QCoh(\mathcal{O}_X)\). By Theorem 09IY there is at least one of these. Combining Lemma 08F3 with Proposition 09M8 and with Derived Categories, Proposition 09SR we see that \(G\) is a classical generator for \(D_{perf}(\mathcal{O}_X)\).

The following result is a strengthening of Proposition 09M8. Let \(T \subset |X|\) be a closed subset where \(X\) is an algebraic space. As before \(D_T(\mathcal{O}_X)\) denotes the strictly full, saturated, triangulated subcategory consisting of complexes whose cohomology sheaves are supported on \(T\). Since taking direct sums commutes with taking cohomology sheaves, it follows that \(D_T(\mathcal{O}_X)\) has direct sums and that they are equal to direct sums in \(D(\mathcal{O}_X)\).

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset such that \(|X| \setminus T\) is quasi-compact. An object of \(D_{\QCoh, T}(\mathcal{O}_X)\) is compact if and only if it is perfect as an object of \(D(\mathcal{O}_X)\).

Proof

We observe that \(D_{\QCoh, T}(\mathcal{O}_X)\) is a triangulated category with direct sums by the remark preceding the lemma. By Proposition 09M8 the perfect objects define compact objects of \(D(\mathcal{O}_X)\) hence a fortiori of any subcategory preserved under taking direct sums. For the converse we will use there exists a generator \(E \in D_{\QCoh, T}(\mathcal{O}_X)\) which is a perfect complex of \(\mathcal{O}_X\)-modules, see Lemma 0AEC. Hence by the above, \(E\) is compact. Then it follows from Derived Categories, Proposition 09SR that \(E\) is a classical generator of the full subcategory of compact objects of \(D_{\QCoh, T}(\mathcal{O}_X)\). Thus any compact object can be constructed out of \(E\) by a finite sequence of operations consisting of (a) taking shifts, (b) taking finite direct sums, (c) taking cones, and (d) taking direct summands. Each of these operations preserves the property of being perfect and the result follows.

Remark

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset such that \(|X| \setminus T\) is quasi-compact. Let \(G\) be a perfect object of \(D_{\QCoh, T}(\mathcal{O}_X)\) which is a generator for \(D_{\QCoh, T}(\mathcal{O}_X)\). By Lemma 0AEC there is at least one of these. Combining the fact that \(D_{\QCoh, T}(\mathcal{O}_X)\) has direct sums with Lemma 0AED and with Derived Categories, Proposition 09SR we see that \(G\) is a classical generator for \(D_{perf, T}(\mathcal{O}_X)\).

The following lemma is an application of the ideas that go into the proof of the preceding lemma.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset such that the complement \(U \subset X\) is quasi-compact. Let \(\alpha : P \to E\) be a morphism of \(D_\QCoh(\mathcal{O}_X)\) with either

1. \(P\) is perfect and \(E\) supported on \(T\), or

2. \(P\) pseudo-coherent, \(E\) supported on \(T\), and \(E\) bounded below.

Then there exists a perfect complex of \(\mathcal{O}_X\)-modules \(I\) and a map \(I \to \mathcal{O}_X[0]\) such that \(I \otimes^\mathbf{L} P \to E\) is zero and such that \(I|_U \to \mathcal{O}_U[0]\) is an isomorphism.

Proof

Set \(\mathcal{D} = D_{\QCoh, T}(\mathcal{O}_X)\). In both cases the complex \(K = R\SheafHom(P, E)\) is an object of \(\mathcal{D}\). See Lemma 0A8A for quasi-coherence. It is clear that \(K\) is supported on \(T\) as formation of \(R\SheafHom\) commutes with restriction to opens. The map \(\alpha\) defines an element of \(H^0(K) = \Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X[0], K)\). Then it suffices to prove the result for the map \(\alpha : \mathcal{O}_X[0] \to K\).

Let \(E \in \mathcal{D}\) be a perfect generator, see Lemma 0AEC. Write \[K = \text{hocolim} K_n\] as in Derived Categories, Lemma 09SN using the generator \(E\). Since the functor \(\mathcal{D} \to D(\mathcal{O}_X)\) commutes with direct sums, we see that \(K = \text{hocolim} K_n\) holds in \(D(\mathcal{O}_X)\). Since \(\mathcal{O}_X\) is a compact object of \(D(\mathcal{O}_X)\) we find an \(n\) and a morphism \(\alpha_n : \mathcal{O}_X \to K_n\) which gives rise to \(\alpha\), see Derived Categories, Lemma 094A. By Derived Categories, Lemma 09SP applied to the morphism \(\mathcal{O}_X[0] \to K_n\) in the ambient category \(D(\mathcal{O}_X)\) we see that \(\alpha_n\) factors as \(\mathcal{O}_X[0] \to Q \to K_n\) where \(Q\) is an object of \(\langle E \rangle\). We conclude that \(Q\) is a perfect complex supported on \(T\).

Choose a distinguished triangle \[I \to \mathcal{O}_X[0] \to Q \to I[1]\] By construction \(I\) is perfect, the map \(I \to \mathcal{O}_X[0]\) restricts to an isomorphism over \(U\), and the composition \(I \to K\) is zero as \(\alpha\) factors through \(Q\). This proves the lemma.

Derived categories as module categories

The section is the analogue of Derived Categories of Schemes, Section 09M2.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(K^\bullet\) be a complex of \(\mathcal{O}_X\)-modules whose cohomology sheaves are quasi-coherent. Let \((E, d) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)\) be the endomorphism differential graded algebra. Then the functor \[- \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_X)\] of Differential Graded Algebra, Lemma 09LX has image contained in \(D_\QCoh(\mathcal{O}_X)\).

Proof

Let \(P\) be a differential graded \(E\)-module with property \(P\). Let \(F_\bullet\) be a filtration on \(P\) as in Differential Graded Algebra, Section 09KK. Then we have \[P \otimes_E K^\bullet = \text{hocolim}\ F_iP \otimes_E K^\bullet\] Each of the \(F_iP\) has a finite filtration whose graded pieces are direct sums of \(E[k]\). The result follows easily.

The following lemma can be strengthened (there is a uniformity in the vanishing over all \(L\) with nonzero cohomology sheaves only in a fixed range).

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(K\) be a perfect object of \(D(\mathcal{O}_X)\). Then

1. there exist integers \(a \leq b\) such that \(\Hom_{D(\mathcal{O}_X)}(K, L) = 0\) for \(L \in D_\QCoh(\mathcal{O}_X)\) with \(H^i(L) = 0\) for \(i \in [a, b]\), and

2. if \(L\) is bounded, then \(\Ext^n_{D(\mathcal{O}_X)}(K, L)\) is zero for all but finitely many \(n\).

Proof

Part (2) follows from (1) as \(\Ext^n_{D(\mathcal{O}_X)}(K, L) = \Hom_{D(\mathcal{O}_X)}(K, L[n])\). We prove (1). Since \(K\) is perfect we have \[\Ext^i_{D(\mathcal{O}_X)}(K, L) = H^i(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] where \(K^\vee\) is the “dual” perfect complex to \(K\), see Cohomology on Sites, Lemma 08JJ. Note that \(P = K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L\) is in \(D_\QCoh(X)\) by Lemmas 08F5 and 08JL (to see that a perfect complex has quasi-coherent cohomology sheaves). Say \(K^\vee\) has tor amplitude in \([a, b]\). Then the spectral sequence \[E_1^{p, q} = H^p(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} H^q(L)) \Rightarrow H^{p + q}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] shows that \(H^j(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\) is zero if \(H^q(L) = 0\) for \(q \in [j - b, j - a]\). Let \(N\) be the integer \(\max(d_p + p)\) of Cohomology of Spaces, Lemma 072C. Then \(H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\) vanishes if the cohomology sheaves \[H^{-N}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L), \ H^{-N + 1}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L), \ \ldots, \ H^0(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] are zero. Namely, by the lemma cited and Lemma 08F7, we have \[H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L) = H^0(X, \tau_{\geq -N}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L))\] and by the vanishing of cohomology sheaves, this is equal to \(H^0(X, \tau_{\geq 1}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L))\) which is zero by Derived Categories, Lemma 05TC. It follows that \(\Hom_{D(\mathcal{O}_X)}(K, L)\) is zero if \(H^i(L) = 0\) for \(i \in [-b - N, -a]\).

The following is the analogue of Derived Categories of Schemes, Theorem 09M5.

Theorem

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Then there exist a differential graded algebra \((E, \text{d})\) with only a finite number of nonzero cohomology groups \(H^i(E)\) such that \(D_\QCoh(\mathcal{O}_X)\) is equivalent to \(D(E, \text{d})\).

Proof

Let \(K^\bullet\) be a K-injective complex of \(\mathcal{O}\)-modules which is perfect and generates \(D_\QCoh(\mathcal{O}_X)\). Such a thing exists by Theorem 09IY and the existence of K-injective resolutions. We will show the theorem holds with \[(E, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)\] where \(\text{Comp}^{dg}(\mathcal{O}_X)\) is the differential graded category of complexes of \(\mathcal{O}\)-modules. Please see Differential Graded Algebra, Section 09LU. Since \(K^\bullet\) is K-injective we have [09MD]\[\begin{equation} H^n(E) = \Ext^n_{D(\mathcal{O}_X)}(K^\bullet, K^\bullet) \end{equation}\] for all \(n \in \mathbf{Z}\). Only a finite number of these Exts are nonzero by Lemma 09MB. Consider the functor \[- \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_X)\] of Differential Graded Algebra, Lemma 09LX. Since \(K^\bullet\) is perfect, it defines a compact object of \(D(\mathcal{O}_X)\), see Proposition 09M8. Combined with (09MD) the functor above is fully faithful as follows from Differential Graded Algebra, Lemmas 09LZ. It has a right adjoint \[R\Hom(K^\bullet, - ) : D(\mathcal{O}_X) \longrightarrow D(E, \text{d})\] by Differential Graded Algebra, Lemmas 09LY which is a left quasi-inverse functor by generalities on adjoint functors. On the other hand, it follows from Lemma 09MA that we obtain \[- \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D_\QCoh(\mathcal{O}_X)\] and by our choice of \(K^\bullet\) as a generator of \(D_\QCoh(\mathcal{O}_X)\) the kernel of the adjoint restricted to \(D_\QCoh(\mathcal{O}_X)\) is zero. A formal argument shows that we obtain the desired equivalence, see Derived Categories, Lemma 09J1.

Remark

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space. Let \(T \subset |X|\) be a closed subset such that \(|X| \setminus T\) is quasi-compact. The analogue of Theorem 09MC holds for \(D_{\QCoh, T}(\mathcal{O}_X)\). This follows from the exact same argument as in the proof of the theorem, using Lemmas 0AEC and 0AED and a variant of Lemma 09MA with supports. If we ever need this, we will precisely state the result here and give a detailed proof.

Remark

Let \(X\) be a quasi-compact and quasi-separated algebraic space over a ring \(R\). By the construction of the proof of Theorem 09MC there exists a differential graded algebra \((A, \text{d})\) over \(R\) such that \(D_\QCoh(X)\) is \(R\)-linearly equivalent to \(D(A, \text{d})\) as a triangulated category. One may ask: how unique is \((A, \text{d})\)? The answer is (only) slightly better than just saying that \((A, \text{d})\) is well defined up to derived equivalence. Namely, suppose that \((B, \text{d})\) is a second such pair. Then we have \[(A, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)\] and \[(B, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, L^\bullet)\] for some K-injective complexes \(K^\bullet\) and \(L^\bullet\) of \(\mathcal{O}_X\)-modules corresponding to perfect generators of \(D_\QCoh(\mathcal{O}_X)\). Set \[\Omega = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, L^\bullet) \quad \Omega' = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, K^\bullet)\] Then \(\Omega\) is a differential graded \(B^{opp} \otimes_R A\)-module and \(\Omega'\) is a differential graded \(A^{opp} \otimes_R B\)-module. Moreover, the equivalence \[D(A, \text{d}) \to D_\QCoh(\mathcal{O}_X) \to D(B, \text{d})\] is given by the functor \(- \otimes_A^\mathbf{L} \Omega'\) and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark 09ST. If we ever need this remark we will provide a precise statement with a detailed proof here.

Characterizing pseudo-coherent complexes, I

This material will be continued in More on Morphisms of Spaces, Section 0CTP. We can characterize pseudo-coherent objects as derived homotopy limits of approximations by perfect objects.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(K \in D(\mathcal{O}_X)\). The following are equivalent

1. \(K\) is pseudo-coherent, and

2. \(K = \text{hocolim} K_n\) where \(K_n\) is perfect and \(\tau_{\geq -n}K_n \to \tau_{\geq -n}K\) is an isomorphism for all \(n\).

Proof

The implication (2) \(\Rightarrow\) (1) is true on any ringed site. Namely, assume (2) holds. Recall that a perfect object of the derived category is pseudo-coherent, see Cohomology on Sites, Lemma 08G8. Then it follows from the definitions that \(\tau_{\geq -n}K_n\) is \((-n + 1)\)-pseudo-coherent and hence \(\tau_{\geq -n}K\) is \((-n + 1)\)-pseudo-coherent, hence \(K\) is \((-n + 1)\)-pseudo-coherent. This is true for all \(n\), hence \(K\) is pseudo-coherent, see Cohomology on Sites, Definition 08FT.

Assume (1). We start by choosing an approximation \(K_1 \to K\) of \((X, K, -2)\) by a perfect complex \(K_1\), see Definitions 08HI and 08HJ and Theorem 08HP. Suppose by induction we have \[K_1 \to K_2 \to \ldots \to K_n \to K\] with \(K_i\) perfect such that such that \(\tau_{\geq -i}K_i \to \tau_{\geq -i}K\) is an isomorphism for all \(1 \leq i \leq n\). Then we pick \(a \leq b\) as in Lemma 09MB for the perfect object \(K_n\). Choose an approximation \(K_{n + 1} \to K\) of \((X, K, \min(a - 1, -n - 1))\). Choose a distinguished triangle \[K_{n + 1} \to K \to C \to K_{n + 1}[1]\] Then we see that \(C \in D_\QCoh(\mathcal{O}_X)\) has \(H^i(C) = 0\) for \(i \geq a\). Thus by our choice of \(a, b\) we see that \(\Hom_{D(\mathcal{O}_X)}(K_n, C) = 0\). Hence the composition \(K_n \to K \to C\) is zero. Hence by Derived Categories, Lemma 0149 we can factor \(K_n \to K\) through \(K_{n + 1}\) proving the induction step.

We still have to prove that \(K = \text{hocolim} K_n\). This follows by an application of Derived Categories, Lemma 0CRK to the functors \(H^i( - ) : D(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) and our choice of \(K_n\).

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is quasi-compact. Let \(K \in D(\mathcal{O}_X)\) supported on \(T\). The following are equivalent

1. \(K\) is pseudo-coherent, and

2. \(K = \text{hocolim} K_n\) where \(K_n\) is perfect, supported on \(T\), and \(\tau_{\geq -n}K_n \to \tau_{\geq -n}K\) is an isomorphism for all \(n\).

Proof

The proof of this lemma is exactly the same as the proof of Lemma 0DKB except that in the choice of the approximations we use the triples \((T, K, m)\).

The coherator revisited

In Section 08GX we constructed and studied the right adjoint \(RQ_X\) to the canonical functor \(D(\QCoh(\mathcal{O}_X)) \to D(\mathcal{O}_X)\). It was constructed as the right derived extension of the coherator \(Q_X : \textit{Mod}(\mathcal{O}_X) \to \QCoh(\mathcal{O}_X)\). In this section, we study when the inclusion functor \[D_\QCoh(\mathcal{O}_X) \longrightarrow D(\mathcal{O}_X)\] has a right adjoint. If this right adjoint exists, we will denote² it \[DQ_X : D(\mathcal{O}_X) \longrightarrow D_\QCoh(\mathcal{O}_X)\] It turns out that quasi-compact and quasi-separated algebraic spaces have such a right adjoint.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). The inclusion functor \(D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_X)\) has a right adjoint.

Proof

We will use the induction principle in Lemma 08GP to prove this. If \(D(\QCoh(\mathcal{O}_X)) \to D_\QCoh(\mathcal{O}_X)\) is an equivalence, then the lemma is true because the functor \(RQ_X\) of Section 08GX is a right adjoint to the functor \(D(\QCoh(\mathcal{O}_X)) \to D(\mathcal{O}_X)\). In particular, our lemma is true for affine algebraic spaces, see Lemma 08H0. Thus we see that it suffices to show: if \((U \subset X, f : V \to X)\) is an elementary distinguished square with \(U\) quasi-compact and \(V\) affine and the lemma holds for \(U\), \(V\), and \(U \times_X V\), then the lemma holds for \(X\).

The adjoint exists if and only if for every object \(K\) of \(D(\mathcal{O}_X)\) we can find a distinguished triangle \[E' \to E \to K \to E'[1]\] in \(D(\mathcal{O}_X)\) such that \(E'\) is in \(D_\QCoh(\mathcal{O}_X)\) and such that \(\Hom(M, K) = 0\) for all \(M\) in \(D_\QCoh(\mathcal{O}_X)\). See Derived Categories, Lemma 0CQS. Consider the distinguished triangle \[E \to Rj_{U, *}E|_U \oplus Rj_{V, *}E|_V \to Rj_{U \times_X V, *}E|_{U \times_X V} \to E[1]\] in \(D(\mathcal{O}_X)\) of Lemma 08GW. By Derived Categories, Lemma 0CQR it suffices to construct the desired distinguished triangles for \(Rj_{U, *}E|_U\), \(Rj_{V, *}E|_V\), and \(Rj_{U \times_X V, *}E|_{U \times_X V}\). This reduces us to the statement discussed in the next paragraph.

Let \(j : U \to X\) be an étale morphism corresponding with \(U\) quasi-compact and quasi-separated and the lemma is true for \(U\). Let \(L\) be an object of \(D(\mathcal{O}_U)\). Then there exists a distinguished triangle \[E' \to Rj_*L \to K \to E'[1]\] in \(D(\mathcal{O}_X)\) such that \(E'\) is in \(D_\QCoh(\mathcal{O}_X)\) and such that \(\Hom(M, K) = 0\) for all \(M\) in \(D_\QCoh(\mathcal{O}_X)\). To see this we choose a distinguished triangle \[L' \to L \to Q \to L'[1]\] in \(D(\mathcal{O}_U)\) such that \(L'\) is in \(D_\QCoh(\mathcal{O}_U)\) and such that \(\Hom(N, Q) = 0\) for all \(N\) in \(D_\QCoh(\mathcal{O}_U)\). This is possible because the statement in Derived Categories, Lemma 0CQS is an if and only if. We obtain a distinguished triangle \[Rj_*L' \to Rj_*L \to Rj_*Q \to Rj_*L'[1]\] in \(D(\mathcal{O}_X)\). Observe that \(Rj_*L'\) is in \(D_\QCoh(\mathcal{O}_X)\) by Lemma 08FA. On the other hand, if \(M\) in \(D_\QCoh(\mathcal{O}_X)\), then \[\Hom(M, Rj_*Q) = \Hom(Lj^*M, Q) = 0\] because \(Lj^*M\) is in \(D_\QCoh(\mathcal{O}_U)\) by Lemma 08F4. This finishes the proof.

Proof

The adjoint exists by Derived Categories, Proposition 0A8G. The hypotheses are satisfied: First, note that \(D_\QCoh(\mathcal{O}_X)\) has direct sums and direct sums commute with the inclusion functor (Lemma 08F3). On the other hand, \(D_\QCoh(\mathcal{O}_X)\) is compactly generated because it has a perfect generator Theorem 09IY and because perfect objects are compact by Proposition 09M8.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\). If the right adjoints \(DQ_X\) and \(DQ_Y\) of the inclusion functors \(D_\QCoh \to D\) exist for \(X\) and \(Y\), then \[Rf_* \circ DQ_X = DQ_Y \circ Rf_*\]

Proof

The statement makes sense because \(Rf_*\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\) by Lemma 08FA. The statement is true because \(Lf^*\) similarly maps \(D_\QCoh(\mathcal{O}_Y)\) into \(D_\QCoh(\mathcal{O}_X)\) (Lemma 08F4) and hence both \(Rf_* \circ DQ_X\) and \(DQ_Y \circ Rf_*\) are right adjoint to \(Lf^* : D_\QCoh(\mathcal{O}_Y) \to D(\mathcal{O}_X)\).

Remark

Let \(S\) be a scheme. Let \((U \subset X, f : V \to X)\) be an elementary distinguished square of algebraic spaces over \(S\). Assume \(X\), \(U\), \(V\) are quasi-compact and quasi-separated. By Lemma 0CR4 the functors \(DQ_X\), \(DQ_U\), \(DQ_V\), \(DQ_{U \times_X V}\) exist. Moreover, there is a canonical distinguished triangle \[DQ_X(K) \to Rj_{U, *}DQ_U(K|_U) \oplus Rj_{V, *}DQ_V(K|_V) \to Rj_{U \times_X V, *}DQ_{U \times_X V}(K|_{U \times_X V}) \to\] for any \(K \in D(\mathcal{O}_X)\). This follows by applying the exact functor \(DQ_X\) to the distinguished triangle of Lemma 08GW and using Lemma 0CR5 three times.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). The functor \(DQ_X\) of Lemma 0CR4 has the following boundedness property: there exists an integer \(N = N(X)\) such that, if \(K\) in \(D(\mathcal{O}_X)\) with \(H^i(U, K) = 0\) for \(U\) affine étale over \(X\) and \(i \not \in [a, b]\), then the cohomology sheaves \(H^i(DQ_X(K))\) are zero for \(i \not \in [a, b + N]\).

Proof

We will prove this using the induction principle of Lemma 08GP.

If \(X\) is affine, then the lemma is true with \(N = 0\) because then \(RQ_X = DQ_X\) is given by taking the complex of quasi-coherent sheaves associated to \(R\Gamma(X, K)\). See Lemma 08H0.

Let \((U \subset W, f : V \to W)\) be an elementary distinguished square with \(W\) quasi-compact and quasi-separated, \(U \subset W\) quasi-compact open, \(V\) affine such that the lemma holds for \(U\), \(V\), and \(U \times_W V\). Say with integers \(N(U)\), \(N(V)\), and \(N(U \times_W V)\). Now suppose \(K\) is in \(D(\mathcal{O}_X)\) with \(H^i(W, K) = 0\) for all affine \(W\) étale over \(X\) and all \(i \not \in [a, b]\). Then \(K|_U\), \(K|_V\), \(K|_{U \times_W V}\) have the same property. Hence we see that \(RQ_U(K|_U)\) and \(RQ_V(K|_V)\) and \(RQ_{U \cap V}(K|_{U \times_W V})\) have vanishing cohomology sheaves outside the interval \([a, b + \max(N(U), N(V), N(U \times_W V))\). Since the functors \(Rj_{U, *}\), \(Rj_{V, *}\), \(Rj_{U \times_W V, *}\) have finite cohomological dimension on \(D_\QCoh\) by Lemma 08FA we see that there exists an \(N\) such that \(Rj_{U, *}DQ_U(K|_U)\), \(Rj_{V, *}DQ_V(K|_V)\), and \(Rj_{U \cap V, *}DQ_{U \times_W V}(K|_{U \times_W V})\) have vanishing cohomology sheaves outside the interval \([a, b + N]\). Then finally we conclude by the distinguished triangle of Remark 0CR6.

Example

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \((\mathcal{F}_n)\) be an inverse system of quasi-coherent sheaves on \(X\). Since \(DQ_X\) is a right adjoint it commutes with products and therefore with derived limits. Hence we see that \[DQ_X(R\lim \mathcal{F}_n) = (R\lim\text{ in }D_\QCoh(\mathcal{O}_X))(\mathcal{F}_n)\] where the first \(R\lim\) is taken in \(D(\mathcal{O}_X)\). In fact, let’s write \(K = R\lim \mathcal{F}_n\) for this. For any affine \(U\) étale over \(X\) we have \[H^i(U, K) = H^i(R\Gamma(U, R\lim \mathcal{F}_n)) = H^i(R\lim R\Gamma(U, \mathcal{F}_n)) = H^i(R\lim \Gamma(U, \mathcal{F}_n))\] since cohomology commutes with derived limits and since the quasi-coherent sheaves \(\mathcal{F}_n\) have no higher cohomology on affines. By the computation of \(R\lim\) in the category of abelian groups, we see that \(H^i(U, K) = 0\) unless \(i \in [0, 1]\). Then finally we conclude that the \(R\lim\) in \(D_\QCoh(\mathcal{O}_X)\), which is \(DQ_X(K)\) by the above, is in \(D^b_\QCoh(\mathcal{O}_X)\) and has vanishing cohomology sheaves in negative degrees by Lemma 0CSS.

Cohomology and base change, IV

This section is the analogue of Derived Categories of Schemes, Section 08ET.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\). For \(E\) in \(D_\QCoh(\mathcal{O}_X)\) and \(K\) in \(D_\QCoh(\mathcal{O}_Y)\) we have \[Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K)\]

Proof

Without any assumptions there is a map \(Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K \to Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K)\). Namely, it is the adjoint to the canonical map \[Lf^*(Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K) = Lf^*(Rf_*(E)) \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K \longrightarrow E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K\] coming from the map \(Lf^*Rf_*E \to E\). See Cohomology on Sites, Lemmas 07A4 and 07A6. To check it is an isomorphism we may work étale locally on \(Y\). Hence we reduce to the case that \(Y\) is an affine scheme.

Suppose that \(K = \bigoplus K_i\) is a direct sum of some complexes \(K_i \in D_\QCoh(\mathcal{O}_Y)\). If the statement holds for each \(K_i\), then it holds for \(K\). Namely, the functors \(Lf^*\) and \(\otimes^\mathbf{L}\) preserve direct sums by construction and \(Rf_*\) commutes with direct sums (for complexes with quasi-coherent cohomology sheaves) by Lemma 08FB. Moreover, suppose that \(K \to L \to M \to K[1]\) is a distinguished triangle in \(D_\QCoh(Y)\). Then if the statement of the lemma holds for two of \(K, L, M\), then it holds for the third (as the functors involved are exact functors of triangulated categories).

Assume \(Y\) affine, say \(Y = \Spec(A)\). The functor \(\widetilde{\ } : D(A) \to D_\QCoh(\mathcal{O}_Y)\) is an equivalence by Lemma 071Q and Derived Categories of Schemes, Lemma 06Z0. Let \(T\) be the property for \(K \in D(A)\) that the statement of the lemma holds for \(\widetilde{K}\). The discussion above and More on Algebra, Remark 09PB shows that it suffices to prove \(T\) holds for \(A[k]\). This finishes the proof, as the statement of the lemma is clear for shifts of the structure sheaf.

Definition

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(X\), \(Y\) be algebraic spaces over \(B\). We say \(X\) and \(Y\) are Tor independent over \(B\) if and only if for every commutative diagram \[\xymatrix{ \Spec(k) \ar[d]_{\overline{y}} \ar[dr]_{\overline{b}} \ar[r]_-{\overline{x}} & X \ar[d] \\ Y \ar[r] & B }\] of geometric points the rings \(\mathcal{O}_{X, \overline{x}}\) and \(\mathcal{O}_{Y, \overline{y}}\) are Tor independent over \(\mathcal{O}_{B, \overline{b}}\) (see More on Algebra, Definition 0660).

The following lemma shows in particular that this definition agrees with our definition in the case of representable algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(X\), \(Y\) be algebraic spaces over \(B\). The following are equivalent

1. \(X\) and \(Y\) are Tor independent over \(B\),

2. for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & W \ar[d] & V \ar[d] \ar[l] \\ X \ar[r] & B & Y \ar[l] }\] with étale vertical arrows \(U\) and \(V\) are Tor independent over \(W\),

3. for some commutative diagram as in (2) with (a) \(W \to B\) étale surjective, (b) \(U \to X \times_B W\) étale surjective, (c) \(V \to Y \times_B W\) étale surjective, the spaces \(U\) and \(V\) are Tor independent over \(W\), and

4. for some commutative diagram as in (3) with \(U\), \(V\), \(W\) schemes, the schemes \(U\) and \(V\) are Tor independent over \(W\) in the sense of Derived Categories of Schemes, Definition 08IA.

Proof

For an étale morphism \(\varphi : U \to X\) of algebraic spaces and geometric point \(\overline{u}\) the map of local rings \(\mathcal{O}_{X, \varphi(\overline{u})} \to \mathcal{O}_{U, \overline{u}}\) is an isomorphism. Hence the equivalence of (1) and (2) follows. So does the implication (1) \(\Rightarrow\) (3). Assume (3) and pick a diagram of geometric points as in Definition 08IP. The assumptions imply that we can first lift \(\overline{b}\) to a geometric point \(\overline{w}\) of \(W\), then lift the geometric point \((\overline{x}, \overline{b})\) to a geometric point \(\overline{u}\) of \(U\), and finally lift the geometric point \((\overline{y}, \overline{b})\) to a geometric point \(\overline{v}\) of \(V\). Use Properties of Spaces, Lemma 05VN to find the lifts. Using the remark on local rings above we conclude that the condition of the definition is satisfied for the given diagram.

Having made these initial points, it is clear that (4) comes down to the statement that Definition 08IP agrees with Derived Categories of Schemes, Definition 08IA when \(X\), \(Y\), and \(B\) are schemes.

Let \(\overline{x}, \overline{b}, \overline{y}\) be as in Definition 08IP lying over the points \(x, y, b\). Recall that \(\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{X, x}^{sh}\) (Properties of Spaces, Lemma 04KF) and similarly for the other two. By Algebra, Lemma 08HV we see that \(\mathcal{O}_{X, \overline{x}}\) is a strict henselization of \(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{B, \overline{b}}\). In particular, the ring map \[\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{B, \overline{b}} \longrightarrow \mathcal{O}_{X, \overline{x}}\] is flat (More on Algebra, Lemma 07QM). By More on Algebra, Lemma 08HW we see that \[\text{Tor}_i^{\mathcal{O}_{B, b}}(\mathcal{O}_{X, x}, \mathcal{O}_{Y, y}) \otimes_{\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y}} (\mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{B, \overline{b}}} \mathcal{O}_{Y, \overline y}) = \text{Tor}_i^{\mathcal{O}_{B, \overline{b}}}( \mathcal{O}_{X, \overline{x}}, \mathcal{O}_{Y, \overline{y}})\] Hence it follows that if \(X\) and \(Y\) are Tor independent over \(B\) as schemes, then \(X\) and \(Y\) are Tor independent as algebraic spaces over \(B\).

For the converse, we may assume \(X\), \(Y\), and \(B\) are affine. Observe that the ring map \[\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{B, \overline{b}}} \mathcal{O}_{Y, \overline y}\] is flat by the observations given above. Moreover, the image of the map on spectra includes all primes \(\mathfrak s \subset \mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y}\) lying over \(\mathfrak m_x\) and \(\mathfrak m_y\). Hence from this and the displayed formula of Tor’s above we see that if \(X\) and \(Y\) are Tor independent over \(B\) as algebraic spaces, then \[\text{Tor}_i^{\mathcal{O}_{B, b}} (\mathcal{O}_{X, x}, \mathcal{O}_{Y, y})_\mathfrak s = 0\] for all \(i > 0\) and all \(\mathfrak s\) as above. By More on Algebra, Lemma 08HX applied to the ring maps \(\Gamma(B, \mathcal{O}_B) \to \Gamma(X, \mathcal{O}_X)\) and \(\Gamma(B, \mathcal{O}_B) \to \Gamma(X, \mathcal{O}_X)\) this implies that \(X\) and \(Y\) are Tor independent over \(B\).

Lemma

Let \(S\) be a scheme. Let \(g : Y' \to Y\) be a morphism of algebraic spaces over \(S\). Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\). Consider the base change diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] If \(X\) and \(Y'\) are Tor independent over \(Y\), then for all \(E \in D_\QCoh(\mathcal{O}_X)\) we have \(Rf'_*L(g')^*E = Lg^*Rf_*E\).

Proof

For any object \(E\) of \(D(\mathcal{O}_X)\) we can use Cohomology on Sites, Remark 07A7 to get a canonical base change map \(Lg^*Rf_*E \to Rf'_*L(g')^*E\). To check this is an isomorphism we may work étale locally on \(Y'\). Hence we may assume \(g : Y' \to Y\) is a morphism of affine schemes. In particular, \(g\) is affine and it suffices to show that \[Rg_*Lg^*Rf_*E \to Rg_*Rf'_*L(g')^*E = Rf_*(Rg'_* L(g')^* E)\] is an isomorphism, see Lemma 08II (and use Lemmas 08F4, 08F5, and 08FA to see that the objects \(Rf'_*L(g')^*E\) and \(Lg^*Rf_*E\) have quasi-coherent cohomology sheaves). Note that \(g'\) is affine as well (Morphisms of Spaces, Lemma 03WI). By Lemma 08IJ the map becomes a map \[Rf_*E \otimes_{\mathcal{O}_Y}^\mathbf{L} g_*\mathcal{O}_{Y'} \longrightarrow Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'})\] Observe that \(g'_*\mathcal{O}_{X'} = f^*g_*\mathcal{O}_{Y'}\). Thus by Lemma 08IN it suffices to prove that \(Lf^*g_*\mathcal{O}_{Y'} = f^*g_*\mathcal{O}_{Y'}\). This follows from our assumption that \(X\) and \(Y'\) are Tor independent over \(Y\). Namely, to check it we may work étale locally on \(X\), hence we may also assume \(X\) is affine. Say \(X = \Spec(A)\), \(Y = \Spec(R)\) and \(Y' = \Spec(R')\). Our assumption implies that \(A\) and \(R'\) are Tor independent over \(R\) (see Lemma 08IQ and More on Algebra, Lemma 08HX), i.e., \(\text{Tor}_i^R(A, R') = 0\) for \(i > 0\). In other words \(A \otimes_R^\mathbf{L} R' = A \otimes_R R'\) which exactly means that \(Lf^*g_*\mathcal{O}_{Y'} = f^*g_*\mathcal{O}_{Y'}\).

The following lemma will be used in the chapter on dualizing complexes.

Lemma

Let \(g : S' \to S\) be a morphism of affine schemes. Consider a cartesian square \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] of quasi-compact and quasi-separated algebraic spaces. Assume \(g\) and \(f\) Tor independent. Write \(S = \Spec(R)\) and \(S' = \Spec(R')\). For \(M, K \in D(\mathcal{O}_X)\) the canonical map \[R\Hom_X(M, K) \otimes^\mathbf{L}_R R' \longrightarrow R\Hom_{X'}(L(g')^*M, L(g')^*K)\] in \(D(R')\) is an isomorphism in the following two cases

1. \(M \in D(\mathcal{O}_X)\) is perfect and \(K \in D_\QCoh(X)\), or

2. \(M \in D(\mathcal{O}_X)\) is pseudo-coherent, \(K \in D_\QCoh^+(X)\), and \(R'\) has finite tor dimension over \(R\).

Proof

There is a canonical map \(R\Hom_X(M, K) \to R\Hom_{X'}(L(g')^*M, L(g')^*K)\) in \(D(\Gamma(X, \mathcal{O}_X))\) of global hom complexes, see Cohomology on Sites, Section 0B6E. Restricting scalars we can view this as a map in \(D(R)\). Then we can use the adjointness of restriction and \(- \otimes_R^\mathbf{L} R'\) to get the displayed map of the lemma. Having defined the map it suffices to prove it is an isomorphism in the derived category of abelian groups.

The right hand side is equal to \[R\Hom_X(M, R(g')_*L(g')^*K) = R\Hom_X(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'})\] by Lemma 08IJ. In both cases the complex \(R\SheafHom(M, K)\) is an object of \(D_\QCoh(\mathcal{O}_X)\) by Lemma 0A8A. There is a natural map \[R\SheafHom(M, K) \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'} \longrightarrow R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'})\] which is an isomorphism in both cases Lemma 0E4Q. To see that this lemma applies in case (2) we note that \(g'_*\mathcal{O}_{X'} = Rg'_*\mathcal{O}_{X'} = Lf^*g_*\mathcal{O}_X\) the second equality by Lemma 08IR. Using Derived Categories of Schemes, Lemma 08E9, Lemma 08HF, and Cohomology on Sites, Lemma 08H5 we conclude that \(g'_*\mathcal{O}_{X'}\) has finite Tor dimension. Hence, in both cases by replacing \(K\) by \(R\SheafHom(M, K)\) we reduce to proving \[R\Gamma(X, K) \otimes^\mathbf{L}_A A' \longrightarrow R\Gamma(X, K \otimes^\mathbf{L}_{\mathcal{O}_X} g'_*\mathcal{O}_{X'})\] is an isomorphism. Note that the left hand side is equal to \(R\Gamma(X', L(g')^*K)\) by Lemma 08IJ. Hence the result follows from Lemma 08IR.

Remark

With notation as in Lemma 0E4S. The diagram \[\xymatrix{ R\Hom_X(M, Rg'_*L) \otimes_R^\mathbf{L} R' \ar[r] \ar[d]_\mu & R\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \ar[d]^a \\ R\Hom_X(M, R(g')_*L) \ar@{=}[r] & R\Hom_{X'}(L(g')^*M, L) }\] is commutative where the top horizontal arrow is the map from the lemma, \(\mu\) is the multiplication map, and \(a\) comes from the adjunction map \(L(g')^*Rg'_*L \to L\). The multiplication map is the adjunction map \(K' \otimes_R^\mathbf{L} R' \to K'\) for any \(K' \in D(R')\).

Lemma

Let \(S\) be a scheme. Consider a cartesian square of algebraic spaces \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] over \(S\). Assume \(g\) and \(f\) Tor independent.

1. If \(E \in D(\mathcal{O}_X)\) has tor amplitude in \([a, b]\) as a complex of \(f^{-1}\mathcal{O}_Y\)-modules, then \(L(g')^*E\) has tor amplitude in \([a, b]\) as a complex of \(f^{-1}\mathcal{O}_{Y'}\)-modules.

2. If \(\mathcal{G}\) is an \(\mathcal{O}_X\)-module flat over \(Y\), then \(L(g')^*\mathcal{G} = (g')^*\mathcal{G}\).

Proof

We can compute tor dimension at stalks, see Cohomology on Sites, Lemma 0DJJ and Properties of Spaces, Theorem 04K5. If \(\overline{x}'\) is a geometric point of \(X'\) with image \(\overline{x}\) in \(X\), then \[(L(g')^*E)_{\overline{x}'} = E_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}}^\mathbf{L} \mathcal{O}_{X', \overline{x}'}\] Let \(\overline{y}'\) in \(Y'\) and \(\overline{y}\) in \(Y\) be the image of \(\overline{x}'\) and \(\overline{x}\). Since \(X\) and \(Y'\) are tor independent over \(Y\), we can apply More on Algebra, Lemma 0661 to see that the right hand side of the displayed formula is equal to \(E_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}}^\mathbf{L} \mathcal{O}_{Y', \overline{y}'}\) in \(D(\mathcal{O}_{Y', \overline{y}'})\). Thus (1) follows from More on Algebra, Lemma 066L. To see (2) observe that flatness of \(\mathcal{G}\) is equivalent to the condition that \(\mathcal{G}[0]\) has tor amplitude in \([0, 0]\). Applying (1) we conclude.

Cohomology and base change, V

This section is the analogue of Derived Categories of Schemes, Section 0DJ6. In Section 08IM we saw a base change theorem holds when the morphisms are tor independent. Even in the affine case there cannot be a base change theorem without such a condition, see More on Algebra, Section 065Y. In this section we analyze when one can get a base change result “one complex at a time”.

To make this work, let \(S\) be a base scheme and suppose we have a commutative diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] of algebraic spaces over \(S\) (usually we will assume it is cartesian). Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). For a geometric point \(\overline{x}'\) of \(X'\) consider the geometric points \(\overline{x} = g'(\overline{x}')\), \(\overline{y}' = f'(\overline{x}')\), \(\overline{y} = f(\overline{x}) = g(\overline{y}')\) of \(X\), \(Y'\), \(Y\). Then we can consider the maps \[K_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}}^\mathbf{L} \mathcal{O}_{Y', \overline{y}'} \to K_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}}^\mathbf{L} \mathcal{O}_{X', \overline{x}'} \to K'_{\overline{x}'}\] where the first arrow is More on Algebra, Equation (065Z) and the second comes from \((L(g')^*K)_{\overline{x}'} = K_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}}^\mathbf{L} \mathcal{O}_{X', \overline{x}'}\) and the given map \(L(g')^*K \to K'\). For each \(i \in \mathbf{Z}\) we obtain a \(\mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{Y', \overline{y}'}\)-module structure on \(H^i(K_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}}^\mathbf{L} \mathcal{O}_{Y', \overline{y}'})\). Putting everything together we obtain canonical maps [0DKF]\[\begin{equation} H^i(K_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}}^\mathbf{L} \mathcal{O}_{Y', \overline{y}'}) \otimes_{(\mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{Y', \overline{y}'})} \mathcal{O}_{X', \overline{x}'} \longrightarrow H^i(K'_{\overline{x}'}) \end{equation}\] of \(\mathcal{O}_{X', \overline{x}'}\)-modules.

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of algebraic spaces over \(S\). Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). The following are equivalent

1. for any \(x' \in X'\) and \(i \in \mathbf{Z}\) the map (0DKF) is an isomorphism,

2. for any commutative diagram \[\xymatrix{ & U \ar[d] \ar[rd]^a \\ V' \ar[r] \ar[rd]^c & V \ar[rd]^b & X \ar[d]^f \\ & Y' \ar[r]^g & Y }\] with \(a, b, c\) étale, \(U, V, V'\) schemes, and with \(U' = V' \times_V U\) the equivalent conditions of Derived Categories of Schemes, Lemma 0DJ8 hold for \((U \to X)^*K\) and \((U' \to X')^*K'\), and

3. there is some diagram as in (2) with \(U' \to X'\) surjective.

Proof

Observe that (1) is étale local on \(X'\). Working through formal implications of what is known, we see that it suffices to prove condition (1) of this lemma is equivalent to condition (1) of Derived Categories of Schemes, Lemma 0DJ8 if \(X, Y, Y', X'\) are representable by schemes \(X_0, Y_0, Y'_0, X'_0\). Denote \(f_0, g_0, g'_0, f'_0\) the morphisms between these schemes corresponding to \(f, g, g', f'\). We may assume \(K = \epsilon^*K_0\) and \(K' = \epsilon^*K'_0\) for some objects \(K_0 \in D_\QCoh(\mathcal{O}_{X_0})\) and \(K'_0 \in D_\QCoh(\mathcal{O}_{X'_0})\), see Lemma 071Q. Moreover, the map \(Lg^*K \to K'\) is the pullback of a map \(L(g_0)^*K_0 \to K'_0\) with notation as in Remark 08GH. Recall that \(\mathcal{O}_{X, \overline{x}}\) is the strict henselization of \(\mathcal{O}_{X, x}\) (Properties of Spaces, Lemma 04KF) and that we have \[K_{\overline{x}} = K_{0, x} \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X, \overline{x}} \quad\text{and}\quad K'_{\overline{x}'} = K'_{0, x'} \otimes_{\mathcal{O}_{X', x'}}^\mathbf{L} \mathcal{O}_{X', \overline{x}'}\] (akin to Properties of Spaces, Lemma 05VP). Consider the commutative diagram \[\xymatrix{ H^i(K_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}}^\mathbf{L} \mathcal{O}_{Y', \overline{y}'}) \otimes_{(\mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{Y', \overline{y}'})} \mathcal{O}_{X', \overline{x}'} \ar[r] & H^i(K'_{\overline{x}'}) \\ H^i(K_{0, x} \otimes_{\mathcal{O}_{Y, y}}^\mathbf{L} \mathcal{O}_{Y', y'}) \otimes_{(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{Y', y'})} \mathcal{O}_{X', x'} \ar[u] \ar[r] & H^i(K'_{0, x'}) \ar[u] }\] We have to show that the lower horizontal arrow is an isomorphism if and only if the upper horizontal arrow is an isomorphism. Since \(\mathcal{O}_{X', x'} \to \mathcal{O}_{X', \overline{x}'}\) is faithfully flat (More on Algebra, Lemma 07QM) it suffices to show that the top arrow is the base change of the bottom arrow by this map. This follows immediately from the relationships between stalks given above for the objects on the right. For the objects on the left it suffices to show that \[\begin{align*} & H^i\left( (K_{0, x} \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X, \overline{x}}) \otimes_{\mathcal{O}_{Y, \overline{y}}}^\mathbf{L} \mathcal{O}_{Y', \overline{y}'}\right) \\ & = H^i(K_{0, x} \otimes_{\mathcal{O}_{Y, y}}^\mathbf{L} \mathcal{O}_{Y', y'}) \otimes_{(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{Y', y'})} (\mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{Y', \overline{y}'}) \end{align*}\] This follows from More on Algebra, Lemma 0DJD. The flatness assumptions of this lemma hold by what was said above as well as Algebra, Lemma 08HV implying that \(\mathcal{O}_{X, \overline{x}}\) is the strict henselization of \(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{Y, \overline{y}}\) and that \(\mathcal{O}_{Y', \overline{y}'}\) is the strict henselization of \(\mathcal{O}_{Y', y'} \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{Y, \overline{y}}\).

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of algebraic spaces over \(S\). Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). If

1. the equivalent conditions of Lemma 0DKG hold, and

2. \(f\) is quasi-compact and quasi-separated,

then the composition \(Lg^*Rf_*K \to Rf'_*L(g')^*K \to Rf'_*K'\) is an isomorphism.

Proof

To check the map is an isomorphism we may work étale locally on \(Y'\). Hence we may assume \(g : Y' \to Y\) is a morphism of affine schemes. In this case, we will use the induction principle of Lemma 08GP to prove that for a quasi-compact and quasi-separated algebraic space \(U\) étale over \(X\) the similarly constructed map \(Lg^*R(U \to Y)_*K|_U \to R(U' \to Y')_*K'|_{U'}\) is an isomorphism. Here \(U' = X' \times_{g', X} U = Y' \times_{g, Y} U\).

If \(U\) is a scheme (for example affine), then the result holds. Namely, then \(Y, Y', U, U'\) are schemes, \(K\) and \(K'\) come from objects of the derived category of the underlying schemes by Lemma 071Q and the condition of Derived Categories of Schemes, Lemma 0DJ8 holds for these complexes by Lemma 0DKG. Thus (by the compatibilities explained in Remark 08GH) we can apply the result in the case of schemes which is Derived Categories of Schemes, Lemma 0DJ9.

The induction step. Let \((U \subset W, V \to W)\) be an elementary distinguished square with \(W\) a quasi-compact and quasi-separated algebraic space étale over \(X\), with \(U\) quasi-compact, \(V\) affine and the result holds for \(U\), \(V\), and \(U \times_W V\). To easy notation we replace \(W\) by \(X\) (this is permissible at this point). Denote \(a : U \to Y\), \(b : V \to Y\), and \(c : U \times_X V \to Y\) the obvious morphisms. Let \(a' : U' \to Y'\), \(b' : V' \to Y'\) and \(c' : U' \times_{X'} V' \to Y'\) be the base changes of \(a\), \(b\), and \(c\). Using the distinguished triangles from relative Mayer-Vietoris (Lemma 08JK) we obtain a commutative diagram \[\xymatrix{ Lg^*Rf_*K \ar[r] \ar[d] & Rf'_*K' \ar[d] \\ Lg^*Ra_*K|_U \oplus Lg^*Rb_*K|_V \ar[r] \ar[d] & Ra'_* K'|_{U'} \oplus Rb'_* K'|_{V'} \ar[d] \\ Lg^*Rc_*K|_{U \times_X V} \ar[r] \ar[d] & Rc'_*K'|_{U' \times_{X'} V'} \ar[d] \\ Lg^*Rf_* K[1] \ar[r] & Rf'_* K'[1] }\] Since the 2nd and 3rd horizontal arrows are isomorphisms so is the first (Derived Categories, Lemma 014A) and the proof of the lemma is finished.

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] be a cartesian diagram of algebraic spaces over \(S\). Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). If the equivalent conditions of Lemma 0DKG hold, then

1. for \(E \in D_\QCoh(\mathcal{O}_X)\) the equivalent conditions of Lemma 0DKG hold for \(L(g')^*(E \otimes^\mathbf{L} K) \to L(g')^*E \otimes^\mathbf{L} K'\),

2. if \(E\) in \(D(\mathcal{O}_X)\) is perfect the equivalent conditions of Lemma 0DKG hold for \(L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')\), and

3. if \(K\) is bounded below and \(E\) in \(D(\mathcal{O}_X)\) pseudo-coherent the equivalent conditions of Lemma 0DKG hold for \(L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')\).

Proof

The statement makes sense as the complexes involved have quasi-coherent cohomology sheaves by Lemmas 08F4, 08F5, and 0A8A and Cohomology on Sites, Lemmas 08H4 and 08H6. Having said this, we can check the maps (0DKF) are isomorphisms in case (1) by computing the source and target of (0DKF) using the transitive property of tensor product, see More on Algebra, Lemma 0BYJ. The map in (2) and (3) is the composition \[L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, L(g')^*K) \to R\SheafHom(L(g')^*E, K')\] where the first arrow is Cohomology on Sites, Remark 08JF and the second arrow comes from the given map \(L(g')^*K \to K'\). To prove the maps (0DKF) are isomorphisms one represents \(E_x\) by a bounded complex of finite projective \(\mathcal{O}_{X. x}\)-modules in case (2) or by a bounded above complex of finite free modules in case (3) and computes the source and target of the arrow. Some details omitted.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\). Let \(E \in D_\QCoh(\mathcal{O}_X)\). Let \(\mathcal{G}^\bullet\) be a bounded above complex of quasi-coherent \(\mathcal{O}_X\)-modules flat over \(Y\). Then formation of \[Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet)\] commutes with arbitrary base change (see proof for precise statement).

Proof

The statement means the following. Let \(g : Y' \to Y\) be a morphism of algebraic spaces and consider the base change diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] in other words \(X' = Y' \times_Y X\). The lemma asserts that \[Lg^*Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) \longrightarrow Rf'_*(L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{X'}} (g')^*\mathcal{G}^\bullet)\] is an isomorphism. Observe that on the right hand side we do not use derived pullback on \(\mathcal{G}^\bullet\). To prove this, we apply Lemmas 0DKH and 0DKI to see that it suffices to prove the canonical map \[L(g')^*\mathcal{G}^\bullet \to (g')^*\mathcal{G}^\bullet\] satisfies the equivalent conditions of Lemma 0DKG. This follows by checking the condition on stalks, where it immediately follows from the fact that \(\mathcal{G}^\bullet_{\overline{x}} \otimes_{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{Y', \overline{y}'}\) computes the derived tensor product by our assumptions on the complex \(\mathcal{G}^\bullet\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\). Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(\mathcal{G}^\bullet\) be a complex of quasi-coherent \(\mathcal{O}_X\)-modules. If

1. \(E\) is perfect, \(\mathcal{G}^\bullet\) is a bounded above, and \(\mathcal{G}^n\) is flat over \(Y\), or

2. \(E\) is pseudo-coherent, \(\mathcal{G}^\bullet\) is bounded, and \(\mathcal{G}^n\) is flat over \(Y\),

then formation of \[Rf_*R\SheafHom(E, \mathcal{G}^\bullet)\] commutes with arbitrary base change (see proof for precise statement).

Proof

The statement means the following. Let \(g : Y' \to Y\) be a morphism of algebraic spaces and consider the base change diagram \[\xymatrix{ X' \ar[r]_h \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] in other words \(X' = Y' \times_Y X\). The lemma asserts that \[Lg^*Rf_*R\SheafHom(E, \mathcal{G}^\bullet) \longrightarrow R(f')_*R\SheafHom(L(g')^*E, (g')^*\mathcal{G}^\bullet)\] is an isomorphism. Observe that on the right hand side we do not use the derived pullback on \(\mathcal{G}^\bullet\). To prove this, we apply Lemmas 0DKH and 0DKI to see that it suffices to prove the canonical map \[L(g')^*\mathcal{G}^\bullet \to (g')^*\mathcal{G}^\bullet\] satisfies the equivalent conditions of Lemma 0DKG. This was shown in the proof of Lemma 0A1K.

Producing perfect complexes

The following lemma is our main technical tool for producing perfect complexes. Later versions of this result will reduce to this by Noetherian approximation.

Lemma

Let \(S\) be a scheme. Let \(Y\) be a Noetherian algebraic space over \(S\). Let \(f : X \to Y\) be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let \(E \in D(\mathcal{O}_X)\) such that

1. \(E \in D^b_{\textit{Coh}}(\mathcal{O}_X)\),

2. the support of \(H^i(E)\) is proper over \(Y\) for all \(i\),

3. \(E\) has finite tor dimension as an object of \(D(f^{-1}\mathcal{O}_Y)\).

Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_Y)\).

Proof

By Lemma 08GK we see that \(Rf_*E\) is an object of \(D^b_{\textit{Coh}}(\mathcal{O}_Y)\). Hence \(Rf_*E\) is pseudo-coherent (Lemma 08IK). Hence it suffices to show that \(Rf_*E\) has finite tor dimension, see Cohomology on Sites, Lemma 08G8. By Lemma 08IL it suffices to check that \(Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}\) has universally bounded cohomology for all quasi-coherent sheaves \(\mathcal{F}\) on \(Y\). Bounded from above is clear as \(Rf_*(E)\) is bounded from above. Let \(T \subset |X|\) be the union of the supports of \(H^i(E)\) for all \(i\). Then \(T\) is proper over \(Y\) by assumptions (1) and (2) and Lemma 0CZH. In particular there exists a quasi-compact open subspace \(X' \subset X\) containing \(T\). Setting \(f' = f|_{X'}\) we have \(Rf_*(E) = Rf'_*(E|_{X'})\) because \(E\) restricts to zero on \(X \setminus T\). Thus we may replace \(X\) by \(X'\) and assume \(f\) is quasi-compact. We have assumed \(f\) is quasi-separated. Thus \[Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F} = Rf_*\left(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}\right) = Rf_*\left(E \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} f^{-1}\mathcal{F}\right)\] by Lemma 08IN and Cohomology on Sites, Lemma 08I6. By assumption (3) the complex \(E \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} f^{-1}\mathcal{F}\) has cohomology sheaves in a given finite range, say \([a, b]\). Then \(Rf_*\) of it has cohomology in the range \([a, \infty)\) and we win.

Lemma

Let \(S\) be a scheme. Let \(B\) be a Noetherian algebraic space over \(S\). Let \(f : X \to B\) be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let \(E \in D(\mathcal{O}_X)\) be perfect. Let \(\mathcal{G}^\bullet\) be a bounded complex of coherent \(\mathcal{O}_X\)-modules flat over \(B\) with support proper over \(B\). Then \(K = Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet)\) is a perfect object of \(D(\mathcal{O}_B)\).

Proof

The object \(K\) is perfect by Lemma 08IS. We check the lemma applies: Locally \(E\) is isomorphic to a finite complex of finite free \(\mathcal{O}_X\)-modules. Hence locally \(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet\) is isomorphic to a finite complex whose terms are of the form \[\bigoplus\nolimits_{i = a, \ldots, b} (\mathcal{G}^i)^{\oplus r_i}\] for some integers \(a, b, r_a, \ldots, r_b\). This immediately implies the cohomology sheaves \(H^i(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G})\) are coherent. The hypothesis on the tor dimension also follows as \(\mathcal{G}^i\) is flat over \(f^{-1}\mathcal{O}_Y\).

Lemma

Let \(S\) be a scheme. Let \(B\) be a Noetherian algebraic space over \(S\). Let \(f : X \to B\) be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let \(E \in D(\mathcal{O}_X)\) be perfect. Let \(\mathcal{G}^\bullet\) be a bounded complex of coherent \(\mathcal{O}_X\)-modules flat over \(B\) with support proper over \(B\). Then \(K = Rf_*R\SheafHom(E, \mathcal{G})\) is a perfect object of \(D(\mathcal{O}_B)\).

Proof

Since \(E\) is a perfect complex there exists a dual perfect complex \(E^\vee\), see Cohomology on Sites, Lemma 08JJ. Observe that \(R\SheafHom(E, \mathcal{G}^\bullet) = E^\vee \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet\). Thus the perfectness of \(K\) follows from Lemma 0DKJ.

A projection formula for Ext

Lemma 08JR (or similar results in the literature) is sometimes useful to verify properties of an obstruction theory needed to verify one of Artin’s criteria for Quot functors, Hilbert schemes, and other moduli problems. Suppose that \(f : X \to Y\) is a proper, flat, finitely presented morphism of algebraic spaces and \(E \in D(\mathcal{O}_X)\) is perfect. Here the lemma says \[\Ext^i_X(E, f^*\mathcal{F}) = \Ext^i_Y((Rf_*E^\vee)^\vee, \mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(Y\). Writing it this way makes it look like a projection formula for Ext and indeed the result follows rather easily from Lemma 08IN.

Lemma

Assumptions and notation as in Lemma 0DKJ. Then there are functorial isomorphisms \[H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \longrightarrow H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}))\] for \(\mathcal{F}\) quasi-coherent on \(B\) compatible with boundary maps (see proof).

Proof

We have \[\mathcal{G}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F} = \mathcal{G}^\bullet \otimes_{f^{-1}\mathcal{O}_B}^\mathbf{L} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes_{f^{-1}\mathcal{O}_B} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}\] the first equality by Cohomology on Sites, Lemma 08I6, the second as \(\mathcal{G}^n\) is a flat \(f^{-1}\mathcal{O}_B\)-module, and the third by definition of pullbacks. Hence we obtain \[\begin{align*} H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})) & = H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}) \\ & = H^i(B, Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet \otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*\mathcal{F})) \\ & = H^i(B, Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \\ & = H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \end{align*}\] The first equality by the above, the second by Leray (Cohomology on Sites, Remark 08J6), and the third equality by Lemma 08IN. The statement on boundary maps means the following: Given a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) then the isomorphisms fit into commutative diagrams \[\xymatrix{ H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3) \ar[r] \ar[d]_\delta & H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3)) \ar[d]^\delta \\ H^{i + 1}(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1) \ar[r] & H^{i + 1}(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1)) }\] where the boundary maps come from the distinguished triangle \[K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_2 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1[1]\] and the distinguished triangle in \(D(\mathcal{O}_X)\) associated to the short exact sequence \[0 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3 \to 0\] of complexes. This sequence is exact because \(\mathcal{G}^n\) is flat over \(B\). We omit the verification of the commutativity of the displayed diagram.

Lemma

Assumption and notation as in Lemma 0DKK. Then there are functorial isomorphisms \[H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(B\) compatible with boundary maps (see proof).

Proof

As in the proof of Lemma 0DKK let \(E^\vee\) be the dual perfect complex and recall that \(K = Rf_*(E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\). Since we also have \[\Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) = H^i(X, E^\vee \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}))\] by construction of \(E^\vee\), the existence of the isomorphisms follows from Lemma 0A1M applied to \(E^\vee\) and \(\mathcal{G}^\bullet\). The statement on boundary maps means the following: Given a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) then the isomorphisms fit into commutative diagrams \[\xymatrix{ H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3) \ar[r] \ar[d]_\delta & \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3) \ar[d]^\delta \\ H^{i + 1}(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1) \ar[r] & \Ext^{i + 1}_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1) }\] where the boundary maps come from the distinguished triangle \[K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_2 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3 \to K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1[1]\] and the distinguished triangle in \(D(\mathcal{O}_X)\) associated to the short exact sequence \[0 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3 \to 0\] of complexes. This sequence is exact because \(\mathcal{G}^n\) is flat over \(B\). We omit the verification of the commutativity of the displayed diagram.

Lemma

Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\), \(E \in D(\mathcal{O}_X)\), and \(\mathcal{F}^\bullet\) a complex of \(\mathcal{O}_X\)-modules. Assume

1. \(B\) is Noetherian,

2. \(f\) is locally of finite type and quasi-separated,

3. \(E \in D^-_{\textit{Coh}}(\mathcal{O}_X)\),

4. \(\mathcal{G}^\bullet\) is a bounded complex of coherent \(\mathcal{O}_X\)-module flat over \(B\) with support proper over \(B\).

Then the following two statements are true

1. for every \(m \in \mathbf{Z}\) there exists a perfect object \(K\) of \(D(\mathcal{O}_B)\) and functorial maps \[\alpha^i_\mathcal{F} : \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(B\) compatible with boundary maps (see proof) such that \(\alpha^i_\mathcal{F}\) is an isomorphism for \(i \leq m\), and

2. there exists a pseudo-coherent \(L \in D(\mathcal{O}_B)\) and functorial isomorphisms \[\Ext^i_{\mathcal{O}_B}(L, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(B\) compatible with boundary maps.

Proof

Proof of (A). Suppose \(\mathcal{G}^i\) is nonzero only for \(i \in [a, b]\). We may replace \(X\) by a quasi-compact open neighbourhood of the union of the supports of \(\mathcal{G}^i\). Hence we may assume \(X\) is Noetherian. In this case \(X\) and \(f\) are quasi-compact and quasi-separated. Choose an approximation \(P \to E\) by a perfect complex \(P\) of \((X, E, -m - 1 + a)\) (possible by Theorem 08HP). Then the induced map \[\Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(P, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] is an isomorphism for \(i \leq m\). Namely, the kernel, resp. cokernel of this map is a quotient, resp. submodule of \[\Ext^i_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \quad\text{resp.}\quad \Ext^{i + 1}_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] where \(C\) is the cone of \(P \to E\). Since \(C\) has vanishing cohomology sheaves in degrees \(\geq -m - 1 + a\) these \(\Ext\)-groups are zero for \(i \leq m + 1\) by Derived Categories, Lemma 06XS. This reduces us to the case that \(E\) is a perfect complex which is Lemma 08JN. The statement on boundaries is explained in the proof of Lemma 08JN.

Proof of (B). As in the proof of (A) we may assume \(X\) is Noetherian. Observe that \(E\) is pseudo-coherent by Lemma 08IK. By Lemma 0DKB we can write \(E = \text{hocolim} E_n\) with \(E_n\) perfect and \(E_n \to E\) inducing an isomorphism on truncations \(\tau_{\geq -n}\). Let \(E_n^\vee\) be the dual perfect complex (Cohomology on Sites, Lemma 08JJ). We obtain an inverse system \(\ldots \to E_3^\vee \to E_2^\vee \to E_1^\vee\) of perfect objects. This in turn gives rise to an inverse system \[\ldots \to K_3 \to K_2 \to K_1\quad\text{with}\quad K_n = Rf_*(E_n^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\] perfect on \(Y\), see Lemma 0DKJ. By Lemma 08JN and its proof and by the arguments in the previous paragraph (with \(P = E_n\)) for any quasi-coherent \(\mathcal{F}\) on \(Y\) we have functorial canonical maps \[\xymatrix{ & \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \ar[ld] \ar[rd] \\ H^i(Y, K_{n + 1} \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) \ar[rr] & & H^i(Y, K_n \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}) }\] which are isomorphisms for \(i \leq n + a\). Let \(L_n = K_n^\vee\) be the dual perfect complex. Then we see that \(L_1 \to L_2 \to L_3 \to \ldots\) is a system of perfect objects in \(D(\mathcal{O}_Y)\) such that for any quasi-coherent \(\mathcal{F}\) on \(Y\) the maps \[\Ext^i_{\mathcal{O}_Y}(L_{n + 1}, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_Y}(L_n, \mathcal{F})\] are isomorphisms for \(i \leq n + a - 1\). This implies that \(L_n \to L_{n + 1}\) induces an isomorphism on truncations \(\tau_{\geq -n - a + 2}\) (hint: take cone of \(L_n \to L_{n + 1}\) and look at its last nonvanishing cohomology sheaf). Thus \(L = \text{hocolim} L_n\) is pseudo-coherent, see Lemma 0DKB. The mapping property of homotopy colimits gives that \(\Ext^i_{\mathcal{O}_Y}(L, \mathcal{F}) = \Ext^i_{\mathcal{O}_Y}(L_n, \mathcal{F})\) for \(i \leq n + a - 3\) which finishes the proof.

Remark

The pseudo-coherent complex \(L\) of part (B) of Lemma 08JR is canonically associated to the situation. For example, formation of \(L\) as in (B) is compatible with base change. In other words, given a cartesian diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] of schemes we have canonical functorial isomorphisms \[\Ext^i_{\mathcal{O}_{Y'}}(Lg^*L, \mathcal{F}') \longrightarrow \Ext^i_{\mathcal{O}_X}(L(g')^*E, (g')^*\mathcal{G}^\bullet \otimes_{\mathcal{O}_{X'}} (f')^*\mathcal{F}')\] for \(\mathcal{F}'\) quasi-coherent on \(Y'\). Obsere that we do not use derived pullback on \(\mathcal{G}^\bullet\) on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof.

Limits and derived categories

In this section we collect some results about the derived category of an algebraic space which is the limit of an inverse system of algebraic spaces. More precisely, we will work in the following setting.

Situation

Let \(S\) be a scheme. Let \(X = \lim_{i \in I} X_i\) be a limit of a directed system of algebraic spaces over \(S\) with affine transition morphisms \(f_{i'i} : X_{i'} \to X_i\). We denote \(f_i : X \to X_i\) the projection. We assume that \(X_i\) is quasi-compact and quasi-separated for all \(i \in I\). We also choose an element \(0 \in I\).

Lemma

In Situation 09RH. Let \(E_0\) and \(K_0\) be objects of \(D(\mathcal{O}_{X_0})\). Set \(E_i = Lf_{i0}^*E_0\) and \(K_i = Lf_{i0}^*K_0\) for \(i \geq 0\) and set \(E = Lf_0^*E_0\) and \(K = Lf_0^*K_0\). Then the map \[\colim_{i \geq 0} \Hom_{D(\mathcal{O}_{X_i})}(E_i, K_i) \longrightarrow \Hom_{D(\mathcal{O}_X)}(E, K)\] is an isomorphism if either

1. \(E_0\) is perfect and \(K_0 \in D_\QCoh(\mathcal{O}_{X_0})\), or

2. \(E_0\) is pseudo-coherent and \(K_0 \in D_\QCoh(\mathcal{O}_{X_0})\) has finite tor dimension.

Proof

For every quasi-compact and quasi-separated object \(U_0\) of \((X_0)_{spaces, \etale}\) consider the condition \(P\) that the canonical map \[\colim_{i \geq 0} \Hom_{D(\mathcal{O}_{U_i})}(E_i|_{U_i}, K_i|_{U_i}) \longrightarrow \Hom_{D(\mathcal{O}_U)}(E|_U, K|_U)\] is an isomorphism, where \(U = X \times_{X_0} U_0\) and \(U_i = X_i \times_{X_0} U_0\). We will prove \(P\) holds for each \(U_0\) by the induction principle of Lemma 08GP. Condition (2) of this lemma follows immediately from Mayer-Vietoris for hom in the derived category, see Lemma 08H9. Thus it suffices to prove the lemma when \(X_0\) is affine.

If \(X_0\) is affine, then the result follows from the case of schemes, see Derived Categories of Schemes, Lemma 09RE. To see this use the equivalence of Lemma 071Q and use the translation of properties explained in Lemmas 08HE, 08HF, and 08HG.

Lemma

In Situation 09RH the category of perfect objects of \(D(\mathcal{O}_X)\) is the colimit of the categories of perfect objects of \(D(\mathcal{O}_{X_i})\).

Proof

For every quasi-compact and quasi-separated object \(U_0\) of \((X_0)_{spaces, \etale}\) consider the condition \(P\) that the functor \[\colim_{i \geq 0} D_{perf}(\mathcal{O}_{U_i}) \longrightarrow D_{perf}(\mathcal{O}_U)\] is an equivalence where \({}_{perf}\) indicates the full subcategory of perfect objects and where \(U = X \times_{X_0} U_0\) and \(U_i = X_i \times_{X_0} U_0\). We will prove \(P\) holds for every \(U_0\) by the induction principle of Lemma 08GP. First, we observe that we already know the functor is fully faithful by Lemma 09RI. Thus it suffices to prove essential surjectivity.

We first check condition (2) of the induction principle. Thus suppose that we have an elementary distinguished square \((U_0 \subset X_0, V_0 \to X_0)\) and that \(P\) holds for \(U_0\), \(V_0\), and \(U_0 \times_{X_0} V_0\). Let \(E\) be a perfect object of \(D(\mathcal{O}_X)\). We can find \(i \geq 0\) and \(E_{U, i}\) perfect on \(U_i\) and \(E_{V, i}\) perfect on \(V_i\) whose pullback to \(U\) and \(V\) are isomorphic to \(E|_U\) and \(E|_V\). Denote \[a : E_{U, i} \to (R(X \to X_i)_*E)|_{U_i} \quad\text{and}\quad b : E_{V, i} \to (R(X \to X_i)_*E)|_{V_i}\] the maps adjoint to the isomorphisms \(L(U \to U_i)^*E_{U, i} \to E|_U\) and \(L(V \to V_i)^*E_{V, i} \to E|_V\). By fully faithfulness, after increasing \(i\), we can find an isomorphism \(c : E_{U, i}|_{U_i \times_{X_i} V_i} \to E_{V, i}|_{U_i \times_{X_i} V_i}\) which pulls back to the identifications \[L(U \to U_i)^*E_{U, i}|_{U \times_X V} \to E|_{U \times_X V} \to L(V \to V_i)^*E_{V, i}|_{U \times_X V}.\] Apply Lemma 08HB to get an object \(E_i\) on \(X_i\) and a map \(d : E_i \to R(X \to X_i)_*E\) which restricts to the maps \(a\) and \(b\) over \(U_i\) and \(V_i\). Then it is clear that \(E_i\) is perfect and that \(d\) is adjoint to an isomorphism \(L(X \to X_i)^*E_i \to E\).

Finally, we check condition (1) of the induction principle, in other words, we check the lemma holds when \(X_0\) is affine. This follows from the case of schemes, see Derived Categories of Schemes, Lemma 09RF. To see this use the equivalence of Lemma 071Q and use the translation of Lemma 08HG.

Cohomology and base change, VI

A final section on cohomology and base change continuing the discussion of Sections 08IM, 0DKE, and 0A1L. An easy to grok special case is given in Remark 0A1Q.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of finite presentation between algebraic spaces over \(S\). Let \(E \in D(\mathcal{O}_X)\) be a perfect object. Let \(\mathcal{G}^\bullet\) be a bounded complex of finitely presented \(\mathcal{O}_X\)-modules, flat over \(Y\), with support proper over \(Y\). Then \[K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\] is a perfect object of \(D(\mathcal{O}_Y)\) and its formation commutes with arbitrary base change.

Proof

The statement on base change is Lemma 0A1K. Thus it suffices to show that \(K\) is a perfect object. If \(Y\) is Noetherian, then this follows from Lemma 0DKJ. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on \(Y\), hence we may assume \(Y\) is affine. Say \(Y = \Spec(R)\). We write \(R = \colim R_i\) as a filtered colimit of Noetherian rings \(R_i\). By Limits of Spaces, Lemma 07SK there exists an \(i\) and an algebraic space \(X_i\) of finite presentation over \(R_i\) whose base change to \(R\) is \(X\). By Limits of Spaces, Lemma 07V7 we may assume after increasing \(i\), that there exists a bounded complex of finitely presented \(\mathcal{O}_{X_i}\)-modules \(\mathcal{G}_i^\bullet\) whose pullback to \(X\) is \(\mathcal{G}^\bullet\). After increasing \(i\) we may assume \(\mathcal{G}_i^n\) is flat over \(R_i\), see Limits of Spaces, Lemma 08K0. After increasing \(i\) we may assume the support of \(\mathcal{G}_i^n\) is proper over \(R_i\), see Limits of Spaces, Lemma 08K2. Finally, by Lemma 09RJ we may, after increasing \(i\), assume there exists a perfect object \(E_i\) of \(D(\mathcal{O}_{X_i})\) whose pullback to \(X\) is \(E\). By Lemma 0DKJ we have that \(K_i = Rf_{i, *}(E_i \otimes_{\mathcal{O}_{X_i}}^\mathbf{L} \mathcal{G}_i^\bullet)\) is perfect on \(\Spec(R_i)\) where \(f_i : X_i \to \Spec(R_i)\) is the structure morphism. By the base change result (Lemma 0A1K) the pullback of \(K_i\) to \(Y = \Spec(R)\) is \(K\) and we conclude.

Remark

Let \(R\) be a ring. Let \(X\) be an algebraic space of finite presentation over \(R\). Let \(\mathcal{G}\) be a finitely presented \(\mathcal{O}_X\)-module flat over \(R\) with support proper over \(R\). By Lemma 0A1P there exists a finite complex of finite projective \(R\)-modules \(M^\bullet\) such that we have \[R\Gamma(X_{R'}, \mathcal{G}_{R'}) = M^\bullet \otimes_R R'\] functorially in the \(R\)-algebra \(R'\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of finite presentation between algebraic spaces over \(S\). Let \(E \in D(\mathcal{O}_X)\) be a pseudo-coherent object. Let \(\mathcal{G}^\bullet\) be a bounded above complex of finitely presented \(\mathcal{O}_X\)-modules, flat over \(Y\), with support proper over \(Y\). Then \[K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\] is a pseudo-coherent object of \(D(\mathcal{O}_Y)\) and its formation commutes with arbitrary base change.

Proof

The statement on base change is Lemma 0A1K. Thus it suffices to show that \(K\) is a pseudo-coherent object. This will follow from Lemma 0A1P by approximation by perfect complexes. We encourage the reader to skip the rest of the proof.

The question is étale local on \(Y\), hence we may assume \(Y\) is affine. Then \(X\) is quasi-compact and quasi-separated. Moreover, there exists an integer \(N\) such that total direct image \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) has cohomological dimension \(N\) as explained in Lemma 08FA. Choose an integer \(b\) such that \(\mathcal{G}^i = 0\) for \(i > b\). It suffices to show that \(K\) is \(m\)-pseudo-coherent for every \(m\). Choose an approximation \(P \to E\) by a perfect complex \(P\) of \((X, E, m - N - 1 - b)\). This is possible by Theorem 08HP. Choose a distinguished triangle \[P \to E \to C \to P[1]\] in \(D_\QCoh(\mathcal{O}_X)\). The cohomology sheaves of \(C\) are zero in degrees \(\geq m - N - 1 - b\). Hence the cohomology sheaves of \(C \otimes^\mathbf{L} \mathcal{G}^\bullet\) are zero in degrees \(\geq m - N - 1\). Thus the cohomology sheaves of \(Rf_*(C \otimes^\mathbf{L} \mathcal{G})\) are zero in degrees \(\geq m - 1\). Hence \[Rf_*(P \otimes^\mathbf{L} \mathcal{G}) \to Rf_*(E \otimes^\mathbf{L} \mathcal{G})\] is an isomorphism on cohomology sheaves in degrees \(\geq m\). Next, suppose that \(H^i(P) = 0\) for \(i > a\). Then \(P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet \longrightarrow P \otimes^\mathbf{L} \mathcal{G}^\bullet\) is an isomorphism on cohomology sheaves in degrees \(\geq m - N - 1\). Thus again we find that \[Rf_*(P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet) \to Rf_*(P \otimes^\mathbf{L} \mathcal{G}^\bullet)\] is an isomorphism on cohomology sheaves in degrees \(\geq m\). By Lemma 0A1P the source is a perfect complex. We conclude that \(K\) is \(m\)-pseudo-coherent as desired.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of finite presentation of algebraic spaces over \(S\).

1. Let \(E \in D(\mathcal{O}_X)\) be perfect and \(f\) flat. Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_Y)\) and its formation commutes with arbitrary base change.

2. Let \(\mathcal{G}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\). Then \(Rf_*\mathcal{G}\) is a perfect object of \(D(\mathcal{O}_Y)\) and its formation commutes with arbitrary base change.

Proof

Special cases of Lemma 0A1P applied with (1) \(\mathcal{G}^\bullet\) equal to \(\mathcal{O}_X\) in degree \(0\) and (2) \(E = \mathcal{O}_X\) and \(\mathcal{G}^\bullet\) consisting of \(\mathcal{G}\) sitting in degree \(0\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a flat proper morphism of finite presentation of algebraic spaces over \(S\). Let \(E \in D(\mathcal{O}_X)\) be pseudo-coherent. Then \(Rf_*E\) is a pseudo-coherent object of \(D(\mathcal{O}_Y)\) and its formation commutes with arbitrary base change.

More generally, if \(f : X \to Y\) is proper and \(E\) on \(X\) is pseudo-coherent relative to \(Y\) (More on Morphisms of Spaces, Definition 0CSY), then \(Rf_*E\) is pseudo-coherent (but formation does not commute with base change in this generality). The case of this for schemes is proved in [Kiehl].

Proof

Special case of Lemma 0CTL applied with \(\mathcal{G} = \mathcal{O}_X\).

Lemma

Let \(R\) be a ring. Let \(X\) be an algebraic space and let \(f : X \to \Spec(R)\) be proper, flat, and of finite presentation. Let \((M_n)\) be an inverse system of \(R\)-modules with surjective transition maps. Then the canonical map \[\mathcal{O}_X \otimes_R (\lim M_n) \longrightarrow \lim \mathcal{O}_X \otimes_R M_n\] induces an isomorphism from the source to \(DQ_X\) applied to the target.

Proof

The statement means that for any object \(E\) of \(D_\QCoh(\mathcal{O}_X)\) the induced map \[\Hom(E, \mathcal{O}_X \otimes_R (\lim M_n)) \longrightarrow \Hom(E, \lim \mathcal{O}_X \otimes_R M_n)\] is an isomorphism. Since \(D_\QCoh(\mathcal{O}_X)\) has a perfect generator (Theorem 09IY) it suffices to check this for perfect \(E\). By Lemma 0D3E we have \(\lim \mathcal{O}_X \otimes_R M_n = R\lim \mathcal{O}_X \otimes_R M_n\). The exact functor \(R\Hom_X(E, -) : D_\QCoh(\mathcal{O}_X) \to D(R)\) of Cohomology on Sites, Section 0B6E commutes with products and hence with derived limits, whence \[R\Hom_X(E, \lim \mathcal{O}_X \otimes_R M_n) = R\lim R\Hom_X(E, \mathcal{O}_X \otimes_R M_n)\] Let \(E^\vee\) be the dual perfect complex, see Cohomology on Sites, Lemma 08JJ. We have \[R\Hom_X(E, \mathcal{O}_X \otimes_R M_n) = R\Gamma(X, E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*M_n) = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n\] by Lemma 08IN. From Lemma 0CTM we see \(R\Gamma(X, E^\vee)\) is a perfect complex of \(R\)-modules. In particular it is a pseudo-coherent complex and by More on Algebra, Lemma 0D2L we obtain \[R\lim R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} \lim M_n\] as desired.

Lemma

Let \(A\) be a ring. Let \(X\) be an algebraic space over \(A\) which is quasi-compact and quasi-separated. Let \(K \in D^-_\QCoh(\mathcal{O}_X)\). If \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every perfect \(E\) in \(D(\mathcal{O}_X)\), then \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E\) in \(D(\mathcal{O}_X)\).

Proof

There exists an integer \(N\) such that \(R\Gamma(X, -) : D_\QCoh(\mathcal{O}_X) \to D(A)\) has cohomological dimension \(N\) as explained in Lemma 08FA. Let \(b \in \mathbf{Z}\) be such that \(H^i(K) = 0\) for \(i > b\). Let \(E\) be pseudo-coherent on \(X\). It suffices to show that \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is \(m\)-pseudo-coherent for every \(m\). Choose an approximation \(P \to E\) by a perfect complex \(P\) of \((X, E, m - N - 1 - b)\). This is possible by Theorem 08HP. Choose a distinguished triangle \[P \to E \to C \to P[1]\] in \(D_\QCoh(\mathcal{O}_X)\). The cohomology sheaves of \(C\) are zero in degrees \(\geq m - N - 1 - b\). Hence the cohomology sheaves of \(C \otimes^\mathbf{L} K\) are zero in degrees \(\geq m - N - 1\). Thus the cohomology of \(R\Gamma(X, C \otimes^\mathbf{L} K)\) are zero in degrees \(\geq m - 1\). Hence \[R\Gamma(X, P \otimes^\mathbf{L} K) \to R\Gamma(X, E \otimes^\mathbf{L} K)\] is an isomorphism on cohomology in degrees \(\geq m\). By assumption the source is pseudo-coherent. We conclude that \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is \(m\)-pseudo-coherent as desired.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of finite presentation between algebraic spaces over \(S\). Let \(E \in D(\mathcal{O}_X)\) be a perfect object. Let \(\mathcal{G}^\bullet\) be a bounded complex of finitely presented \(\mathcal{O}_X\)-modules, flat over \(Y\), with support proper over \(Y\). Then \[K = Rf_*R\SheafHom(E, \mathcal{G}^\bullet)\] is a perfect object of \(D(\mathcal{O}_Y)\) and its formation commutes with arbitrary base change.

Proof

The statement on base change is Lemma 08JQ. Thus it suffices to show that \(K\) is a perfect object. If \(Y\) is Noetherian, then this follows from Lemma 0DKK. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on \(Y\), hence we may assume \(Y\) is affine. Say \(Y = \Spec(R)\). We write \(R = \colim R_i\) as a filtered colimit of Noetherian rings \(R_i\). By Limits of Spaces, Lemma 07SK there exists an \(i\) and an algebraic space \(X_i\) of finite presentation over \(R_i\) whose base change to \(R\) is \(X\). By Limits of Spaces, Lemma 07V7 we may assume after increasing \(i\), that there exists a bounded complex of finitely presented \(\mathcal{O}_{X_i}\)-module \(\mathcal{G}_i^\bullet\) whose pullback to \(X\) is \(\mathcal{G}\). After increasing \(i\) we may assume \(\mathcal{G}_i^n\) is flat over \(R_i\), see Limits of Spaces, Lemma 08K0. After increasing \(i\) we may assume the support of \(\mathcal{G}_i^n\) is proper over \(R_i\), see Limits of Spaces, Lemma 08K2. Finally, by Lemma 08HG we may, after increasing \(i\), assume there exists a perfect object \(E_i\) of \(D(\mathcal{O}_{X_i})\) whose pullback to \(X\) is \(E\). Applying Lemma 08JN to \(X_i \to \Spec(R_i)\), \(E_i\), \(\mathcal{G}_i^\bullet\) and using the base change property already shown we obtain the result.

Perfect complexes

We first talk about jumping loci for betti numbers of perfect complexes. First we have to define betti numbers.

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(x \in |X|\). We want to define \(\beta_i(x) \in \{0, 1, 2, \ldots \} \cup \{\infty\}\). To do this, choose a morphism \(f : \Spec(k) \to X\) in the equivalence class of \(x\). Then \(Lf^*E\) is an object of \(D(\Spec(k)_\etale, \mathcal{O})\). By Étale Cohomology, Lemma 0D1W and Theorem 03OJ we find that \(D(\Spec(k)_\etale, \mathcal{O}) = D(k)\) is the derived category of \(k\)-vector spaces. Hence \(Lf^*E\) is a complex of \(k\)-vector spaces and we can take \(\beta_i(x) = \dim_k H^i(Lf^*E)\). It is easy to see that this does not depend on the choice of the representative in \(x\). Moreover, if \(X\) is a scheme, this is the same as the notion used in Derived Categories of Schemes, Section 0BDH.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E \in D(\mathcal{O}_X)\) be pseudo-coherent (for example perfect). For any \(i \in \mathbf{Z}\) consider the function \[\beta_i : |X| \longrightarrow \{0, 1, 2, \ldots\}\] defined above. Then we have

1. formation of \(\beta_i\) commutes with arbitrary base change,

2. the functions \(\beta_i\) are upper semi-continuous, and

3. the level sets of \(\beta_i\) are étale locally constructible.

Proof

Choose a scheme \(U\) and a surjective étale morphism \(\varphi : U \to X\). Then \(L\varphi^*E\) is a pseudo-coherent complex on the scheme \(U\) (use Lemma 08HE) and we can apply the result for schemes, see Derived Categories of Schemes, Lemma 0BDI. The meaning of part (3) is that the inverse image of the level sets to \(U\) are locally constructible, see Properties of Spaces, Definition 0ECU.

Lemma

Let \(Y\) be a scheme and let \(X\) be an algebraic space over \(Y\) such that the structure morphism \(f : X \to Y\) is flat, proper, and of finite presentation. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(Y\). For fixed \(i \in \mathbf{Z}\) consider the function \[\beta_i : |Y| \to \{0, 1, 2, \ldots\},\quad y \longmapsto \dim_{\kappa(y)} H^i(X_y, \mathcal{F}_y)\] Then we have

1. formation of \(\beta_i\) commutes with arbitrary base change,

2. the functions \(\beta_i\) are upper semi-continuous, and

3. the level sets of \(\beta_i\) are locally constructible in \(Y\).

Proof

By cohomology and base change (more precisely by Lemma 0CTM) the object \(K = Rf_*\mathcal{F}\) is a perfect object of the derived category of \(Y\) whose formation commutes with arbitrary base change. In particular we have \[H^i(X_y, \mathcal{F}_y) = H^i(K \otimes_{\mathcal{O}_Y}^\mathbf{L} \kappa(y))\] Thus the lemma follows from Lemma 0D1Y.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E \in D(\mathcal{O}_X)\) be perfect. The function \[\chi_E : |X| \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^i \beta_i(x)\] is locally constant on \(X\).

Proof

Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 0BDJ.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E \in D(\mathcal{O}_X)\) be perfect. Given \(i, r \in \mathbf{Z}\), there exists an open subspace \(U \subset X\) characterized by the following

1. \(E|_U \cong H^i(E|_U)[-i]\) and \(H^i(E|_U)\) is a locally free \(\mathcal{O}_U\)-module of rank \(r\),

2. a morphism \(f : Y \to X\) factors through \(U\) if and only if \(Lf^*E\) is isomorphic to a locally free module of rank \(r\) placed in degree \(i\).

Proof

Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 0BDK.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) which is proper, flat, and of finite presentation. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(Y\). Fix \(i, r \in \mathbf{Z}\). Then there exists an open subspace \(V \subset Y\) with the following property: A morphism \(T \to Y\) factors through \(V\) if and only if \(Rf_{T, *}\mathcal{F}_T\) is isomorphic to a finite locally free module of rank \(r\) placed in degree \(i\).

Proof

By cohomology and base change ( Lemma 0CTM) the object \(K = Rf_*\mathcal{F}\) is a perfect object of the derived category of \(Y\) whose formation commutes with arbitrary base change. Thus this lemma follows immediately from Lemma 0D20.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(E \in D(\mathcal{O}_X)\) be perfect of tor-amplitude in \([a, b]\) for some \(a, b \in \mathbf{Z}\). Let \(r \geq 0\). Then there exists a locally closed subspace \(j : Z \to X\) characterized by the following

1. \(H^a(Lj^*E)\) is a locally free \(\mathcal{O}_Z\)-module of rank \(r\), and

2. a morphism \(f : Y \to X\) factors through \(Z\) if and only if for all morphisms \(g : Y' \to Y\) the \(\mathcal{O}_{Y'}\)-module \(H^a(L(f \circ g)^*E)\) is locally free of rank \(r\).

Moreover, \(j : Z \to X\) is of finite presentation and we have

1. if \(f : Y \to X\) factors as \(Y \xrightarrow{g} Z \to X\), then \(H^a(Lf^*E) = g^*H^a(Lj^*E)\),

2. if \(\beta_a(x) \leq r\) for all \(x \in |X|\), then \(j\) is a closed immersion and given \(f : Y \to X\) the following are equivalent

   1. \(f : Y \to X\) factors through \(Z\),

   2. \(H^0(Lf^*E)\) is a locally free \(\mathcal{O}_Y\)-module of rank \(r\),

   and if \(r = 1\) these are also equivalent to

   1. \(\mathcal{O}_Y \to \SheafHom_{\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))\) is injective.

Proof

Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 0BDL.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume

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

2. for a morphism \(\Spec(k) \to Y\) where \(k\) is a field, we have \(k = H^0(X_k, \mathcal{O}_{X_k})\).

Then we have

1. \(f_*\mathcal{O}_X = \mathcal{O}_Y\) and this holds after any base change,

2. étale locally on \(Y\) we have \[Rf_*\mathcal{O}_X = \mathcal{O}_Y \oplus P\] in \(D(\mathcal{O}_Y)\) where \(P\) is perfect of tor amplitude in \([1, \infty)\).

Proof

It suffices to prove (a) and (b) étale locally on \(Y\), thus we may and do assume \(Y\) is an affine scheme. By cohomology and base change (Lemma 0CTM) the complex \(E = Rf_*\mathcal{O}_X\) is perfect and its formation commutes with arbitrary base change. In particular, for \(y \in Y\) we see that \(H^0(E \otimes^\mathbf{L} \kappa(y)) = H^0(X_y, \mathcal{O}_{X_y}) = \kappa(y)\). Thus \(\beta_0(y) \leq 1\) for all \(y \in Y\) with notation as in Lemma 0D1Y. Apply Lemma 0D21 with \(a = 0\) and \(r = 1\). We obtain a universal closed subscheme \(j : Z \to Y\) with \(H^0(Lj^*E)\) invertible characterized by the equivalence of (4)(a), (b), and (c) of the lemma. Since formation of \(E\) commutes with base change, we have \[Lf^*E = R\text{pr}_{1, *}\mathcal{O}_{X \times_Y X}\] The morphism \(\text{pr}_1 : X \times_Y X\) has a section namely the diagonal morphism \(\Delta\) for \(X\) over \(Y\). We obtain maps \[\mathcal{O}_X \longrightarrow R\text{pr}_{1, *}\mathcal{O}_{X \times_Y X} \longrightarrow \mathcal{O}_X\] in \(D(\mathcal{O}_X)\) whose composition is the identity. Thus \(R\text{pr}_{1, *}\mathcal{O}_{X \times_Y X} = \mathcal{O}_X \oplus E'\) in \(D(\mathcal{O}_X)\). Thus \(\mathcal{O}_X\) is a direct summand of \(H^0(Lf^*E)\) and we conclude that \(X \to Y\) factors through \(Z\) by the equivalence of (4)(c) and (4)(a) of the lemma cited above. Since \(\{X \to Y\}\) is an fppf covering, we have \(Z = Y\). Thus \(f_*\mathcal{O}_X\) is an invertible \(\mathcal{O}_Y\)-module. We conclude \(\mathcal{O}_Y \to f_*\mathcal{O}_X\) is an isomorphism because a ring map \(A \to B\) such that \(B\) is invertible as an \(A\)-module is an isomorphism. Since the assumptions are preserved under base change, we see that (a) is true.

Proof of (b). Above we have seen that for every \(y \in Y\) the map \(\mathcal{O}_Y \to H^0(E \otimes^\mathbf{L} \kappa(y))\) is surjective. Thus we may apply More on Algebra, Lemma 0A1U to see that in an open neighbourhood of \(y\) we have a decomposition \(Rf_*\mathcal{O}_X = \mathcal{O}_Y \oplus P\)

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume

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

2. the geometric fibres of \(f\) are reduced and connected.

Then \(f_*\mathcal{O}_X = \mathcal{O}_Y\) and this holds after any base change.

Proof

By Lemma 0E6B it suffices to show that \(k = H^0(X_k, \mathcal{O}_{X_k})\) for all morphisms \(\Spec(k) \to Y\) where \(k\) is a field. This follows from Spaces over Fields, Lemma 0DMY and the fact that \(X_k\) is geometrically connected and geometrically reduced.

Other applications

In this section we state and prove some results that can be deduced from the theory worked out above.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(K\) be an object of \(D_\QCoh(\mathcal{O}_X)\) such that the cohomology sheaves \(H^i(K)\) have countable sets of sections over affine schemes étale over \(X\). Then for any quasi-compact and quasi-separated étale morphism \(U \to X\) and any perfect object \(E\) in \(D(\mathcal{O}_X)\) the sets \[H^i(U, K \otimes^\mathbf{L} E),\quad \Ext^i(E|_U, K|_U)\] are countable.

Proof

Using Cohomology on Sites, Lemma 08JJ we see that it suffices to prove the result for the groups \(H^i(U, K \otimes^\mathbf{L} E)\). We will use the induction principle to prove the lemma, see Lemma 08GP.

When \(U = \Spec(A)\) is affine the result follows from the case of schemes, see Derived Categories of Schemes, Lemma 0CRP.

To finish the proof it suffices to show: if \((U \subset W, V \to W)\) is an elementary distinguished triangle and the result holds for \(U\), \(V\), and \(U \times_W V\), then the result holds for \(W\). This is an immediate consequence of the Mayer-Vietoris sequence, see Lemma 0CRS.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Assume the sets of sections of \(\mathcal{O}_X\) over affines étale over \(X\) are countable. Let \(K\) be an object of \(D_\QCoh(\mathcal{O}_X)\). The following are equivalent

1. \(K = \text{hocolim} E_n\) with \(E_n\) a perfect object of \(D(\mathcal{O}_X)\), and

2. the cohomology sheaves \(H^i(K)\) have countable sets of sections over affines étale over \(X\).

Proof

If (1) is true, then (2) is true because homotopy colimits commutes with taking cohomology sheaves (by Derived Categories, Lemma 0CRK) and because a perfect complex is locally isomorphic to a finite complex of finite free \(\mathcal{O}_X\)-modules and therefore satisfies (2) by assumption on \(X\).

Assume (2). Choose a K-injective complex \(\mathcal{K}^\bullet\) representing \(K\). Choose a perfect generator \(E\) of \(D_\QCoh(\mathcal{O}_X)\) and represent it by a K-injective complex \(\mathcal{I}^\bullet\). According to Theorem 09MC and its proof there is an equivalence of triangulated categories \(F : D_\QCoh(\mathcal{O}_X) \to D(A, \text{d})\) where \((A, \text{d})\) is the differential graded algebra \[(A, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{I}^\bullet)\] which maps \(K\) to the differential graded module \[M = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{K}^\bullet)\] Note that \(H^i(A) = \Ext^i(E, E)\) and \(H^i(M) = \Ext^i(E, K)\). Moreover, since \(F\) is an equivalence it and its quasi-inverse commute with homotopy colimits. Therefore, it suffices to write \(M\) as a homotopy colimit of compact objects of \(D(A, \text{d})\). By Differential Graded Algebra, Lemma 0CRM it suffices show that \(\Ext^i(E, E)\) and \(\Ext^i(E, K)\) are countable for each \(i\). This follows from Lemma 0CRU.

Lemma

Let \(A\) be a ring. Let \(f : U \to X\) be a flat morphism of algebraic spaces of finite presentation over \(A\). Then

1. there exists an inverse system of perfect objects \(L_n\) of \(D(\mathcal{O}_X)\) such that \[R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Hom_X(L_n, K)\] in \(D(A)\) functorially in \(K\) in \(D_\QCoh(\mathcal{O}_X)\), and

2. there exists a system of perfect objects \(E_n\) of \(D(\mathcal{O}_X)\) such that \[R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Gamma(X, E_n \otimes^\mathbf{L} K)\] in \(D(A)\) functorially in \(K\) in \(D_\QCoh(\mathcal{O}_X)\).

Proof

By Lemma 08IN we have \[R\Gamma(U, Lf^*K) = R\Gamma(X, Rf_*\mathcal{O}_U \otimes^\mathbf{L} K)\] functorially in \(K\). Observe that \(R\Gamma(X, -)\) commutes with homotopy colimits because it commutes with direct sums by Lemma 08FB. Similarly, \(- \otimes^\mathbf{L} K\) commutes with derived colimits because \(- \otimes^\mathbf{L} K\) commutes with direct sums (because direct sums in \(D(\mathcal{O}_X)\) are given by direct sums of representing complexes). Hence to prove (2) it suffices to write \(Rf_*\mathcal{O}_U = \text{hocolim} E_n\) for a system of perfect objects \(E_n\) of \(D(\mathcal{O}_X)\). Once this is done we obtain (1) by setting \(L_n = E_n^\vee\), see Cohomology on Sites, Lemma 08JJ.

Write \(A = \colim A_i\) with \(A_i\) of finite type over \(\mathbf{Z}\). By Limits of Spaces, Lemma 07SK we can find an \(i\) and morphisms \(U_i \to X_i \to \Spec(A_i)\) of finite presentation whose base change to \(\Spec(A)\) recovers \(U \to X \to \Spec(A)\). After increasing \(i\) we may assume that \(f_i : U_i \to X_i\) is flat, see Limits of Spaces, Lemma 08K0. By Lemma 08IR the derived pullback of \(Rf_{i, *}\mathcal{O}_{U_i}\) by \(g : X \to X_i\) is equal to \(Rf_*\mathcal{O}_U\). Since \(Lg^*\) commutes with derived colimits, it suffices to prove what we want for \(f_i\). Hence we may assume that \(U\) and \(X\) are of finite type over \(\mathbf{Z}\).

Assume \(f : U \to X\) is a morphism of algebraic spaces of finite type over \(\mathbf{Z}\). To finish the proof we will show that \(Rf_*\mathcal{O}_U\) is a homotopy colimit of perfect complexes. To see this we apply Lemma 0CRV. Thus it suffices to show that \(R^if_*\mathcal{O}_U\) has countable sets of sections over affines étale over \(X\). This follows from Lemma 0CRU applied to the structure sheaf.

The resolution property

This section is the analogue of Derived Categories of Schemes, Section 0F85 for algebraic spaces; please read that section first. It is currently not known if a smooth proper algebraic space over a field always has the resolution property or if this is false. If you know the answer to this question, please email stacks.project@gmail.com.

We can make the following definition although it scarcely makes sense to consider it for general algebraic spaces.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). We say \(X\) has the resolution property if every quasi-coherent \(\mathcal{O}_X\)-module of finite type is the quotient of a finite locally free \(\mathcal{O}_X\)-module.

If \(X\) is a quasi-compact and quasi-separated algebraic space, then it suffices to check every \(\mathcal{O}_X\)-module module of finite presentation (automatically quasi-coherent) is the quotient of a finite locally free \(\mathcal{O}_X\)-module, see Limits of Spaces, Lemma 086Y. If \(X\) is a Noetherian algebraic space, then finite type quasi-coherent modules are exactly the coherent \(\mathcal{O}_X\)-modules, see Cohomology of Spaces, Lemma 07UB.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume

1. \(Y\) is quasi-compact and quasi-separated and has the resolution property,

2. there exists an \(f\)-ample invertible module on \(X\) (Divisors on Spaces, Definition 0D31).

Then \(X\) has the resolution property.

Proof

Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{L}\) be an \(f\)-ample invertible module. Choose an affine scheme \(V\) and a surjective étale morphism \(V \to Y\). Set \(U = V \times_Y X\). Then \(\mathcal{L}|_U\) is ample on \(U\). By Properties, Proposition 01Q3 we know there exists finitely many maps \(s_i : \mathcal{L}^{\otimes n_i}|_U \to \mathcal{F}|_U\) which are jointly surjective. Consider the quasi-coherent \(\mathcal{O}_Y\)-modules \[\mathcal{H}_n = f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\] We may think of \(s_i\) as a section over \(V\) of the sheaf \(\mathcal{H}_{-n_i}\). Suppose we can find finite locally free \(\mathcal{O}_Y\)-modules \(\mathcal{E}_i\) and maps \(\mathcal{E}_i \to \mathcal{H}_{-n_i}\) such that \(s_i\) is in the image. Then the corresponding maps \[f^*\mathcal{E}_i \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n_i} \longrightarrow \mathcal{F}\] are going to be jointly surjective and the lemma is proved. By Limits of Spaces, Lemma 0829 for each \(i\) we can find a finite type quasi-coherent submodule \(\mathcal{H}'_i \subset \mathcal{H}_{-n_i}\) which contains the section \(s_i\) over \(V\). Thus the resolution property of \(Y\) produces surjections \(\mathcal{E}_i \to \mathcal{H}'_i\) and we conclude.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine or quasi-affine morphism of algebraic spaces over \(S\) with \(Y\) quasi-compact and quasi-separated. If \(Y\) has the resolution property, so does \(X\).

Proof

By Divisors on Spaces, Lemma 0GUQ this is a special case of Lemma 0GUT.

Here is a case where one can prove the resolution property goes down.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a surjective finite locally free morphism of algebraic spaces over \(S\). If \(X\) has the resolution property, so does \(Y\).

Proof

The condition means that \(f\) is affine and that \(f_*\mathcal{O}_X\) is a finite locally free \(\mathcal{O}_Y\)-module of positive rank. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module of finite type. By assumption there exists a surjection \(\mathcal{E} \to f^*\mathcal{G}\) for some finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\). Since \(f_*\) is exact (Cohomology of Spaces, Section 0DK2) we get a surjection \[f_*\mathcal{E} \longrightarrow f_*f^*\mathcal{G} = \mathcal{G} \otimes_{\mathcal{O}_Y} f_*\mathcal{O}_X\] Taking duals we get a surjection \[f_*\mathcal{E} \otimes_{\mathcal{O}_Y} \SheafHom_{\mathcal{O}_Y}(f_*\mathcal{O}_X, \mathcal{O}_Y) \longrightarrow \mathcal{G}\] Since \(f_*\mathcal{E}\) is finite locally free, we conclude.

For more on the resolution property of algebraic spaces, please see More on Morphisms of Spaces, Section 0GUX.

Detecting Boundedness

In this section, we show that compact generators of \(D_\QCoh\) of a quasi-compact, quasi-separated scheme, as constructed in Section 09IU, have a special property. We recommend reading that section first as it is very similar to this one.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(P \in D_{perf}(\mathcal{O}_X)\) and \(E \in D_{\QCoh}(\mathcal{O}_X)\). Let \(a \in \mathbf{Z}\). The following are equivalent

1. \(\Hom_{D(\mathcal{O}_X)}(P[-i], E) = 0\) for \(i \gg 0\), and

2. \(\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{\geq a} E) = 0\) for \(i \gg 0\).

Proof

Using the triangle \(\tau_{< a} E \to E \to \tau_{\geq a} E \to\) we see that the equivalence follows if we can show \[\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{< a} E) = \Hom_{D(\mathcal{O}_X)}(P, (\tau_{< a} E)[i]) = 0\] for \(i \gg 0\). As \(P\) is perfect this is true by Lemma 09MB.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(P \in D_{perf}(\mathcal{O}_X)\) and \(E \in D_{\QCoh}(\mathcal{O}_X)\). Let \(a \in \mathbf{Z}\). The following are equivalent

1. \(\Hom_{D(\mathcal{O}_X)}(P[-i], E) = 0\) for \(i \ll 0\), and

2. \(\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{\leq a} E) = 0\) for \(i \ll 0\).

Proof

Using the triangle \(\tau_{\leq a} E \to E \to \tau_{> a} E \to\) we see that the equivalence follows if we can show \[\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{> a} E) = \Hom_{D(\mathcal{O}_X)}(P, (\tau_{> a} E)[i]) = 0\] for \(i \ll 0\). As \(P\) is perfect this is true by Lemma 09MB.

Proposition

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(G \in D_{perf}(\mathcal{O}_X)\) be a perfect complex which generates \(D_\QCoh (\mathcal{O}_X)\). Let \(E \in D_\QCoh (\mathcal{O}_X)\). The following are equivalent

1. \(E \in D^-_\QCoh (\mathcal{O}_X)\),

2. \(\Hom_{D(\mathcal{O}_X)}(G[-i], E) = 0\) for \(i \gg 0\),

3. \(\Ext^i_X(G, E) = 0\) for \(i \gg 0\),

4. \(R\Hom_X(G, E)\) is in \(D^-(\mathbf{Z})\),

5. \(H^i(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0\) for \(i \gg 0\),

6. \(R\Gamma(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E)\) is in \(D^-(\mathbf{Z})\),

7. for every perfect object \(P\) of \(D(\mathcal{O}_X)\)

   1. the assertions (2), (3), (4) hold with \(G\) replaced by \(P\), and

   2. \(H^i(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0\) for \(i \gg 0\),

   3. \(R\Gamma(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E)\) is in \(D^-(\mathbf{Z})\).

Proof

Assume (1). Since \(\Hom_{D(\mathcal{O}_X)}(G[-i], E) = \Hom_{D(\mathcal{O}_X)}(G, E[i])\) we see that this is zero for \(i \gg 0\) by Lemma 09MB. This proves that (1) implies (2).

Parts (2), (3), (4) are equivalent by the discussion in Cohomology on Sites, Section 0B6E. Part (5) and (6) are equivalent as \(H^i(X, -) = H^i(R\Gamma(X, -))\) by definition. The equivalent conditions (2), (3), (4) are equivalent to the equivalent conditions (5), (6) by Cohomology on Sites, Lemma 08JJ and the fact that \((G[-i])^\vee = G^\vee[i]\).

It is clear that (7) implies (2). Conversely, let us prove that the equivalent conditions (2) – (6) imply (7). Recall that \(G\) is a classical generator for \(D_{perf}(\mathcal{O}_X)\) by Remark 0GFC. For \(P \in D_{perf}(\mathcal{O}_X)\) let \(T(P)\) be the assertion that \(R\Hom_X(P, E)\) is in \(D^-(\mathbf{Z})\). Clearly, \(T\) is inherited by direct sums, satisfies the 2-out-of-three property for distinguished triangles, is inherited by direct summands, and is preserved by shifts. Hence by Derived Categories, Remark 0ATH we see that (4) implies \(T\) holds on all of \(D_{perf}(\mathcal{O}_X)\). The same argument works for all other properties, except that for property (7)(b) and (7)(c) we also use that \(P \mapsto P^\vee\) is a self equivalence of \(D_{perf}(\mathcal{O}_X)\). Small detail omitted.

We will prove the equivalent conditions (2) – (7) imply (1) using the induction principle of Lemma 08GP.

First, we prove (2) – (7) \(\Rightarrow\) (1) if \(X\) is affine. This follows from the case of schemes, see Derived Categories of Schemes, Proposition 0GEN.

Now assume \((U \subset X, j : V \to X)\) is an elementary distinguished square of quasi-compact and quasi-separated algebraic spaces over \(S\) and assume the implication (2) – (7) \(\Rightarrow\) (1) is known for \(U\), \(V\), and \(U \times_X V\). To finish the proof we have to show the implication (2) – (7) \(\Rightarrow\) (1) for \(X\). Suppose \(E \in D_\QCoh(\mathcal{O}_X)\) satisfies (2) – (7). By Lemma 09IX and Theorem 09IY there exists a perfect complex \(Q\) on \(X\) such that \(Q|_U\) generates \(D_\QCoh (\mathcal{O}_U)\).

Say \(V = \Spec(A)\). Let \(Z \subset V\) be the reduced closed subscheme which is the inverse image of \(X \setminus U\) and maps isomorphically to it (see Definition 08GM). This is a retrocompact closed subset of \(V\). Choose \(f_1, \ldots, f_r \in A\) such that \(Z = V(f_1, \ldots, f_r)\). Let \(K \in D(\mathcal{O}_V)\) be the perfect object corresponding to the Koszul complex on \(f_1, \ldots, f_r\) over \(A\). Note that since \(K\) is supported on \(Z\), the pushforward \(K' = Rj_*K\) is a perfect object of \(D(\mathcal{O}_X)\) whose restriction to \(V\) is \(K\) (see Lemmas 08HK and 08GG). By assumption, we know \(R\Hom_{\mathcal{O}_X}(Q, E)\) and \(R\Hom_{\mathcal{O}_X}(K', E)\) are bounded above.

By Lemma 08GG we have \(K' = j_!K\) and hence \[\Hom_{D(\mathcal{O}_X)}(K'[-i], E) = \Hom_{D(\mathcal{O}_V)}(K[-i], E|_V) = 0\] for \(i \gg 0\). Therefore, we may apply Derived Categories of Schemes, Lemma 0GEJ to \(E|_V\) to obtain an integer \(a\) such that \(\tau_{\geq a}(E|_V) = \tau_{\geq a} R (U \times_X V \to V)_* (E|_{U \times_X V})\). Then \(\tau_{\geq a} E = \tau_{\geq a} R (U \to X)_* (E |_U)\) (check that the canonical map is an isomorphism after restricting to \(U\) and to \(V\)). Hence using Lemma 0GFF twice we see that \[\Hom_{D(\mathcal{O}_U)}(Q|_U [-i], E|_U) = \Hom_{D(\mathcal{O}_X)}(Q[-i], R (U \to X)_* (E|_U)) = 0\] for \(i \gg 0\). Since the Proposition holds for \(U\) and the generator \(Q|_U\), we have \(E|_U \in D^-_\QCoh(\mathcal{O}_U)\). But then since the functor \(R (U \to X)_*\) preserves \(D^-_\QCoh\) (by Lemma 08FA), we get \(\tau_{\geq a}E \in D^-_\QCoh(\mathcal{O}_X)\). Thus \(E \in D^-_\QCoh (\mathcal{O}_X)\).

Proposition

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(G \in D_{perf}(\mathcal{O}_X)\) be a perfect complex which generates \(D_\QCoh (\mathcal{O}_X)\). Let \(E \in D_\QCoh (\mathcal{O}_X)\). The following are equivalent

1. \(E \in D^+_\QCoh (\mathcal{O}_X)\),

2. \(\Hom_{D(\mathcal{O}_X)}(G[-i], E) = 0\) for \(i \ll 0\),

3. \(\Ext^i_X(G, E) = 0\) for \(i \ll 0\),

4. \(R\Hom_X(G, E)\) is in \(D^+(\mathbf{Z})\),

5. \(H^i(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0\) for \(i \ll 0\),

6. \(R\Gamma(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E)\) is in \(D^+(\mathbf{Z})\),

7. for every perfect object \(P\) of \(D(\mathcal{O}_X)\)

   1. the assertions (2), (3), (4) hold with \(G\) replaced by \(P\), and

   2. \(H^i(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0\) for \(i \ll 0\),

   3. \(R\Gamma(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E)\) is in \(D^+(\mathbf{Z})\).

Proof

Assume (1). Since \(\Hom_{D(\mathcal{O}_X)}(G[-i], E) = \Hom_{D(\mathcal{O}_X)}(G, E[i])\) we see that this is zero for \(i \ll 0\) by Lemma 09MB. This proves that (1) implies (2).

Parts (2), (3), (4) are equivalent by the discussion in Cohomology on Sites, Section 0B6E. Part (5) and (6) are equivalent as \(H^i(X, -) = H^i(R\Gamma(X, -))\) by definition. The equivalent conditions (2), (3), (4) are equivalent to the equivalent conditions (5), (6) by Cohomology on Sites, Lemma 08JJ and the fact that \((G[-i])^\vee = G^\vee[i]\).

It is clear that (7) implies (2). Conversely, let us prove that the equivalent conditions (2) – (6) imply (7). Recall that \(G\) is a classical generator for \(D_{perf}(\mathcal{O}_X)\) by Remark 0GFC. For \(P \in D_{perf}(\mathcal{O}_X)\) let \(T(P)\) be the assertion that \(R\Hom_X(P, E)\) is in \(D^+(\mathbf{Z})\). Clearly, \(T\) is inherited by direct sums, satisfies the 2-out-of-three property for distinguished triangles, is inherited by direct summands, and is preserved by shifts. Hence by Derived Categories, Remark 0ATH we see that (4) implies \(T\) holds on all of \(D_{perf}(\mathcal{O}_X)\). The same argument works for all other properties, except that for property (7)(b) and (7)(c) we also use that \(P \mapsto P^\vee\) is a self equivalence of \(D_{perf}(\mathcal{O}_X)\). Small detail omitted.

We will prove the equivalent conditions (2) – (7) imply (1) using the induction principle of Lemma 08GP.

First, we prove (2) – (7) \(\Rightarrow\) (1) if \(X\) is affine. This follows from the case of schemes, see Derived Categories of Schemes, Proposition 0GEQ.

Now assume \((U \subset X, j : V \to X)\) is an elementary distinguished square of quasi-compact and quasi-separated algebraic spaces over \(S\) and assume the implication (2) – (7) \(\Rightarrow\) (1) is known for \(U\), \(V\), and \(U \times_X V\). To finish the proof we have to show the implication (2) – (7) \(\Rightarrow\) (1) for \(X\). Suppose \(E \in D_\QCoh(\mathcal{O}_X)\) satisfies (2) – (7). By Lemma 09IX and Theorem 09IY there exists a perfect complex \(Q\) on \(X\) such that \(Q|_U\) generates \(D_\QCoh (\mathcal{O}_U)\).

Say \(V = \Spec(A)\). Let \(Z \subset V\) be the reduced closed subscheme which is the inverse image of \(X \setminus U\) and maps isomorphically to it (see Definition 08GM). This is a retrocompact closed subset of \(V\). Choose \(f_1, \ldots, f_r \in A\) such that \(Z = V(f_1, \ldots, f_r)\). Let \(K \in D(\mathcal{O}_V)\) be the perfect object corresponding to the Koszul complex on \(f_1, \ldots, f_r\) over \(A\). Note that since \(K\) is supported on \(Z\), the pushforward \(K' = Rj_*K\) is a perfect object of \(D(\mathcal{O}_X)\) whose restriction to \(V\) is \(K\) (see Lemmas 08HK and 08GG). By assumption, we know \(R\Hom_{\mathcal{O}_X}(Q, E)\) and \(R\Hom_{\mathcal{O}_X}(K', E)\) are bounded below.

By Lemma 08GG we have \(K' = j_!K\) and hence \[\Hom_{D(\mathcal{O}_X)}(K'[-i], E) = \Hom_{D(\mathcal{O}_V)}(K[-i], E|_V) = 0\] for \(i \ll 0\). Therefore, we may apply Derived Categories of Schemes, Lemma 0GEK to \(E|_V\) to obtain an integer \(a\) such that \(\tau_{\leq a}(E|_V) = \tau_{\leq a} R (U \times_X V \to V)_* (E|_{U \times_X V})\). Then \(\tau_{\leq a} E = \tau_{\leq a} R (U \to X)_* (E |_U)\) (check that the canonical map is an isomorphism after restricting to \(U\) and to \(V\)). Hence using Lemma 0GFG twice we see that \[\Hom_{D(\mathcal{O}_U)}(Q|_U [-i], E|_U) = \Hom_{D(\mathcal{O}_X)}(Q[-i], R (U \to X)_* (E|_U)) = 0\] for \(i \ll 0\). Since the Proposition holds for \(U\) and the generator \(Q|_U\), we have \(E|_U \in D^+_\QCoh(\mathcal{O}_U)\). But then since the functor \(R (U \to X)_*\) preserves \(D^+_\QCoh\) (by Lemma 08FA), we get \(\tau_{\leq a}E \in D^+_\QCoh(\mathcal{O}_X)\). Thus \(E \in D^+_\QCoh (\mathcal{O}_X)\).

Quasi-coherent objects in the derived category

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Recall that \(X_{affine, \etale}\) denotes the category of affine objects of \(X_\etale\) with topology given by standard étale coverings, see Properties of Spaces, Definition 0H01. We remind the reader that the topos of \(X_{affine, \etale}\) is the small étale topos of \(X\), see Properties of Spaces, Lemma 04JS. The site \(X_\etale\) comes with a structure sheaf \(\mathcal{O}_X\) whose restriction to \(X_{affine, \etale}\) we also denote \(\mathcal{O}_X\). Then there is an equivalence of ringed topoi \[(\Sh(X_{affine, \etale}), \mathcal{O}_X) \longrightarrow (\Sh(X_\etale), \mathcal{O}_X)\] See Descent on Spaces, Equation (0H03) and the discussion in Descent on Spaces, Section 0H02.

In this section we denote \(X_{affine}\) the underlying category of \(X_{affine, \etale}\) endowed with the chaotic topology, i.e., such that sheaves agree with presheaves. In particular, the structure sheaf \(\mathcal{O}_X\) becomes a sheaf on \(X_{affine}\) as well. We obtain a morphisms of ringed sites \[\epsilon : (X_{affine, \etale}, \mathcal{O}_X) \longrightarrow (X_{affine}, \mathcal{O}_X)\] as in Cohomology on Sites, Section 0EWK. In this section we will identify \(D_\QCoh(\mathcal{O}_X)\) with the category \(\mathit{QC}(X_{affine}, \mathcal{O}_X)\) introduced in Cohomology on Sites, Section 0GYU.

Lemma

In the situation above there are canonical exact equivalences between the following triangulated categories

1. \(D_\QCoh(\mathcal{O}_X)\),

2. \(D_\QCoh(X_{affine, \etale}, \mathcal{O}_X)\),

3. \(D_\QCoh(X_{affine}, \mathcal{O}_X)\), and

4. \(\mathit{QC}(X_{affine}, \mathcal{O}_X)\).

Proof

If \(U \to V \to X\) are étale morphisms with \(U\) and \(V\) affine, then the ring map \(\mathcal{O}_X(V) \to \mathcal{O}_X(U)\) is flat. Hence the equivalence between (3) and (4) is a special case of Cohomology on Sites, Lemma 0GZR (the proof also clarifies the statement).

The discussion preceding the lemma shows that we have an equivalence of ringed topoi \((\Sh(X_{affine, \etale}), \mathcal{O}_X) \to (\Sh(X_\etale), \mathcal{O}_X)\) and hence an equivalence between abelian categories of modules. Since the notion of quasi-coherent modules is intrinsic (Modules on Sites, Lemma 03DM) we see that this equivalence preserves the subcategories of quasi-coherent modules. Thus we get a canonical exact equivalence between the triangulated categories in (1) and (2).

To get an exact equivalence between the triangulated categories in (2) and (3) we will apply Cohomology on Sites, Lemma 07A8 to the morphism \(\epsilon : (X_{affine, \etale}, \mathcal{O}_X) \to (X_{affine}, \mathcal{O}_X)\) above. We take \(\mathcal{B} = \Ob(X_{affine})\) and we take \(\mathcal{A} \subset \textit{PMod}(X_{affine}, \mathcal{O})\) to be the full subcategory of those presheaves \(\mathcal{F}\) such that \(\mathcal{F}(V) \otimes_{\mathcal{O}_X(V)} \mathcal{O}_X(U) \to \mathcal{F}(U)\) is an isomorphism. Observe that by Descent on Spaces, Lemma 0H04 objects of \(\mathcal{A}\) are exactly those sheaves in the étale topology which are quasi-coherent modules on \((X_{affine, \etale}, \mathcal{O}_X)\). On the other hand, by Modules on Sites, Lemma 0GZN, the objects of \(\mathcal{A}\) are exactly the quasi-coherent modules on \((X_{affine}, \mathcal{O}_X)\), i.e., in the chaotic topology. Thus if we show that Cohomology on Sites, Lemma 07A8 applies, then we do indeed get the canonical equivalence between the categories of (2) and (3) using \(\epsilon^*\) and \(R\epsilon_*\).

We have to verify 4 conditions:

1. Every object of \(\mathcal{A}\) is a sheaf for the étale topology. This we have seen above.

2. \(\mathcal{A}\) is a weak Serre subcategory of \(\textit{Mod}(X_{affine, \etale}, \mathcal{O}_X)\). Above we have seen that \(\mathcal{A} = \QCoh(X_{affine, \etale}, \mathcal{O}_X)\) and we have seen above that these, via the equivalence \(\textit{Mod}(X_{affine, \etale}, \mathcal{O}) = \textit{Mod}(X_\etale, \mathcal{O}_X)\), correspond to the quasi-coherent modules on \(X\). Thus the result by Properties of Spaces, Lemma 03M1 and Homology, Lemma 0754.

3. Every object of \(X_{affine}\) has a covering in the chaotic topology whose members are elements of \(\mathcal{B}\). This holds because \(\mathcal{B}\) contains all objects.

4. For every object \(U\) of \(X_{affine}\) and \(\mathcal{F}\) in \(\mathcal{A}\) we have \(H^p_{Zar}(U, \mathcal{F}) = 0\) for \(p > 0\). This holds by the vanishing of cohomology of quasi-coherent modules on affines, see discussion in Cohomology of Spaces, Section 071Y and Cohomology of Schemes, Lemma 01XB.

This finishes the proof.

Remark

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). We will later show that also \(\mathit{QC}((\textit{Aff}/X), \mathcal{O})\) is canonically equivalent to \(D_\QCoh(\mathcal{O}_X)\). See Sheaves on Stacks, Proposition 0H0K.