Cohomology of Algebraic Spaces

Introduction

In this chapter we write about cohomology of algebraic spaces. Although we prove some results on cohomology of abelian sheaves, we focus mainly on cohomology of quasi-coherent sheaves, i.e., we prove analogues of the results in the chapter “Cohomology of Schemes”. Some of the results in this chapter can be found in [Kn].

An important missing ingredient in this chapter is the induction principle, i.e., the analogue for quasi-compact and quasi-separated algebraic spaces of Cohomology of Schemes, Lemma 08DR. This is formulated precisely and proved in detail in Derived Categories of Spaces, Section 08GL. Instead of the induction principle, in this chapter we use the alternating Čech complex, see Section 0721. It is designed to prove vanishing statements such as Proposition 072B, but in some cases the induction principle is a more powerful and perhaps more “standard” tool. We encourage the reader to take a look at the induction principle after reading some of the material in this section.

Conventions

The standing assumption is that all schemes are contained in a big fppf site \(\Sch_{fppf}\). And all rings \(A\) considered have the property that \(\Spec(A)\) is (isomorphic) to an object of this big site.

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). In this chapter and the following we will write \(X \times_S X\) for the product of \(X\) with itself (in the category of algebraic spaces over \(S\)), instead of \(X \times X\).

Higher direct images

Let \(S\) be a scheme. Let \(X\) be a representable algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent module on \(X\) (see Properties of Spaces, Section 03G5). By Descent, Proposition 03DW the cohomology groups \(H^i(X, \mathcal{F})\) agree with the usual cohomology group computed in the Zariski topology of the corresponding quasi-coherent module on the scheme representing \(X\).

More generally, let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of representable algebraic spaces \(X\) and \(Y\). Let \(\mathcal{F}\) be a quasi-coherent module on \(X\). By Descent, Lemma 071N the sheaf \(R^if_*\mathcal{F}\) agrees with the usual higher direct image computed for the Zariski topology of the quasi-coherent module on the scheme representing \(X\) mapping to the scheme representing \(Y\).

More generally still, suppose \(f : X \to Y\) is a representable, quasi-compact, and quasi-separated morphism of algebraic spaces over \(S\). Let \(V\) be a scheme and let \(V \to Y\) be an étale surjective morphism. Let \(U = V \times_Y X\) and let \(f' : U \to V\) be the base change of \(f\). Then for any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have [071Z]\[\begin{equation} R^if'_*(\mathcal{F}|_U) = (R^if_*\mathcal{F})|_V, \end{equation}\] see Properties of Spaces, Lemma 03LX. And because \(f' : U \to V\) is a quasi-compact and quasi-separated morphism of schemes, by the remark of the preceding paragraph we may compute \(R^if'_*(\mathcal{F}|_U)\) by thinking of \(\mathcal{F}|_U\) as a quasi-coherent sheaf on the scheme \(U\), and \(f'\) as a morphism of schemes. We will frequently use this without further mention.

Next, we prove that higher direct images of quasi-coherent sheaves are quasi-coherent for any quasi-compact and quasi-separated morphism of algebraic spaces. In the proof we use a trick; a “better” proof would use a relative Čech complex, as discussed in Sheaves on Stacks, Sections 06X3 and 06X7 ff.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is quasi-compact and quasi-separated, then \(R^if_*\) transforms quasi-coherent \(\mathcal{O}_X\)-modules into quasi-coherent \(\mathcal{O}_Y\)-modules.

Proof

Let \(V \to Y\) be an étale morphism where \(V\) is an affine scheme. Set \(U = V \times_Y X\) and denote \(f' : U \to V\) the induced morphism. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. By Properties of Spaces, Lemma 03LX we have \(R^if'_*(\mathcal{F}|_U) = (R^if_*\mathcal{F})|_V\). Since the property of being a quasi-coherent module is local in the étale topology on \(Y\) (see Properties of Spaces, Lemma 03M0) we may replace \(Y\) by \(V\), i.e., we may assume \(Y\) is an affine scheme.

Assume \(Y\) is affine. Since \(f\) is quasi-compact we see that \(X\) is quasi-compact. Thus we may choose an affine scheme \(U\) and a surjective étale morphism \(g : U \to X\), see Properties of Spaces, Lemma 03H6. Picture \[\xymatrix{ U \ar[r]_g \ar[rd]_{f \circ g} & X \ar[d]^f \\ & Y }\] The morphism \(g : U \to X\) is representable, separated and quasi-compact because \(X\) is quasi-separated. Hence the lemma holds for \(g\) (by the discussion above the lemma). It also holds for \(f \circ g : U \to Y\) (as this is a morphism of affine schemes).

In the situation described in the previous paragraph we will show by induction on \(n\) that \(IH_n\): for any quasi-coherent sheaf \(\mathcal{F}\) on \(X\) the sheaves \(R^if\mathcal{F}\) are quasi-coherent for \(i \leq n\). The case \(n = 0\) follows from Morphisms of Spaces, Lemma 03M9. Assume \(IH_n\). In the rest of the proof we show that \(IH_{n + 1}\) holds.

Let \(\mathcal{H}\) be a quasi-coherent \(\mathcal{O}_U\)-module. Consider the Leray spectral sequence \[E_2^{p, q} = R^pf_* R^qg_* \mathcal{H} \Rightarrow R^{p + q}(f \circ g)_*\mathcal{H}\] Cohomology on Sites, Lemma 0734. As \(R^qg_*\mathcal{H}\) is quasi-coherent by \(IH_n\) all the sheaves \(R^pf_*R^qg_*\mathcal{H}\) are quasi-coherent for \(p \leq n\). The sheaves \(R^{p + q}(f \circ g)_*\mathcal{H}\) are all quasi-coherent (in fact zero for \(p + q > 0\) but we do not need this). Looking in degrees \(\leq n + 1\) the only module which we do not yet know is quasi-coherent is \(E_2^{n + 1, 0} = R^{n + 1}f_*g_*\mathcal{H}\). Moreover, the differentials \(d_r^{n + 1, 0} : E_r^{n + 1, 0} \to E_r^{n + 1 + r, 1 - r}\) are zero as the target is zero. Using that \(\QCoh(\mathcal{O}_X)\) is a weak Serre subcategory of \(\textit{Mod}(\mathcal{O}_X)\) (Properties of Spaces, Lemma 03M1) it follows that \(R^{n + 1}f_*g_*\mathcal{H}\) is quasi-coherent (details omitted).

Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Set \(\mathcal{H} = g^*\mathcal{F}\). The adjunction mapping \(\mathcal{F} \to g_*g^*\mathcal{F} = g_*\mathcal{H}\) is injective as \(U \to X\) is surjective étale. Consider the exact sequence \[0 \to \mathcal{F} \to g_*\mathcal{H} \to \mathcal{G} \to 0\] where \(\mathcal{G}\) is the cokernel of the first map and in particular quasi-coherent. Applying the long exact cohomology sequence we obtain \[R^nf_*g_*\mathcal{H} \to R^nf_*\mathcal{G} \to R^{n + 1}f_*\mathcal{F} \to R^{n + 1}f_*g_*\mathcal{H} \to R^{n + 1}f_*\mathcal{G}\] The cokernel of the first arrow is quasi-coherent and we have seen above that \(R^{n + 1}f_*g_*\mathcal{H}\) is quasi-coherent. Thus \(R^{n + 1}f_*\mathcal{F}\) has a \(2\)-step filtration where the first step is quasi-coherent and the second a submodule of a quasi-coherent sheaf. Since \(\mathcal{F}\) is an arbitrary quasi-coherent \(\mathcal{O}_X\)-module, this result also holds for \(\mathcal{G}\). Thus we can choose an exact sequence \(0 \to \mathcal{A} \to R^{n + 1}f_*\mathcal{G} \to \mathcal{B}\) with \(\mathcal{A}\), \(\mathcal{B}\) quasi-coherent \(\mathcal{O}_Y\)-modules. Then the kernel \(\mathcal{K}\) of \(R^{n + 1}f_*g_*\mathcal{H} \to R^{n + 1}f_*\mathcal{G} \to \mathcal{B}\) is quasi-coherent, whereupon we obtain a map \(\mathcal{K} \to \mathcal{A}\) whose kernel \(\mathcal{K}'\) is quasi-coherent too. Hence \(R^{n + 1}f_*\mathcal{F}\) sits in an exact sequence \[R^nf_*g_*\mathcal{H} \to R^nf_*\mathcal{G} \to R^{n + 1}f_*\mathcal{F} \to \mathcal{K}' \to 0\] with all modules quasi-coherent except for possibly \(R^{n + 1}f_*\mathcal{F}\). We conclude that \(R^{n + 1}f_*\mathcal{F}\) is quasi-coherent, i.e., \(IH_{n + 1}\) holds as desired.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-separated and quasi-compact morphism of algebraic spaces over \(S\). For any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) and any affine object \(V\) of \(Y_\etale\) we have \[H^q(V \times_Y X, \mathcal{F}) = H^0(V, R^qf_*\mathcal{F})\] for all \(q \in \mathbf{Z}\).

Proof

Since formation of \(Rf_*\) commutes with étale localization (Properties of Spaces, Lemma 03LX) we may replace \(Y\) by \(V\) and assume \(Y = V\) is affine. Consider the Leray spectral sequence \(E_2^{p, q} = H^p(Y, R^qf_*\mathcal{F})\) converging to \(H^{p + q}(X, \mathcal{F})\), see Cohomology on Sites, Lemma 0732. By Lemma 0720 we see that the sheaves \(R^qf_*\mathcal{F}\) are quasi-coherent. By Cohomology of Schemes, Lemma 01XB we see that \(E_2^{p, q} = 0\) when \(p > 0\). Hence the spectral sequence degenerates at \(E_2\) and we win.

Finite morphisms

Here are some results which hold for all abelian sheaves (in particular also quasi-coherent modules). We warn the reader that these lemmas do not hold for finite morphisms of schemes and the Zariski topology.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an integral (for example finite) morphism of algebraic spaces. Then \(f_* : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)\) is an exact functor and \(R^pf_* = 0\) for \(p > 0\).

Proof

By Properties of Spaces, Lemma 03LR we may compute the higher direct images on an étale cover of \(Y\). Hence we may assume \(Y\) is a scheme. This implies that \(X\) is a scheme (Morphisms of Spaces, Lemma 03ZQ). In this case we may apply Étale Cohomology, Lemma 04C2. For the finite case the reader may wish to consult the less technical Étale Cohomology, Proposition 03QP.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a finite morphism of algebraic spaces over \(S\). Let \(\overline{y}\) be a geometric point of \(Y\) with lifts \(\overline{x}_1, \ldots, \overline{x}_n\) in \(X\). Then \[(f_*\mathcal{F})_{\overline{y}} = \prod\nolimits_{i = 1, \ldots, n} \mathcal{F}_{\overline{x}_i}\] for any sheaf \(\mathcal{F}\) on \(X_\etale\).

Proof

Choose an étale neighbourhood \((V, \overline{v})\) of \(\overline{y}\). Then the stalk \((f_*\mathcal{F})_{\overline{y}}\) is the stalk of \(f_*\mathcal{F}|_V\) at \(\overline{v}\). By Properties of Spaces, Lemma 03LR we may replace \(Y\) by \(V\) and \(X\) by \(X \times_Y V\). Then \(X \to Y\) is a finite morphism of schemes and the result is Étale Cohomology, Proposition 03QP.

Lemma

Let \(S\) be a scheme. Let \(\pi : X \to Y\) be a finite morphism of algebraic spaces over \(S\). Let \(\mathcal{A}\) be a sheaf of rings on \(X_\etale\). Let \(\mathcal{B}\) be a sheaf of rings on \(Y_\etale\). Let \(\varphi : \mathcal{B} \to \pi_*\mathcal{A}\) be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi \[f = (\pi, \varphi) : (\Sh(X_\etale), \mathcal{A}) \longrightarrow (\Sh(Y_\etale), \mathcal{B}).\] For a sheaf of \(\mathcal{A}\)-modules \(\mathcal{F}\) and a sheaf of \(\mathcal{B}\)-modules \(\mathcal{G}\) the canonical map \[\mathcal{G} \otimes_\mathcal{B} f_*\mathcal{F} \longrightarrow f_*(f^*\mathcal{G} \otimes_\mathcal{A} \mathcal{F}).\] is an isomorphism.

Proof

The map is the map adjoint to the map \[f^*\mathcal{G} \otimes_\mathcal{A} f^* f_*\mathcal{F} = f^*(\mathcal{G} \otimes_\mathcal{B} f_*\mathcal{F}) \longrightarrow f^*\mathcal{G} \otimes_\mathcal{A} \mathcal{F}\] coming from \(\text{id} : f^*\mathcal{G} \to f^*\mathcal{G}\) and the adjunction map \(f^* f_*\mathcal{F} \to \mathcal{F}\). To see this map is an isomorphism, we may check on stalks (Properties of Spaces, Theorem 04K5). Let \(\overline{y}\) be a geometric point of \(Y\) and let \(\overline{x}_1, \ldots, \overline{x}_n\) be the geometric points of \(X\) lying over \(\overline{y}\). Working out what our maps does on stalks, we see that we have to show \[\mathcal{G}_{\overline{y}} \otimes_{\mathcal{B}_{\overline{y}}} \left( \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{F}_{\overline{x}_i} \right) = \bigoplus\nolimits_{i = 1, \ldots, n} (\mathcal{G}_{\overline{y}} \otimes_{\mathcal{B}_{\overline{y}}} \mathcal{A}_{\overline{x}_i}) \otimes_{\mathcal{A}_{\overline{x}_i}} \mathcal{F}_{\overline{x}_i}\] which holds true. Here we have used that taking tensor products commutes with taking stalks, the behaviour of stalks under pullback Properties of Spaces, Lemma 04K2, and the behaviour of stalks under pushforward along a finite morphism Lemma 0DK3.

We end this section with an insanely general projection formula for finite morphisms.

Lemma

With \(S\), \(X\), \(Y\), \(\pi\), \(\mathcal{A}\), \(\mathcal{B}\), \(\varphi\), and \(f\) as in Lemma 0DK4 we have \[K \otimes_\mathcal{B}^\mathbf{L} Rf_*M = Rf_*(Lf^*K \otimes_\mathcal{A}^\mathbf{L} M)\] in \(D(\mathcal{B})\) for any \(K \in D(\mathcal{B})\) and \(M \in D(\mathcal{A})\).

Proof

Since \(f_*\) is exact (Lemma 0A4K) the functor \(Rf_*\) is computed by applying \(f_*\) to any representative complex. Choose a complex \(\mathcal{K}^\bullet\) of \(\mathcal{B}\)-modules representing \(K\) which is K-flat with flat terms, see Cohomology on Sites, Lemma 06YS. Then \(f^*\mathcal{K}^\bullet\) is K-flat with flat terms, see Cohomology on Sites, Lemma 0G7E. Choose any complex \(\mathcal{M}^\bullet\) of \(\mathcal{A}\)-modules representing \(M\). Then we have to show \[\text{Tot}(\mathcal{K}^\bullet \otimes_\mathcal{B} f_*\mathcal{M}^\bullet) = f_*\text{Tot}(f^*\mathcal{K}^\bullet \otimes_\mathcal{A} \mathcal{M}^\bullet)\] because by our choices these complexes represent the right and left hand side of the formula in the lemma. Since \(f_*\) commutes with direct sums (for example by the description of the stalks in Lemma 0DK3), this reduces to the equalities \[\mathcal{K}^n \otimes_\mathcal{B} f_*\mathcal{M}^m = f_*(f^*\mathcal{K}^n \otimes_\mathcal{A} \mathcal{M}^m)\] which are true by Lemma 0DK4.

Colimits and cohomology

The following lemma in particular applies to diagrams of quasi-coherent sheaves.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). If \(X\) is quasi-compact and quasi-separated, then \[\colim_i H^p(X, \mathcal{F}_i) \longrightarrow H^p(X, \colim_i \mathcal{F}_i)\] is an isomorphism for every filtered diagram of abelian sheaves on \(X_\etale\).

Proof

This follows from Cohomology on Sites, Lemma 0739. Namely, let \(\mathcal{B} \subset \Ob(X_{spaces, \etale})\) be the set of quasi-compact and quasi-separated spaces étale over \(X\). Note that if \(U \in \mathcal{B}\) then, because \(U\) is quasi-compact, the collection of finite coverings \(\{U_i \to U\}\) with \(U_i \in \mathcal{B}\) is cofinal in the set of coverings of \(U\) in \(X_{spaces, \etale}\). By Morphisms of Spaces, Lemma 073B the set \(\mathcal{B}\) satisfies all the assumptions of Cohomology on Sites, Lemma 0739. Since \(X \in \mathcal{B}\) we win.

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 \(\mathcal{F} = \colim \mathcal{F}_i\) be a filtered colimit of abelian sheaves on \(X_\etale\). Then for any \(p \geq 0\) we have \[R^pf_*\mathcal{F} = \colim R^pf_*\mathcal{F}_i.\]

Proof

We will use that the morphism of topoi \(f_{small} : X_{small} \to Y_{small}\) comes from the morphism of sites \(f_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}\) corresponding to the continuous functor \(V \longmapsto X \times_Y V\), see Properties of Spaces, Lemma 03G2. We will apply Cohomology on Sites, Lemma 0H7B to this morphism of sites. Since every object of \(Y_{spaces, \etale}\) has a covering by affine objects, it suffices to show that for \(V\) affine and étale over \(Y\) we have \(H^p(X \times_Y V, \mathcal{F}) = \colim H^p(X \times_Y V, \mathcal{F}_i)\). Since \(V\) is affine, the algebraic space \(X \times_Y V\) is quasi-compact and quasi-separated. Hence we can apply Lemma 073E to conclude.

The following lemma tells us that finitely presented modules behave as expected in quasi-compact and quasi-separated algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(I\) be a directed set and let \((\mathcal{F}_i, \varphi_{ii'})\) be a system over \(I\) of \(\mathcal{O}_X\)-modules. Let \(\mathcal{G}\) be an \(\mathcal{O}_X\)-module of finite presentation. Then we have \[\colim_i \Hom_X(\mathcal{G}, \mathcal{F}_i) = \Hom_X(\mathcal{G}, \colim_i \mathcal{F}_i).\] In particular, \(\Hom_X(\mathcal{G}, -)\) commutes with filtered colimits in \(\QCoh(\mathcal{O}_X)\).

Proof

The displayed equality is a special case of Modules on Sites, Lemma 0GN0. In order to apply it, we need to check the hypotheses of Sites, Lemma 0GMR part (4) for the site \(X_\etale\). In order to do this, we will check hypotheses (2)(a), (2)(b), (2)(c) of Sites, Remark 0GMS. Namely, let \(\mathcal{B} \subset \Ob(X_\etale)\) be the set of affine objects. Then

1. Since \(X\) is quasi-compact, there exists a \(U \in \mathcal{B}\) such that \(U \to X\) is surjective (Properties of Spaces, Lemma 03H6), hence \(h_U^\# \to *\) is surjective.

2. For \(U \in \mathcal{B}\) every étale covering \(\{U_i \to U\}_{i \in I}\) of \(U\) can be refined by a finite étale covering \(\{U_j \to U\}_{j = 1, \ldots, m}\) with \(U_j \in \mathcal{B}\) (Topologies, Lemma 0218).

3. For \(U, U' \in \Ob(X_\etale)\) we have \(h_U^\# \times h_{U'}^\# = h_{U \times_X U'}^\#\). If \(U, U' \in \mathcal{B}\), then \(U \times_X U'\) is quasi-compact because \(X\) is quasi-separated, see Morphisms of Spaces, Lemma 073B for example. Hence we can find a surjective étale morphism \(U'' \to U \times_X U'\) with \(U'' \in \mathcal{B}\) (Properties of Spaces, Lemma 03H6). In other words, we have morphisms \(U'' \to U\) and \(U'' \to U'\) such that the map \(h_{U''}^\# \to h_U^\# \times h_{U'}^\#\) is surjective.

For the final statement, observe that the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) commutes with colimits and that finitely presented modules are quasi-coherent. See Properties of Spaces, Lemma 03M1.

The alternating Čech complex

Let \(S\) be a scheme. Let \(f : U \to X\) be an étale morphism of algebraic spaces over \(S\). The functor \[j : U_{spaces, \etale} \longrightarrow X_{spaces, \etale},\quad V/U \longmapsto V/X\] induces an equivalence of \(U_{spaces, \etale}\) with the localization \(X_{spaces, \etale}/U\), see Properties of Spaces, Section 04LX. Hence there exist functors \[f_! : \textit{Ab}(U_\etale) \longrightarrow \textit{Ab}(X_\etale),\quad f_! : \textit{Mod}(\mathcal{O}_U) \longrightarrow \textit{Mod}(\mathcal{O}_X),\] which are left adjoint to \[f^{-1} : \textit{Ab}(X_\etale) \longrightarrow \textit{Ab}(U_\etale),\quad f^* : \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_U)\] see Modules on Sites, Section 03DH. Warning: This functor, a priori, has nothing to do with cohomology with compact supports! We dubbed this functor “extension by zero” in the reference above. Note that the two versions of \(f_!\) agree as \(f^* = f^{-1}\) for sheaves of \(\mathcal{O}_X\)-modules.

As we are going to use this construction below let us recall some of its properties. Given an abelian sheaf \(\mathcal{G}\) on \(U_\etale\) the sheaf \(f_!\mathcal{G}\) is the sheafification of the presheaf \[V/X \longmapsto f_!\mathcal{G}(V) = \bigoplus\nolimits_{\varphi \in \Mor_X(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U),\] see Modules on Sites, Lemma 03DI. Moreover, if \(\mathcal{G}\) is an \(\mathcal{O}_U\)-module, then \(f_!\mathcal{G}\) is the sheafification of the exact same presheaf of abelian groups which is endowed with an \(\mathcal{O}_X\)-module structure in an obvious way (see loc. cit.). Let \(\overline{x} : \Spec(k) \to X\) be a geometric point. Then there is a canonical identification \[(f_!\mathcal{G})_{\overline{x}} = \bigoplus\nolimits_{\overline{u}} \mathcal{G}_{\overline{u}}\] where the sum is over all \(\overline{u} : \Spec(k) \to U\) such that \(f \circ \overline{u} = \overline{x}\), see Modules on Sites, Lemma 0710 and Properties of Spaces, Lemma 04K6. In the following we are going to study the sheaf \(f_!\underline{\mathbf{Z}}\). Here \(\underline{\mathbf{Z}}\) denotes the constant sheaf on \(X_\etale\) or \(U_\etale\).

Lemma

Let \(S\) be a scheme. Let \(f_i : U_i \to X\) be étale morphisms of algebraic spaces over \(S\). Then there are isomorphisms \[f_{1, !}\underline{\mathbf{Z}} \otimes_{\mathbf{Z}} f_{2, !}\underline{\mathbf{Z}} \longrightarrow f_{12, !}\underline{\mathbf{Z}}\] where \(f_{12} : U_1 \times_X U_2 \to X\) is the structure morphism and \[(f_1 \amalg f_2)_! \underline{\mathbf{Z}} \longrightarrow f_{1, !}\underline{\mathbf{Z}} \oplus f_{2, !}\underline{\mathbf{Z}}\]

Proof

Once we have defined the map it will be an isomorphism by our description of stalks above. To define the map it suffices to work on the level of presheaves. Thus we have to define a map \[\left(\bigoplus\nolimits_{\varphi_1 \in \Mor_X(V, U_1)} \mathbf{Z}\right) \otimes_{\mathbf{Z}} \left(\bigoplus\nolimits_{\varphi_2 \in \Mor_X(V, U_2)} \mathbf{Z}\right) \longrightarrow \bigoplus\nolimits_{\varphi \in \Mor_X(V, U_1 \times_X U_2)} \mathbf{Z}\] We map the element \(1_{\varphi_1} \otimes 1_{\varphi_2}\) to the element \(1_{\varphi_1 \times \varphi_2}\) with obvious notation. We omit the proof of the second equality.

Another important feature is the trace map \[\text{Tr}_f : f_!\underline{\mathbf{Z}} \longrightarrow \underline{\mathbf{Z}}.\] The trace map is adjoint to the map \(\mathbf{Z} \to f^{-1}\underline{\mathbf{Z}}\) (which is an isomorphism). If \(\overline{x}\) is above, then \(\text{Tr}_f\) on stalks at \(\overline{x}\) is the map \[(\text{Tr}_f)_{\overline{x}} : (f_!\underline{\mathbf{Z}})_{\overline{x}} = \bigoplus\nolimits_{\overline{u}} \mathbf{Z} \longrightarrow \mathbf{Z} = \underline{\mathbf{Z}}_{\overline{x}}\] which sums the given integers. This is true because it is adjoint to the map \(1 : \mathbf{Z} \to f^{-1}\underline{\mathbf{Z}}\). In particular, if \(f\) is surjective as well as étale then \(\text{Tr}_f\) is surjective.

Assume that \(f : U \to X\) is a surjective étale morphism of algebraic spaces. Consider the Koszul complex associated to the trace map we discussed above \[\ldots \to \wedge^3f_!\underline{\mathbf{Z}} \to \wedge^2f_!\underline{\mathbf{Z}} \to f_!\underline{\mathbf{Z}} \to \underline{\mathbf{Z}} \to 0\] Here the exterior powers are over the sheaf of rings \(\underline{\mathbf{Z}}\). The maps are defined by the rule \[e_1 \wedge \ldots \wedge e_n \longmapsto \sum\nolimits_{i = 1, \ldots, n} (-1)^{i + 1} \text{Tr}_f(e_i) e_1 \wedge \ldots \wedge \widehat{e_i} \wedge \ldots \wedge e_n\] where \(e_1, \ldots, e_n\) are local sections of \(f_!\underline{\mathbf{Z}}\). Let \(\overline{x}\) be a geometric point of \(X\) and set \(M_{\overline{x}} = (f_!\underline{\mathbf{Z}})_{\overline{x}} = \bigoplus_{\overline{u}} \mathbf{Z}\). Then the stalk of the complex above at \(\overline{x}\) is the complex \[\ldots \to \wedge^3 M_{\overline{x}} \to \wedge^2 M_{\overline{x}} \to M_{\overline{x}} \to \mathbf{Z} \to 0\] which is exact because \(M_{\overline{x}} \to \mathbf{Z}\) is surjective, see More on Algebra, Lemma 0626. Hence if we let \(K^\bullet = K^\bullet(f)\) be the complex with \(K^i = \wedge^{i + 1}f_!\underline{\mathbf{Z}}\), then we obtain a quasi-isomorphism [0723]\[\begin{equation} K^\bullet \longrightarrow \underline{\mathbf{Z}}[0] \end{equation}\] We use the complex \(K^\bullet\) to define what we call the alternating Čech complex associated to \(f : U \to X\).

Definition

Let \(S\) be a scheme. Let \(f : U \to X\) be a surjective étale morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be an object of \(\textit{Ab}(X_\etale)\). The alternating Čech complex¹ \(\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F})\) associated to \(\mathcal{F}\) and \(f\) is the complex \[\Hom(K^0, \mathcal{F}) \to \Hom(K^1, \mathcal{F}) \to \Hom(K^2, \mathcal{F}) \to \ldots\] with Hom groups computed in \(\textit{Ab}(X_\etale)\).

The reader may verify that if \(U = \coprod U_i\) and \(f|_{U_i} : U_i \to X\) is the open immersion of a subspace, then \(\check{\mathcal{C}}_{alt}^\bullet(f, \mathcal{F})\) agrees with the complex introduced in Cohomology, Section 01FG for the Zariski covering \(X = \bigcup U_i\) and the restriction of \(\mathcal{F}\) to the Zariski site of \(X\). What is more important however, is to relate the cohomology of the alternating Čech complex to the cohomology.

Lemma

Let \(S\) be a scheme. Let \(f : U \to X\) be a surjective étale morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be an object of \(\textit{Ab}(X_\etale)\). There exists a canonical map \[\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F}) \longrightarrow R\Gamma(X, \mathcal{F})\] in \(D(\textit{Ab})\). Moreover, there is a spectral sequence with \(E_1\)-page \[E_1^{p, q} = \Ext_{\textit{Ab}(X_\etale)}^q(K^p, \mathcal{F})\] converging to \(H^{p + q}(X, \mathcal{F})\) where \(K^p = \wedge^{p + 1}f_!\underline{\mathbf{Z}}\).

Proof

Recall that we have the quasi-isomorphism \(K^\bullet \to \underline{\mathbf{Z}}[0]\), see (0723). Choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\) in \(\textit{Ab}(X_\etale)\). Consider the double complex \(\Hom(K^\bullet, \mathcal{I}^\bullet)\) with terms \(\Hom(K^p, \mathcal{I}^q)\). The differential \(d_1^{p, q} : A^{p, q} \to A^{p + 1, q}\) is the one coming from the differential \(K^{p + 1} \to K^p\) and the differential \(d_2^{p, q} : A^{p, q} \to A^{p, q + 1}\) is the one coming from the differential \(\mathcal{I}^q \to \mathcal{I}^{q + 1}\). Denote \(\text{Tot}(\Hom(K^\bullet, \mathcal{I}^\bullet))\) the associated total complex, see Homology, Section 0FNB. We will use the two spectral sequences \(({}'E_r, {}'d_r)\) and \(({}''E_r, {}''d_r)\) associated to this double complex, see Homology, Section 012X.

Because \(K^\bullet\) is a resolution of \(\underline{\mathbf{Z}}\) we see that the complexes \[\Hom(K^\bullet, \mathcal{I}^q) : \Hom(K^0, \mathcal{I}^q) \to \Hom(K^1, \mathcal{I}^q) \to \Hom(K^2, \mathcal{I}^q) \to \ldots\] are acyclic in positive degrees and have \(H^0\) equal to \(\Gamma(X, \mathcal{I}^q)\). Hence by Homology, Lemma 0133 the natural map \[\mathcal{I}^\bullet(X) \longrightarrow \text{Tot}(\Hom(K^\bullet, \mathcal{I}^\bullet))\] is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that \(H^n(\text{Tot}(\Hom(K^\bullet, \mathcal{I}^\bullet))) = H^n(X, \mathcal{F})\).

The map \(\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F}) \to R\Gamma(X, \mathcal{F})\) of the lemma is the composition of \(\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F}) \to \text{Tot}(\Hom(K^\bullet, \mathcal{I}^\bullet))\) with the inverse of the displayed quasi-isomorphism.

Finally, consider the spectral sequence \(({}'E_r, {}'d_r)\). We have \[E_1^{p, q} = q\text{th cohomology of } \Hom(K^p, \mathcal{I}^0) \to \Hom(K^p, \mathcal{I}^1) \to \Hom(K^p, \mathcal{I}^2) \to \ldots\] This proves the lemma.

It follows from the lemma that it is important to understand the ext groups \(\Ext_{\textit{Ab}(X_\etale)}(K^p, \mathcal{F})\), i.e., the right derived functors of \(\mathcal{F} \mapsto \Hom(K^p, \mathcal{F})\).

Lemma

Let \(S\) be a scheme. Let \(f : U \to X\) be a surjective, étale, and separated morphism of algebraic spaces over \(S\). For \(p \geq 0\) set \[W_p = U \times_X \ldots \times_X U \setminus \text{all diagonals}\] where the fibre product has \(p + 1\) factors. There is a free action of \(S_{p + 1}\) on \(W_p\) over \(X\) and \[\Hom(K^p, \mathcal{F}) = S_{p + 1}\text{-anti-invariant elements of } \mathcal{F}(W_p)\] functorially in \(\mathcal{F}\) where \(K^p = \wedge^{p + 1}f_!\underline{\mathbf{Z}}\).

Proof

Because \(U \to X\) is separated the diagonal \(U \to U \times_X U\) is a closed immersion. Since \(U \to X\) is étale the diagonal \(U \to U \times_X U\) is an open immersion, see Morphisms of Spaces, Lemmas 06CR and 05W1. Hence \(W_p\) is an open and closed subspace of \(U^{p + 1} = U \times_X \ldots \times_X U\). The action of \(S_{p + 1}\) on \(W_p\) is free as we’ve thrown out the fixed points of the action. By Lemma 0722 we see that \[(f_!\underline{\mathbf{Z}})^{\otimes p + 1} = f^{p + 1}_!\underline{\mathbf{Z}} = (W_p \to X)_!\underline{\mathbf{Z}} \oplus Rest\] where \(f^{p + 1} : U^{p + 1} \to X\) is the structure morphism. Looking at stalks over a geometric point \(\overline{x}\) of \(X\) we see that \[\left( \bigoplus\nolimits_{\overline{u} \mapsto \overline{x}} \mathbf{Z} \right)^{\otimes p + 1} \longrightarrow (W_p \to X)_!\underline{\mathbf{Z}}_{\overline{x}}\] is the quotient whose kernel is generated by all tensors \(1_{\overline{u}_0} \otimes \ldots \otimes 1_{\overline{u}_p}\) where \(\overline{u}_i = \overline{u}_j\) for some \(i \not = j\). Thus the quotient map \[(f_!\underline{\mathbf{Z}})^{\otimes p + 1} \longrightarrow \wedge^{p + 1}f_!\underline{\mathbf{Z}}\] factors through \((W_p \to X)_!\underline{\mathbf{Z}}\), i.e., we get \[(f_!\underline{\mathbf{Z}})^{\otimes p + 1} \longrightarrow (W_p \to X)_!\underline{\mathbf{Z}} \longrightarrow \wedge^{p + 1}f_!\underline{\mathbf{Z}}\] This already proves that \(\Hom(K^p, \mathcal{F})\) is (functorially) a subgroup of \[\Hom((W_p \to X)_!\underline{\mathbf{Z}}, \mathcal{F}) = \mathcal{F}(W_p)\] To identify it with the \(S_{p + 1}\)-anti-invariants we have to prove that the surjection \((W_p \to X)_!\underline{\mathbf{Z}} \to \wedge^{p + 1}f_!\underline{\mathbf{Z}}\) is the maximal \(S_{p + 1}\)-anti-invariant quotient. In other words, we have to show that \(\wedge^{p + 1}f_!\underline{\mathbf{Z}}\) is the quotient of \((W_p \to X)_!\underline{\mathbf{Z}}\) by the subsheaf generated by the local sections \(s - \text{sign}(\sigma)\sigma(s)\) where \(s\) is a local section of \((W_p \to X)_!\underline{\mathbf{Z}}\). This can be checked on the stalks, where it is clear.

Lemma

Let \(S\) be a scheme. Let \(W\) be an algebraic space over \(S\). Let \(G\) be a finite group acting freely on \(W\). Let \(U = W/G\), see Properties of Spaces, Lemma 071S. Let \(\chi : G \to \{+1, -1\}\) be a character. Then there exists a rank 1 locally free sheaf of \(\mathbf{Z}\)-modules \(\underline{\mathbf{Z}}(\chi)\) on \(U_\etale\) such that for every abelian sheaf \(\mathcal{F}\) on \(U_\etale\) we have \[H^0(W, \mathcal{F}|_W)^\chi = H^0(U, \mathcal{F} \otimes_{\mathbf{Z}} \underline{\mathbf{Z}}(\chi))\]

Proof

The quotient morphism \(q : W \to U\) is a \(G\)-torsor, i.e., there exists a surjective étale morphism \(U' \to U\) such that \(W \times_U U' = \coprod_{g \in G} U'\) as spaces with \(G\)-action over \(U'\). (Namely, \(U' = W\) works.) Hence \(q_*\underline{\mathbf{Z}}\) is a finite locally free \(\mathbf{Z}\)-module with an action of \(G\). For any geometric point \(\overline{u}\) of \(U\), then we get \(G\)-equivariant isomorphisms \[(q_*\underline{\mathbf{Z}})_{\overline{u}} = \bigoplus\nolimits_{\overline{w} \mapsto \overline{u}} \mathbf{Z} = \bigoplus\nolimits_{g \in G} \mathbf{Z} = \mathbf{Z}[G]\] where the second \(=\) uses a geometric point \(\overline{w}_0\) lying over \(\overline{u}\) and maps the summand corresponding to \(g \in G\) to the summand corresponding to \(g(\overline{w}_0)\). We have \[H^0(W, \mathcal{F}|_W) = H^0(U, \mathcal{F} \otimes_\mathbf{Z} q_*\underline{\mathbf{Z}})\] because \(q_*\mathcal{F}|_W = \mathcal{F} \otimes_\mathbf{Z} q_*\underline{\mathbf{Z}}\) as one can check by restricting to \(U'\). Let \[\underline{\mathbf{Z}}(\chi) = (q_*\underline{\mathbf{Z}})^\chi \subset q_*\underline{\mathbf{Z}}\] be the subsheaf of sections that transform according to \(\chi\). For any geometric point \(\overline{u}\) of \(U\) we have \[\underline{\mathbf{Z}}(\chi)_{\overline{u}} = \mathbf{Z} \cdot \sum\nolimits_g \chi(g) g \subset \mathbf{Z}[G] = (q_*\underline{\mathbf{Z}})_{\overline{u}}\] It follows that \(\underline{\mathbf{Z}}(\chi)\) is locally free of rank 1 (more precisely, this should be checked after restricting to \(U'\)). Note that for any \(\mathbf{Z}\)-module \(M\) the \(\chi\)-semi-invariants of \(M[G]\) are the elements of the form \(m \cdot \sum\nolimits_g \chi(g) g\). Thus we see that for any abelian sheaf \(\mathcal{F}\) on \(U\) we have \[\left(\mathcal{F} \otimes_\mathbf{Z} q_*\underline{\mathbf{Z}}\right)^\chi = \mathcal{F} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi)\] because we have equality at all stalks. The result of the lemma follows by taking global sections.

Now we can put everything together and obtain the following pleasing result.

Lemma

Let \(S\) be a scheme. Let \(f : U \to X\) be a surjective, étale, and separated morphism of algebraic spaces over \(S\). For \(p \geq 0\) set \[W_p = U \times_X \ldots \times_X U \setminus \text{all diagonals}\] (with \(p + 1\) factors) as in Lemma 0726. Let \(\chi_p : S_{p + 1} \to \{+1, -1\}\) be the sign character. Let \(U_p = W_p/S_{p + 1}\) and \(\underline{\mathbf{Z}}(\chi_p)\) be as in Lemma 0727. Then the spectral sequence of Lemma 0725 has \(E_1\)-page \[E_1^{p, q} = H^q(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))\] and converges to \(H^{p + q}(X, \mathcal{F})\).

Proof

Note that since the action of \(S_{p + 1}\) on \(W_p\) is over \(X\) we do obtain a morphism \(U_p \to X\). Since \(W_p \to X\) is étale and since \(W_p \to U_p\) is surjective étale, it follows that also \(U_p \to X\) is étale, see Morphisms of Spaces, Lemma 03XT. Therefore an injective object of \(\textit{Ab}(X_\etale)\) restricts to an injective object of \(\textit{Ab}(U_{p, \etale})\), see Cohomology on Sites, Lemma 03F3. Moreover, the functor \(\mathcal{G} \mapsto \mathcal{G} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)\) is an auto-equivalence of \(\textit{Ab}(U_{p, \etale})\), whence transforms injective objects into injective objects and is exact (because \(\underline{\mathbf{Z}}(\chi_p)\) is an invertible \(\underline{\mathbf{Z}}\)-module). Thus given an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\) in \(\textit{Ab}(X_\etale)\) the complex \[\Gamma(U_p, \mathcal{I}^0|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \to \Gamma(U_p, \mathcal{I}^1|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \to \Gamma(U_p, \mathcal{I}^2|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \to \ldots\] computes \(H^*(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))\). On the other hand, by Lemma 0727 it is equal to the complex of \(S_{p + 1}\)-anti-invariants in \[\Gamma(W_p, \mathcal{I}^0) \to \Gamma(W_p, \mathcal{I}^1) \to \Gamma(W_p, \mathcal{I}^2) \to \ldots\] which by Lemma 0726 is equal to the complex \[\Hom(K^p, \mathcal{I}^0) \to \Hom(K^p, \mathcal{I}^1) \to \Hom(K^p, \mathcal{I}^2) \to \ldots\] which computes \(\Ext^*_{\textit{Ab}(X_\etale)}(K^p, \mathcal{F})\). Putting everything together we win.

Higher vanishing for quasi-coherent sheaves

In this section we show that given a quasi-compact and quasi-separated algebraic space \(X\) there exists an integer \(n = n(X)\) such that the cohomology of any quasi-coherent sheaf on \(X\) vanishes beyond degree \(n\).

Lemma

With \(S\), \(W\), \(G\), \(U\), \(\chi\) as in Lemma 0727. If \(\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_U\)-module, then so is \(\mathcal{F} \otimes_{\mathbf{Z}} \underline{\mathbf{Z}}(\chi)\).

Proof

The \(\mathcal{O}_U\)-module structure is clear. To check that \(\mathcal{F} \otimes_{\mathbf{Z}} \underline{\mathbf{Z}}(\chi)\) is quasi-coherent it suffices to check étale locally. Hence the lemma follows as \(\underline{\mathbf{Z}}(\chi)\) is finite locally free as a \(\underline{\mathbf{Z}}\)-module.

The following proposition is interesting even if \(X\) is a scheme. It is the natural generalization of Cohomology of Schemes, Lemma 01XI. Before we state it, observe that given an étale morphism \(f : U \to X\) from an affine scheme towards a quasi-separated algebraic space \(X\) the fibres of \(f\) are universally bounded, in particular there exists an integer \(d\) such that the fibres of \(|U| \to |X|\) all have size at most \(d\); this is the implication \((\eta) \Rightarrow (\delta)\) of Decent Spaces, Lemma 03JX.

Proposition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Assume \(X\) is quasi-compact and separated. Let \(U\) be an affine scheme, and let \(f : U \to X\) be a surjective étale morphism. Let \(d\) be an upper bound for the size of the fibres of \(|U| \to |X|\). Then for any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(H^q(X, \mathcal{F}) = 0\) for \(q \geq d\).

Proof

We will use the spectral sequence of Lemma 0728. The lemma applies since \(f\) is separated as \(U\) is separated, see Morphisms of Spaces, Lemma 03KR. Since \(X\) is separated the scheme \(U \times_X \ldots \times_X U\) is a closed subscheme of \(U \times_{\Spec(\mathbf{Z})} \ldots \times_{\Spec(\mathbf{Z})} U\) hence is affine. Thus \(W_p\) is affine. Hence \(U_p = W_p/S_{p + 1}\) is an affine scheme by Groupoids, Proposition 03BM. The discussion in Section 071Y shows that cohomology of quasi-coherent sheaves on \(W_p\) (as an algebraic space) agrees with the cohomology of the corresponding quasi-coherent sheaf on the underlying affine scheme, hence vanishes in positive degrees by Cohomology of Schemes, Lemma 01XB. By Lemma 072A the sheaves \(\mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)\) are quasi-coherent. Hence \(H^q(W_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))\) is zero when \(q > 0\). By our definition of the integer \(d\) we see that \(W_p = \emptyset\) for \(p \geq d\). Hence also \(H^0(W_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))\) is zero when \(p \geq d\). This proves the proposition.

In the following lemma we establish that a quasi-compact and quasi-separated algebraic space has finite cohomological dimension for quasi-coherent modules. We are explicit about the bound only because we will use it later to prove a similar result for higher direct images.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Assume \(X\) is quasi-compact and quasi-separated. Then we can choose

1. an affine scheme \(U\),

2. a surjective étale morphism \(f : U \to X\),

3. an integer \(d\) bounding the degrees of the fibres of \(U \to X\),

4. for every \(p = 0, 1, \ldots, d\) a surjective étale morphism \(V_p \to U_p\) from an affine scheme \(V_p\) where \(U_p\) is as in Lemma 0728, and

5. an integer \(d_p\) bounding the degree of the fibres of \(V_p \to U_p\).

Moreover, whenever we have (1) – (5), then for any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(H^q(X, \mathcal{F}) = 0\) for \(q \geq \max(d_p + p)\).

Proof

Since \(X\) is quasi-compact we can find a surjective étale morphism \(U \to X\) with \(U\) affine, see Properties of Spaces, Lemma 03H6. By Decent Spaces, Lemma 03JX the fibres of \(f\) are universally bounded, hence we can find \(d\). We have \(U_p = W_p/S_{p + 1}\) and \(W_p \subset U \times_X \ldots \times_X U\) is open and closed. Since \(X\) is quasi-separated the schemes \(W_p\) are quasi-compact, hence \(U_p\) is quasi-compact. Since \(U\) is separated, the schemes \(W_p\) are separated, hence \(U_p\) is separated by (the absolute version of) Spaces, Lemma 02Z4. By Properties of Spaces, Lemma 03H6 we can find the morphisms \(V_p \to W_p\). By Decent Spaces, Lemma 03JX we can find the integers \(d_p\).

At this point the proof uses the spectral sequence \[E_1^{p, q} = H^q(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \Rightarrow H^{p + q}(X, \mathcal{F})\] see Lemma 0728. By definition of the integer \(d\) we see that \(U_p = 0\) for \(p \geq d\). By Proposition 072B and Lemma 072A we see that \(H^q(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))\) is zero for \(q \geq d_p\) for \(p = 0, \ldots, d\). Whence the lemma.

Vanishing for higher direct images

We apply the results of Section 0729 to obtain vanishing of higher direct images of quasi-coherent sheaves for quasi-compact and quasi-separated morphisms. This is useful because it allows one to argue by descending induction on the cohomological degree in certain situations.

Lemma

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

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

2. \(Y\) is quasi-compact.

Then there exists an integer \(n(X \to Y)\) such that for any algebraic space \(Y'\), any morphism \(Y' \to Y\) and any quasi-coherent sheaf \(\mathcal{F}'\) on \(X' = Y' \times_Y X\) the higher direct images \(R^if'_*\mathcal{F}'\) are zero for \(i \geq n(X \to Y)\).

Proof

Let \(V \to Y\) be a surjective étale morphism where \(V\) is an affine scheme, see Properties of Spaces, Lemma 03H6. Suppose we prove the result for the base change \(f_V : V \times_Y X \to V\). Then the result holds for \(f\) with \(n(X \to Y) = n(X_V \to V)\). Namely, if \(Y' \to Y\) and \(\mathcal{F}'\) are as in the lemma, then \(R^if'_*\mathcal{F}'|_{V \times_Y Y'}\) is equal to \(R^if'_{V, *}\mathcal{F}'|_{X'_V}\) where \(f'_V : X'_V = V \times_Y Y' \times_Y X \to V \times_Y Y' = Y'_V\), see Properties of Spaces, Lemma 03LX. Thus we may assume that \(Y\) is an affine scheme.

Moreover, to prove the vanishing for all \(Y' \to Y\) and \(\mathcal{F}'\) it suffices to do so when \(Y'\) is an affine scheme. In this case, \(R^if'_*\mathcal{F}'\) is quasi-coherent by Lemma 0720. Hence it suffices to prove that \(H^i(X', \mathcal{F}') = 0\), because \(H^i(X', \mathcal{F}') = H^0(Y', R^if'_*\mathcal{F}')\) by Cohomology on Sites, Lemma 0733 and the vanishing of higher cohomology of quasi-coherent sheaves on affine algebraic spaces (Proposition 072B).

Choose \(U \to X\), \(d\), \(V_p \to U_p\) and \(d_p\) as in Lemma 072C. For any affine scheme \(Y'\) and morphism \(Y' \to Y\) denote \(X' = Y' \times_Y X\), \(U' = Y' \times_Y U\), \(V'_p = Y' \times_Y V_p\). Then \(U' \to X'\), \(d' = d\), \(V'_p \to U'_p\) and \(d'_p = d\) is a collection of choices as in Lemma 072C for the algebraic space \(X'\) (details omitted). Hence we see that \(H^i(X', \mathcal{F}') = 0\) for \(i \geq \max(p + d_p)\) and we win.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). Then \(R^if_*\mathcal{F} = 0\) for \(i > 0\) and any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\).

Proof

Recall that an affine morphism of algebraic spaces is representable. Hence this follows from (071Z) and Cohomology of Schemes, Lemma 01XC.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(H^i(X, \mathcal{F}) = H^i(Y, f_*\mathcal{F})\) for all \(i \geq 0\).

Proof

Follows from Lemma 073H and the Leray spectral sequence. See Cohomology on Sites, Lemma 0733.

Cohomology with support in a closed subspace

This section is the analogue of Cohomology, Sections 0A39 and 0G6Y and Étale Cohomology, Section 09XP for abelian sheaves on algebraic spaces.

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\) and let \(Z \subset X\) be a closed subspace. Let \(\mathcal{F}\) be an abelian sheaf on \(X_\etale\). We let \[\Gamma_Z(X, \mathcal{F}) = \{s \in \mathcal{F}(X) \mid \text{Supp}(s) \subset Z\}\] be the sections with support in \(Z\) (Properties of Spaces, Definition 04KA). This is a left exact functor which is not exact in general. Hence we obtain a derived functor \[R\Gamma_Z(X, -) : D(X_\etale) \longrightarrow D(\textit{Ab})\] and cohomology groups with support in \(Z\) defined by \(H^q_Z(X, \mathcal{F}) = R^q\Gamma_Z(X, \mathcal{F})\).

Let \(\mathcal{I}\) be an injective abelian sheaf on \(X_\etale\). Let \(U \subset X\) be the open subspace which is the complement of \(Z\). Then the restriction map \(\mathcal{I}(X) \to \mathcal{I}(U)\) is surjective (Cohomology on Sites, Lemma 093X) with kernel \(\Gamma_Z(X, \mathcal{I})\). It immediately follows that for \(K \in D(X_\etale)\) there is a distinguished triangle \[R\Gamma_Z(X, K) \to R\Gamma(X, K) \to R\Gamma(U, K) \to R\Gamma_Z(X, K)[1]\] in \(D(\textit{Ab})\). As a consequence we obtain a long exact cohomology sequence \[\ldots \to H^i_Z(X, K) \to H^i(X, K) \to H^i(U, K) \to H^{i + 1}_Z(X, K) \to \ldots\] for any \(K\) in \(D(X_\etale)\).

For an abelian sheaf \(\mathcal{F}\) on \(X_\etale\) we can consider the subsheaf of sections with support in \(Z\), denoted \(\mathcal{H}_Z(\mathcal{F})\), defined by the rule \[\mathcal{H}_Z(\mathcal{F})(U) = \{s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset U \times_X Z\}\] Here we use the support of a section from Properties of Spaces, Definition 04KA. Using the equivalence of Morphisms of Spaces, Lemma 04E5 we may view \(\mathcal{H}_Z(\mathcal{F})\) as an abelian sheaf on \(Z_\etale\). Thus we obtain a functor \[\textit{Ab}(X_\etale) \longrightarrow \textit{Ab}(Z_\etale),\quad \mathcal{F} \longmapsto \mathcal{H}_Z(\mathcal{F})\] which is left exact, but in general not exact.

Lemma

Let \(S\) be a scheme. Let \(i : Z \to X\) be a closed immersion of algebraic spaces over \(S\). Let \(\mathcal{I}\) be an injective abelian sheaf on \(X_\etale\). Then \(\mathcal{H}_Z(\mathcal{I})\) is an injective abelian sheaf on \(Z_\etale\).

Proof

Observe that for any abelian sheaf \(\mathcal{G}\) on \(Z_\etale\) we have \[\Hom_Z(\mathcal{G}, \mathcal{H}_Z(\mathcal{F})) = \Hom_X(i_*\mathcal{G}, \mathcal{F})\] because after all any section of \(i_*\mathcal{G}\) has support in \(Z\). Since \(i_*\) is exact (Lemma 0A4K) and as \(\mathcal{I}\) is injective on \(X_\etale\) we conclude that \(\mathcal{H}_Z(\mathcal{I})\) is injective on \(Z_\etale\).

Denote \[R\mathcal{H}_Z : D(X_\etale) \longrightarrow D(Z_\etale)\] the derived functor. We set \(\mathcal{H}^q_Z(\mathcal{F}) = R^q\mathcal{H}_Z(\mathcal{F})\) so that \(\mathcal{H}^0_Z(\mathcal{F}) = \mathcal{H}_Z(\mathcal{F})\). By the lemma above we have a Grothendieck spectral sequence \[E_2^{p, q} = H^p(Z, \mathcal{H}^q_Z(\mathcal{F})) \Rightarrow H^{p + q}_Z(X, \mathcal{F})\]

Lemma

Let \(S\) be a scheme. Let \(i : Z \to X\) be a closed immersion of algebraic spaces over \(S\). Let \(\mathcal{G}\) be an injective abelian sheaf on \(Z_\etale\). Then \(\mathcal{H}^p_Z(i_*\mathcal{G}) = 0\) for \(p > 0\).

Proof

This is true because the functor \(i_*\) is exact (Lemma 0A4K) and transforms injective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma 0730).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an étale morphism of algebraic spaces over \(S\). Let \(Z \subset Y\) be a closed subspace such that \(f^{-1}(Z) \to Z\) is an isomorphism of algebraic spaces. Let \(\mathcal{F}\) be an abelian sheaf on \(X\). Then \[\mathcal{H}^q_Z(\mathcal{F}) = \mathcal{H}^q_{f^{-1}(Z)}(f^{-1}\mathcal{F})\] as abelian sheaves on \(Z = f^{-1}(Z)\) and we have \(H^q_Z(Y, \mathcal{F}) = H^q_{f^{-1}(Z)}(X, f^{-1}\mathcal{F})\).

Proof

Because \(f\) is étale an injective resolution of \(\mathcal{F}\) pulls back to an injective resolution of \(f^{-1}\mathcal{F}\). Hence it suffices to check the equality for \(\mathcal{H}_Z(-)\) which follows from the definitions. The proof for cohomology with supports is the same. Some details omitted.

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset. We denote \(D_T(X_\etale)\) the strictly full saturated triangulated subcategory of \(D(X_\etale)\) consisting of objects whose cohomology sheaves are supported on \(T\).

Lemma

Let \(S\) be a scheme. Let \(i : Z \to X\) be a closed immersion of algebraic spaces over \(S\). The map \(Ri_* = i_* : D(Z_\etale) \to D(X_\etale)\) induces an equivalence \(D(Z_\etale) \to D_{|Z|}(X_\etale)\) with quasi-inverse \[i^{-1}|_{D_Z(X_\etale)} = R\mathcal{H}_Z|_{D_{|Z|}(X_\etale)}\]

Proof

Recall that \(i^{-1}\) and \(i_*\) is an adjoint pair of exact functors such that \(i^{-1}i_*\) is isomorphic to the identify functor on abelian sheaves. See Properties of Spaces, Lemma 04K2 and Morphisms of Spaces, Lemma 04E5. Thus \(i_* : D(Z_\etale) \to D_Z(X_\etale)\) is fully faithful and \(i^{-1}\) determines a left inverse. On the other hand, suppose that \(K\) is an object of \(D_Z(X_\etale)\) and consider the adjunction map \(K \to i_*i^{-1}K\). Using exactness of \(i_*\) and \(i^{-1}\) this induces the adjunction maps \(H^n(K) \to i_*i^{-1}H^n(K)\) on cohomology sheaves. Since these cohomology sheaves are supported on \(Z\) we see these adjunction maps are isomorphisms and we conclude that \(D(Z_\etale) \to D_Z(X_\etale)\) is an equivalence.

To finish the proof we have to show that \(R\mathcal{H}_Z(K) = i^{-1}K\) if \(K\) is an object of \(D_Z(X_\etale)\). To do this we can use that \(K = i_*i^{-1}K\) as we’ve just proved this is the case. Then we can choose a K-injective representative \(\mathcal{I}^\bullet\) for \(i^{-1}K\). Since \(i_*\) is the right adjoint to the exact functor \(i^{-1}\), the complex \(i_*\mathcal{I}^\bullet\) is K-injective (Derived Categories, Lemma 08BJ). We see that \(R\mathcal{H}_Z(K)\) is computed by \(\mathcal{H}_Z(i_*\mathcal{I}^\bullet) = \mathcal{I}^\bullet\) as desired.

Vanishing above the dimension

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). In this case \(|X|\) is a spectral space, see Properties of Spaces, Lemma 0A4G. Moreover, the dimension of \(X\) (as defined in Properties of Spaces, Definition 04N6) is equal to the Krull dimension of \(|X|\), see Decent Spaces, Lemma 0A4J. We will show that for quasi-coherent sheaves on \(X\) we have vanishing of cohomology above the dimension. This result is already interesting for quasi-separated algebraic spaces of finite type over a field.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Assume \(\dim(X) \leq d\) for some integer \(d\). Let \(\mathcal{F}\) be a quasi-coherent sheaf \(\mathcal{F}\) on \(X\).

1. \(H^q(X, \mathcal{F}) = 0\) for \(q > d\),

2. \(H^d(X, \mathcal{F}) \to H^d(U, \mathcal{F})\) is surjective for any quasi-compact open \(U \subset X\),

3. \(H^q_Z(X, \mathcal{F}) = 0\) for \(q > d\) for any closed subspace \(Z \subset X\) whose complement is quasi-compact.

Proof

By Properties of Spaces, Lemma 0A4H every algebraic space \(Y\) étale over \(X\) has dimension \(\leq d\). If \(Y\) is quasi-separated, the dimension of \(Y\) is equal to the Krull dimension of \(|Y|\) by Decent Spaces, Lemma 0A4J. Also, if \(Y\) is a scheme, then étale cohomology of \(\mathcal{F}\) over \(Y\), resp. étale cohomology of \(\mathcal{F}\) with support in a closed subscheme, agrees with usual cohomology of \(\mathcal{F}\), resp. usual cohomology with support in the closed subscheme. See Descent, Proposition 03DW and Étale Cohomology, Lemma 0A46. We will use these facts without further mention.

By Decent Spaces, Lemma 07ST there exist an integer \(n\) and open subspaces \[\emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X\] with the following property: setting \(T_p = U_p \setminus U_{p + 1}\) (with reduced induced subspace structure) there exists a quasi-compact separated scheme \(V_p\) and a surjective étale morphism \(f_p : V_p \to U_p\) such that \(f_p^{-1}(T_p) \to T_p\) is an isomorphism.

As \(U_n = V_n\) is a scheme, our initial remarks imply the cohomology of \(\mathcal{F}\) over \(U_n\) vanishes in degrees \(> d\) by Cohomology, Proposition 0A3G. Suppose we have shown, by induction, that \(H^q(U_{p + 1}, \mathcal{F}|_{U_{p + 1}}) = 0\) for \(q > d\). It suffices to show \(H_{T_p}^q(U_p, \mathcal{F})\) for \(q > d\) is zero in order to conclude the vanishing of cohomology of \(\mathcal{F}\) over \(U_p\) in degrees \(> d\). However, we have \[H^q_{T_p}(U_p, \mathcal{F}) = H^q_{f_p^{-1}(T_p)}(V_p, \mathcal{F})\] by Lemma 0A4P and as \(V_p\) is a scheme we obtain the desired vanishing from Cohomology, Proposition 0A3G. In this way we conclude that (1) is true.

To prove (2) let \(U \subset X\) be a quasi-compact open subspace. Consider the open subspace \(U' = U \cup U_n\). Let \(Z = U' \setminus U\). Then \(g : U_n \to U'\) is an étale morphism such that \(g^{-1}(Z) \to Z\) is an isomorphism. Hence by Lemma 0A4P we have \(H^q_Z(U', \mathcal{F}) = H^q_Z(U_n, \mathcal{F})\) which vanishes in degree \(> d\) because \(U_n\) is a scheme and we can apply Cohomology, Proposition 0A3G. We conclude that \(H^d(U', \mathcal{F}) \to H^d(U, \mathcal{F})\) is surjective. Assume, by induction, that we have reduced our problem to the case where \(U\) contains \(U_{p + 1}\). Then we set \(U' = U \cup U_p\), set \(Z = U' \setminus U\), and we argue using the morphism \(f_p : V_p \to U'\) which is étale and has the property that \(f_p^{-1}(Z) \to Z\) is an isomorphism. In other words, we again see that \[H^q_Z(U', \mathcal{F}) = H^q_{f_p^{-1}(Z)}(V_p, \mathcal{F})\] and we again see this vanishes in degrees \(> d\). We conclude that \(H^d(U', \mathcal{F}) \to H^d(U, \mathcal{F})\) is surjective. Eventually we reach the stage where \(U_1 = X \subset U\) which finishes the proof.

A formal argument shows that (2) implies (3).

Cohomology and base change, I

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Suppose further that \(g : Y' \to Y\) is a morphism of algebraic spaces over \(S\). Denote \(X' = X_{Y'} = Y' \times_Y X\) the base change of \(X\) and denote \(f' : X' \to Y'\) the base change of \(f\). Also write \(g' : X' \to X\) the projection, and set \(\mathcal{F}' = (g')^*\mathcal{F}\). Here is a diagram representing the situation: [073J]\[\begin{equation} \vcenter{ \xymatrix{ \mathcal{F}' = (g')^*\mathcal{F} & X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f & \mathcal{F} \\ Rf'_*\mathcal{F}' & Y' \ar[r]^g & Y & Rf_*\mathcal{F} } } \end{equation}\] Here is the simplest case of the base change property we have in mind.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. In this case \(f_*\mathcal{F} \cong Rf_*\mathcal{F}\) is a quasi-coherent sheaf, and for every diagram (073J) we have \[g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}.\]

Proof

By the discussion surrounding (071Z) this reduces to the case of an affine morphism of schemes which is treated in Cohomology of Schemes, Lemma 02KG.

Lemma

Let \(S\) be a scheme. Consider a cartesian diagram of algebraic spaces \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module with pullback \(\mathcal{F}' = (g')^*\mathcal{F}\). Assume that \(g\) is flat and that \(f\) is quasi-compact and quasi-separated. For any \(i \geq 0\)

1. the base change map of Cohomology on Sites, Lemma 0736 is an isomorphism \[g^*R^if_*\mathcal{F} \longrightarrow R^if'_*\mathcal{F}',\]

2. if \(Y = \Spec(A)\) and \(Y' = \Spec(B)\), then \(H^i(X, \mathcal{F}) \otimes_A B = H^i(X', \mathcal{F}')\).

Proof

The morphism \(g'\) is flat by Morphisms of Spaces, Lemma 03MO. Note that flatness of \(g\) and \(g'\) is equivalent to flatness of the morphisms of small étale ringed sites, see Morphisms of Spaces, Lemma 073C. Hence we can apply Cohomology on Sites, Lemma 0736 to obtain a base change map \[g^*R^pf_*\mathcal{F} \longrightarrow R^pf'_*\mathcal{F}'\] To prove this map is an isomorphism we can work locally in the étale topology on \(Y'\). Thus we may assume that \(Y\) and \(Y'\) are affine schemes. Say \(Y = \Spec(A)\) and \(Y' = \Spec(B)\). In this case we are really trying to show that the map \[H^p(X, \mathcal{F}) \otimes_A B \longrightarrow H^p(X_B, \mathcal{F}_B)\] is an isomorphism where \(X_B = \Spec(B) \times_{\Spec(A)} X\) and \(\mathcal{F}_B\) is the pullback of \(\mathcal{F}\) to \(X_B\). In other words, it suffices to prove (2).

Fix \(A \to B\) a flat ring map and let \(X\) be a quasi-compact and quasi-separated algebraic space over \(A\). Note that \(g' : X_B \to X\) is affine as a base change of \(\Spec(B) \to \Spec(A)\). Hence the higher direct images \(R^i(g')_*\mathcal{F}_B\) are zero by Lemma 073H. Thus \(H^p(X_B, \mathcal{F}_B) = H^p(X, g'_*\mathcal{F}_B)\), see Cohomology on Sites, Lemma 0733. Moreover, we have \[g'_*\mathcal{F}_B = \mathcal{F} \otimes_{\underline{A}} \underline{B}\] where \(\underline{A}\), \(\underline{B}\) denotes the constant sheaf of rings with value \(A\), \(B\). Namely, it is clear that there is a map from right to left. For any affine scheme \(U\) étale over \(X\) we have \[\begin{align*} g'_*\mathcal{F}_B(U) & = \mathcal{F}_B(\Spec(B) \times_{\Spec(A)} U) \\ & = \Gamma(\Spec(B) \times_{\Spec(A)} U, (\Spec(B) \times_{\Spec(A)} U \to U)^*\mathcal{F}|_U) \\ & = B \otimes_A \mathcal{F}(U) \end{align*}\] hence the map is an isomorphism. Write \(B = \colim M_i\) as a filtered colimit of finite free \(A\)-modules \(M_i\) using Lazard’s theorem, see Algebra, Theorem 058G. We deduce that \[\begin{align*} H^p(X, g'_*\mathcal{F}_B) & = H^p(X, \mathcal{F} \otimes_{\underline{A}} \underline{B}) \\ & = H^p(X, \colim_i \mathcal{F} \otimes_{\underline{A}} \underline{M_i}) \\ & = \colim_i H^p(X, \mathcal{F} \otimes_{\underline{A}} \underline{M_i}) \\ & = \colim_i H^p(X, \mathcal{F}) \otimes_A M_i \\ & = H^p(X, \mathcal{F}) \otimes_A \colim_i M_i \\ & = H^p(X, \mathcal{F}) \otimes_A B \end{align*}\] The first equality because \(g'_*\mathcal{F}_B = \mathcal{F} \otimes_{\underline{A}} \underline{B}\) as seen above. The second because \(\otimes\) commutes with colimits. The third equality because cohomology on \(X\) commutes with colimits (see Lemma 073E). The fourth equality because \(M_i\) is finite free (i.e., because cohomology commutes with finite direct sums). The fifth because \(\otimes\) commutes with colimits. The sixth by choice of our system.

Lemma

Let \(f : X \to Y\) be a quasi-compact, separated, étale morphism of algebraic spaces. Then for any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the map \(f^*f_*\mathcal{F} \to \mathcal{F}\) is split.

Proof

Consider the cartesian diagram \[\xymatrix{ X \times_Y X \ar[r]_-p \ar[d]_q & X \ar[d]^f \\ X \ar[r]^f & Y }\] By Lemma 07U8 we have \(f^*f_*\mathcal{F} = q_*p^*\mathcal{F}\). The morphism \(\Delta : X \to X \times_Y X\) is open (as a morphism between algebraic spaces étale over \(Y\)) and closed as \(f\) is separated. Thus we see that \(p^*\mathcal{F}\) is a direct sum of \(\Delta_*\mathcal{F}\) and a quasi-coherent module supported on the (open and closed) complement of \(\Delta(X)\). Tracing the maps the reader verifies that \[\mathcal{F} = q_*\Delta_*\mathcal{F} \to q_*p^*\mathcal{F} = f^*f_*\mathcal{F} \to \mathcal{F}\] is the identity map. Details omitted.

Coherent modules on locally Noetherian algebraic spaces

This section is the analogue of Cohomology of Schemes, Section 01XY. In Modules on Sites, Definition 03DL we have defined coherent modules on any ringed topos. We use this notion to define coherent modules on locally Noetherian algebraic spaces. Although it is possible to work with coherent modules more generally we resist the urge to do so.

Definition

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). A quasi-coherent module \(\mathcal{F}\) on \(X\) is called coherent if \(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module on the site \(X_\etale\) in the sense of Modules on Sites, Definition 03DL.

This definition is compatible with the already existing notion of a coherent module on a locally Noetherian scheme; see assertion (5) of Properties of Spaces, Section 05VR (or more directly Descent, Lemma 05VG). Thus from now on, if \(X\) is a locally Noetherian scheme over \(S\), we will not distinguish between a coherent module on \(X\) viewed as a scheme or a coherent module on \(X\) viewed as an algebraic space; this is compatible with the corresponding identifications of categories of quasi-coherent modules discussed in Properties of Spaces, Section 03G5.

Having said the above, the following lemma gives an understandable characterization of coherent modules on locally Noetherian algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. The following are equivalent

1. \(\mathcal{F}\) is coherent,

2. \(\mathcal{F}\) is a quasi-coherent, finite type \(\mathcal{O}_X\)-module,

3. \(\mathcal{F}\) is a finitely presented \(\mathcal{O}_X\)-module,

4. for any étale morphism \(\varphi : U \to X\) where \(U\) is a scheme the pullback \(\varphi^*\mathcal{F}\) is a coherent module on \(U\), and

5. there exists a surjective étale morphism \(\varphi : U \to X\) where \(U\) is a scheme such that the pullback \(\varphi^*\mathcal{F}\) is a coherent module on \(U\).

In particular \(\mathcal{O}_X\) is coherent, any invertible \(\mathcal{O}_X\)-module is coherent, and more generally any finite locally free \(\mathcal{O}_X\)-module is coherent.

Proof

To be sure, if \(X\) is a locally Noetherian algebraic space and \(U \to X\) is an étale morphism, then \(U\) is locally Noetherian, see Properties of Spaces, Section 03E5. The lemma then follows from the points (1) – (5) made in Properties of Spaces, Section 05VR and the corresponding result for coherent modules on locally Noetherian schemes, see Cohomology of Schemes, Lemma 01XZ.

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). The category of coherent \(\mathcal{O}_X\)-modules is abelian. More precisely, the kernel and cokernel of a map of coherent \(\mathcal{O}_X\)-modules are coherent. Any extension of coherent sheaves is coherent.

Proof

Choose a scheme \(U\) and a surjective étale morphism \(f : U \to X\). Pullback \(f^*\) is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (03LW). By Lemma 07UB we can check whether an \(\mathcal{O}_X\)-module \(\mathcal{F}\) is coherent by checking whether \(f^*\mathcal{F}\) is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma 01Y0.

Coherent modules form a Serre subcategory of the category of quasi-coherent \(\mathcal{O}_X\)-modules. This does not hold for modules on a general ringed topos.

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Any quasi-coherent submodule of \(\mathcal{F}\) is coherent. Any quasi-coherent quotient module of \(\mathcal{F}\) is coherent.

Proof

Choose a scheme \(U\) and a surjective étale morphism \(f : U \to X\). Pullback \(f^*\) is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (03LW). By Lemma 07UB we can check whether an \(\mathcal{O}_X\)-module \(\mathcal{G}\) is coherent by checking whether \(f^*\mathcal{H}\) is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma 01Y1.

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\),. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. The \(\mathcal{O}_X\)-modules \(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}\) and \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) are coherent.

Proof

Via Lemma 07UB this follows from the result for schemes, see Cohomology of Schemes, Lemma 01Y2.

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Let \(\varphi : \mathcal{G} \to \mathcal{F}\) be a homomorphism of \(\mathcal{O}_X\)-modules. Let \(\overline{x}\) be a geometric point of \(X\) lying over \(x \in |X|\).

1. If \(\mathcal{F}_{\overline{x}} = 0\) then there exists an open neighbourhood \(X' \subset X\) of \(x\) such that \(\mathcal{F}|_{X'} = 0\).

2. If \(\varphi_{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}\) is injective, then there exists an open neighbourhood \(X' \subset X\) of \(x\) such that \(\varphi|_{X'}\) is injective.

3. If \(\varphi_{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}\) is surjective, then there exists an open neighbourhood \(X' \subset X\) of \(x\) such that \(\varphi|_{X'}\) is surjective.

4. If \(\varphi_{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}\) is bijective, then there exists an open neighbourhood \(X' \subset X\) of \(x\) such that \(\varphi|_{X'}\) is an isomorphism.

Proof

Let \(\varphi : U \to X\) be an étale morphism where \(U\) is a scheme and let \(u \in U\) be a point mapping to \(x\). By Properties of Spaces, Lemmas 05VP and 04KF as well as More on Algebra, Lemma 07QM we see that \(\varphi_{\overline{x}}\) is injective, surjective, or bijective if and only if \(\varphi_u : \varphi^*\mathcal{F}_u \to \varphi^*\mathcal{G}_u\) has the corresponding property. Thus we can apply the schemes version of this lemma to see that (after possibly shrinking \(U\)) the map \(\varphi^*\mathcal{F} \to \varphi^*\mathcal{G}\) is injective, surjective, or an isomorphism. Let \(X' \subset X\) be the open subspace corresponding to \(|\varphi|(|U|) \subset |X|\), see Properties of Spaces, Lemma 03BZ. Since \(\{U \to X'\}\) is a covering for the étale topology, we conclude that \(\varphi|_{X'}\) is injective, surjective, or an isomorphism as desired. Finally, observe that (1) follows from (2) by looking at the map \(\mathcal{F} \to 0\).

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Let \(i : Z \to X\) be the scheme theoretic support of \(\mathcal{F}\) and \(\mathcal{G}\) the quasi-coherent \(\mathcal{O}_Z\)-module such that \(i_*\mathcal{G} = \mathcal{F}\), see Morphisms of Spaces, Definition 07U1. Then \(\mathcal{G}\) is a coherent \(\mathcal{O}_Z\)-module.

Proof

The statement of the lemma makes sense as a coherent module is in particular of finite type. Moreover, as \(Z \to X\) is a closed immersion it is locally of finite type and hence \(Z\) is locally Noetherian, see Morphisms of Spaces, Lemmas 06ED and 04ZK. Finally, as \(\mathcal{G}\) is of finite type it is a coherent \(\mathcal{O}_Z\)-module by Lemma 07UB

Lemma

Let \(S\) be a scheme. Let \(i : Z \to X\) be a closed immersion of locally Noetherian algebraic spaces over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be the quasi-coherent sheaf of ideals cutting out \(Z\). The functor \(i_*\) induces an equivalence between the category of coherent \(\mathcal{O}_X\)-modules annihilated by \(\mathcal{I}\) and the category of coherent \(\mathcal{O}_Z\)-modules.

Proof

The functor is fully faithful by Morphisms of Spaces, Lemma 04CJ. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module annihilated by \(\mathcal{I}\). By Morphisms of Spaces, Lemma 04CJ we can write \(\mathcal{F} = i_*\mathcal{G}\) for some quasi-coherent sheaf \(\mathcal{G}\) on \(Z\). To check that \(\mathcal{G}\) is coherent we can work étale locally (Lemma 07UB). Choosing an étale covering by a scheme we conclude that \(\mathcal{G}\) is coherent by the case of schemes (Cohomology of Schemes, Lemma 087T). Hence the functor is fully faithful and the proof is done.

Lemma

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

Proof

Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Then \(V \times_Y X \to V\) is a finite morphism of locally Noetherian schemes. By (071Z) we reduce to the case of schemes which is Cohomology of Schemes, Lemma 01Y6.

Coherent sheaves on Noetherian spaces

In this section we mention some properties of coherent sheaves on Noetherian algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. The ascending chain condition holds for quasi-coherent submodules of \(\mathcal{F}\). In other words, given any sequence \[\mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F}\] of quasi-coherent submodules, then \(\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots\) for some \(n \geq 0\).

Proof

Choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\) (see Properties of Spaces, Lemma 03H6). Then \(U\) is a Noetherian scheme (by Morphisms of Spaces, Lemma 04ZK). If \(\mathcal{F}_n|_U = \mathcal{F}_{n + 1}|_U = \ldots\) then \(\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots\). Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma 01Y8.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals corresponding to a closed subspace \(Z \subset X\). Then there is some \(n \geq 0\) such that \(\mathcal{I}^n\mathcal{F} = 0\) if and only if \(\text{Supp}(\mathcal{F}) \subset Z\) (set theoretically).

Proof

Choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\) (see Properties of Spaces, Lemma 03H6). Then \(U\) is a Noetherian scheme (by Morphisms of Spaces, Lemma 04ZK). Note that \(\mathcal{I}^n\mathcal{F}|_U = 0\) if and only if \(\mathcal{I}^n\mathcal{F} = 0\) and similarly for the condition on the support. Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma 01Y9.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Let \(\mathcal{G} \subset \mathcal{F}\) be a quasi-coherent subsheaf. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Then there exists a \(c \geq 0\) such that for all \(n \geq c\) we have \[\mathcal{I}^{n - c}(\mathcal{I}^c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^n\mathcal{F} \cap \mathcal{G}\]

Proof

Choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\) (see Properties of Spaces, Lemma 03H6). Then \(U\) is a Noetherian scheme (by Morphisms of Spaces, Lemma 04ZK). The equality of the lemma holds if and only if it holds after restricting to \(U\). Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma 01YA.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{G}\) be a coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Denote \(Z \subset X\) the corresponding closed subspace and set \(U = X \setminus Z\). There is a canonical isomorphism \[\colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n\mathcal{G}, \mathcal{F}) \longrightarrow \Hom_{\mathcal{O}_U}(\mathcal{G}|_U, \mathcal{F}|_U).\] In particular we have an isomorphism \[\colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n, \mathcal{F}) \longrightarrow \Gamma(U, \mathcal{F}).\]

Proof

Let \(W\) be an affine scheme and let \(W \to X\) be a surjective étale morphism (see Properties of Spaces, Lemma 03H6). Set \(R = W \times_X W\). Then \(W\) and \(R\) are Noetherian schemes, see Morphisms of Spaces, Lemma 04ZK. Hence the result hold for the restrictions of \(\mathcal{F}\), \(\mathcal{G}\), and \(\mathcal{I}\), \(U\), \(Z\) to \(W\) and \(R\) by Cohomology of Schemes, Lemma 01YB. It follows formally that the result holds over \(X\).

Devissage of coherent sheaves

This section is the analogue of Cohomology of Schemes, Section 01YC.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Suppose that \(\text{Supp}(\mathcal{F}) = Z \cup Z'\) with \(Z\), \(Z'\) closed. Then there exists a short exact sequence of coherent sheaves \[0 \to \mathcal{G}' \to \mathcal{F} \to \mathcal{G} \to 0\] with \(\text{Supp}(\mathcal{G}') \subset Z'\) and \(\text{Supp}(\mathcal{G}) \subset Z\).

Proof

Let \(\mathcal{I} \subset \mathcal{O}_X\) be the sheaf of ideals defining the reduced induced closed subspace structure on \(Z\), see Properties of Spaces, Lemma 03IQ. Consider the subsheaves \(\mathcal{G}'_n = \mathcal{I}^n\mathcal{F}\) and the quotients \(\mathcal{G}_n = \mathcal{F}/\mathcal{I}^n\mathcal{F}\). For each \(n\) we have a short exact sequence \[0 \to \mathcal{G}'_n \to \mathcal{F} \to \mathcal{G}_n \to 0\] For every geometric point \(\overline{x}\) of \(Z' \setminus Z\) we have \(\mathcal{I}_{\overline{x}} = \mathcal{O}_{X, \overline{x}}\) and hence \(\mathcal{G}_{n, \overline{x}} = 0\). Thus we see that \(\text{Supp}(\mathcal{G}_n) \subset Z\). Note that \(X \setminus Z'\) is a Noetherian algebraic space. Hence by Lemma 07UK there exists an \(n\) such that \(\mathcal{G}'_n|_{X \setminus Z'} = \mathcal{I}^n\mathcal{F}|_{X \setminus Z'} = 0\). For such an \(n\) we see that \(\text{Supp}(\mathcal{G}'_n) \subset Z'\). Thus setting \(\mathcal{G}' = \mathcal{G}'_n\) and \(\mathcal{G} = \mathcal{G}_n\) works.

In the following we will freely use the scheme theoretic support of finite type modules as defined in Morphisms of Spaces, Definition 07U1.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Assume that the scheme theoretic support of \(\mathcal{F}\) is a reduced \(Z \subset X\) with \(|Z|\) irreducible. Then there exist an integer \(r > 0\), a nonzero sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\), and an injective map of coherent sheaves \[i_*\left(\mathcal{I}^{\oplus r}\right) \to \mathcal{F}\] whose cokernel is supported on a proper closed subspace of \(Z\).

Proof

By assumption there exists a coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\) with support \(Z\) and \(\mathcal{F} \cong i_*\mathcal{G}\), see Lemma 07UG. Hence it suffices to prove the lemma for the case \(Z = X\) and \(i = \text{id}\).

By Properties of Spaces, Proposition 06NH there exists a dense open subspace \(U \subset X\) which is a scheme. Note that \(U\) is a Noetherian integral scheme. After shrinking \(U\) we may assume that \(\mathcal{F}|_U \cong \mathcal{O}_U^{\oplus r}\) (for example by Cohomology of Schemes, Lemma 01YE or by a direct algebra argument). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals whose associated closed subspace is the complement of \(U\) in \(X\) (see for example Properties of Spaces, Section 03IP). By Lemma 07UM there exists an \(n \geq 0\) and a morphism \(\mathcal{I}^n(\mathcal{O}_X^{\oplus r}) \to \mathcal{F}\) which recovers our isomorphism over \(U\). Since \(\mathcal{I}^n(\mathcal{O}_X^{\oplus r}) = (\mathcal{I}^n)^{\oplus r}\) we get a map as in the lemma. It is injective: namely, if \(\sigma\) is a nonzero section of \(\mathcal{I}^{\oplus r}\) over a scheme \(W\) étale over \(X\), then because \(X\) hence \(W\) is reduced the support of \(\sigma\) contains a nonempty open of \(W\). But the kernel of \((\mathcal{I}^n)^{\oplus r} \to \mathcal{F}\) is zero over a dense open, hence \(\sigma\) cannot be a section of the kernel.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent sheaf on \(X\). There exists a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F}\] by coherent subsheaves such that for each \(j = 1, \ldots, m\) there exists a reduced closed subspace \(Z_j \subset X\) with \(|Z_j|\) irreducible and a sheaf of ideals \(\mathcal{I}_j \subset \mathcal{O}_{Z_j}\) such that \[\mathcal{F}_j/\mathcal{F}_{j - 1} \cong (Z_j \to X)_* \mathcal{I}_j\]

Proof

Consider the collection \[\mathcal{T} = \left\{ \begin{matrix} T \subset |X| \text{ closed such that there exists a coherent sheaf } \mathcal{F} \\ \text{ with } \text{Supp}(\mathcal{F}) = T \text{ for which the lemma is wrong} \end{matrix} \right\}\] We are trying to show that \(\mathcal{T}\) is empty. If not, then because \(|X|\) is Noetherian (Properties of Spaces, Lemma 04ZF) we can choose a minimal element \(T \in \mathcal{T}\). This means that there exists a coherent sheaf \(\mathcal{F}\) on \(X\) whose support is \(T\) and for which the lemma does not hold. Clearly \(T \not = \emptyset\) since the only sheaf whose support is empty is the zero sheaf for which the lemma does hold (with \(m = 0\)).

If \(T\) is not irreducible, then we can write \(T = Z_1 \cup Z_2\) with \(Z_1, Z_2\) closed and strictly smaller than \(T\). Then we can apply Lemma 07UP to get a short exact sequence of coherent sheaves \[0 \to \mathcal{G}_1 \to \mathcal{F} \to \mathcal{G}_2 \to 0\] with \(\text{Supp}(\mathcal{G}_i) \subset Z_i\). By minimality of \(T\) each of \(\mathcal{G}_i\) has a filtration as in the statement of the lemma. By considering the induced filtration on \(\mathcal{F}\) we arrive at a contradiction. Hence we conclude that \(T\) is irreducible.

Suppose \(T\) is irreducible. Let \(\mathcal{J}\) be the sheaf of ideals defining the reduced induced closed subspace structure on \(T\), see Properties of Spaces, Lemma 03IQ. By Lemma 07UK we see there exists an \(n \geq 0\) such that \(\mathcal{J}^n\mathcal{F} = 0\). Hence we obtain a filtration \[0 = \mathcal{I}^n\mathcal{F} \subset \mathcal{I}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{I}\mathcal{F} \subset \mathcal{F}\] each of whose successive subquotients is annihilated by \(\mathcal{J}\). Hence if each of these subquotients has a filtration as in the statement of the lemma then also \(\mathcal{F}\) does. In other words we may assume that \(\mathcal{J}\) does annihilate \(\mathcal{F}\).

Assume \(T\) is irreducible and \(\mathcal{J}\mathcal{F} = 0\) where \(\mathcal{J}\) is as above. Then the scheme theoretic support of \(\mathcal{F}\) is \(T\), see Morphisms of Spaces, Lemma 04CJ. Hence we can apply Lemma 07UQ. This gives a short exact sequence \[0 \to i_*(\mathcal{I}^{\oplus r}) \to \mathcal{F} \to \mathcal{Q} \to 0\] where the support of \(\mathcal{Q}\) is a proper closed subset of \(T\). Hence we see that \(\mathcal{Q}\) has a filtration of the desired type by minimality of \(T\). But then clearly \(\mathcal{F}\) does too, which is our final contradiction.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\). Assume

1. For any short exact sequence of coherent sheaves \[0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0\] if \(\mathcal{F}_i\), \(i = 1, 2\) have property \(\mathcal{P}\) then so does \(\mathcal{F}\).

2. For every reduced closed subspace \(Z \subset X\) with \(|Z|\) irreducible and every quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\) we have \(\mathcal{P}\) for \(i_*\mathcal{I}\).

Then property \(\mathcal{P}\) holds for every coherent sheaf on \(X\).

Proof

First note that if \(\mathcal{F}\) is a coherent sheaf with a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F}\] by coherent subsheaves such that each of \(\mathcal{F}_i/\mathcal{F}_{i - 1}\) has property \(\mathcal{P}\), then so does \(\mathcal{F}\). This follows from the property (1) for \(\mathcal{P}\). On the other hand, by Lemma 07UR we can filter any \(\mathcal{F}\) with successive subquotients as in (2). Hence the lemma follows.

Here is a more useful variant of the lemma above.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\). Assume

1. For any short exact sequence of coherent sheaves \[0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0\] if \(\mathcal{F}_i\), \(i = 1, 2\) have property \(\mathcal{P}\) then so does \(\mathcal{F}\).

2. If \(\mathcal{P}\) holds for \(\mathcal{F}^{\oplus r}\) for some \(r \geq 1\), then it holds for \(\mathcal{F}\).

3. For every reduced closed subspace \(i : Z \to X\) with \(|Z|\) irreducible there exists a coherent sheaf \(\mathcal{G}\) on \(Z\) such that

   1. \(\text{Supp}(\mathcal{G}) = Z\),

   2. for every nonzero quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\) there exists a quasi-coherent subsheaf \(\mathcal{G}' \subset \mathcal{I}\mathcal{G}\) such that \(\text{Supp}(\mathcal{G}/\mathcal{G}')\) is proper closed in \(|Z|\) and such that \(\mathcal{P}\) holds for \(i_*\mathcal{G}'\).

Then property \(\mathcal{P}\) holds for every coherent sheaf on \(X\).

Proof

Consider the collection \[\mathcal{T} = \left\{ \begin{matrix} T \subset |X| \text{ nonempty closed such that there exists a coherent sheaf } \\ \mathcal{F} \text{ with } \text{Supp}(\mathcal{F}) = T \text{ for which the lemma is wrong} \end{matrix} \right\}\] We are trying to show that \(\mathcal{T}\) is empty. If not, then because \(|X|\) is Noetherian (Properties of Spaces, Lemma 04ZF) we can choose a minimal element \(T \in \mathcal{T}\). This means that there exists a coherent sheaf \(\mathcal{F}\) on \(X\) whose support is \(T\) and for which the lemma does not hold.

If \(T\) is not irreducible, then we can write \(T = Z_1 \cup Z_2\) with \(Z_1, Z_2\) closed and strictly smaller than \(T\). Then we can apply Lemma 07UP to get a short exact sequence of coherent sheaves \[0 \to \mathcal{G}_1 \to \mathcal{F} \to \mathcal{G}_2 \to 0\] with \(\text{Supp}(\mathcal{G}_i) \subset Z_i\). By minimality of \(T\) each of \(\mathcal{G}_i\) has \(\mathcal{P}\). Hence \(\mathcal{F}\) has property \(\mathcal{P}\) by (1), a contradiction.

Suppose \(T\) is irreducible. Let \(\mathcal{J}\) be the sheaf of ideals defining the reduced induced closed subspace structure on \(T\), see Properties of Spaces, Lemma 03IQ. By Lemma 07UK we see there exists an \(n \geq 0\) such that \(\mathcal{J}^n\mathcal{F} = 0\). Hence we obtain a filtration \[0 = \mathcal{J}^n\mathcal{F} \subset \mathcal{J}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{J}\mathcal{F} \subset \mathcal{F}\] each of whose successive subquotients is annihilated by \(\mathcal{J}\). Hence if each of these subquotients has a filtration as in the statement of the lemma then also \(\mathcal{F}\) does by (1). In other words we may assume that \(\mathcal{J}\) does annihilate \(\mathcal{F}\).

Assume \(T\) is irreducible and \(\mathcal{J}\mathcal{F} = 0\) where \(\mathcal{J}\) is as above. Denote \(i : Z \to X\) the closed subspace corresponding to \(\mathcal{J}\). Then \(\mathcal{F} = i_*\mathcal{H}\) for some coherent \(\mathcal{O}_Z\)-module \(\mathcal{H}\), see Morphisms of Spaces, Lemma 04CJ and Lemma 07UG. Let \(\mathcal{G}\) be the coherent sheaf on \(Z\) satisfying (3)(a) and (3)(b). We apply Lemma 07UQ to get injective maps \[\mathcal{I}_1^{\oplus r_1} \to \mathcal{H} \quad\text{and}\quad \mathcal{I}_2^{\oplus r_2} \to \mathcal{G}\] where the support of the cokernels are proper closed in \(Z\). Hence we find an nonempty open \(V \subset Z\) such that \[\mathcal{H}^{\oplus r_2}_V \cong \mathcal{G}^{\oplus r_1}_V\] Let \(\mathcal{I} \subset \mathcal{O}_Z\) be a quasi-coherent ideal sheaf cutting out \(Z \setminus V\) we obtain (Lemma 07UM) a map \[\mathcal{I}^n\mathcal{G}^{\oplus r_1} \longrightarrow \mathcal{H}^{\oplus r_2}\] which is an isomorphism over \(V\). The kernel is supported on \(Z \setminus V\) hence annihilated by some power of \(\mathcal{I}\), see Lemma 07UK. Thus after increasing \(n\) we may assume the displayed map is injective, see Lemma 07UL. Applying (3)(b) we find \(\mathcal{G}' \subset \mathcal{I}^n\mathcal{G}\) such that \[(i_*\mathcal{G}')^{\oplus r_1} \longrightarrow i_*\mathcal{H}^{\oplus r_2} = \mathcal{F}^{\oplus r_2}\] is injective with cokernel supported in a proper closed subset of \(Z\) and such that property \(\mathcal{P}\) holds for \(i_*\mathcal{G}'\). By (1) property \(\mathcal{P}\) holds for \((i_*\mathcal{G}')^{\oplus r_1}\). By (1) and minimality of \(T = |Z|\) property \(\mathcal{P}\) holds for \(\mathcal{F}^{\oplus r_2}\). And finally by (2) property \(\mathcal{P}\) holds for \(\mathcal{F}\) which is the desired contradiction.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\). Assume

1. For any short exact sequence of coherent sheaves on \(X\) if two out of three have property \(\mathcal{P}\) so does the third.

2. If \(\mathcal{P}\) holds for \(\mathcal{F}^{\oplus r}\) for some \(r \geq 1\), then it holds for \(\mathcal{F}\).

3. For every reduced closed subspace \(i : Z \to X\) with \(|Z|\) irreducible there exists a coherent sheaf \(\mathcal{G}\) on \(X\) whose scheme theoretic support is \(Z\) such that \(\mathcal{P}\) holds for \(\mathcal{G}\).

Then property \(\mathcal{P}\) holds for every coherent sheaf on \(X\).

Proof

We will show that conditions (1) and (2) of Lemma 07US hold. This is clear for condition (1). To show that (2) holds, let \[\mathcal{T} = \left\{ \begin{matrix} i : Z \to X \text{ reduced closed subspace with }|Z|\text{ irreducible such}\\ \text{ that }i_*\mathcal{I}\text{ does not have }\mathcal{P} \text{ for some quasi-coherent }\mathcal{I} \subset \mathcal{O}_Z \end{matrix} \right\}\] If \(\mathcal{T}\) is nonempty, then since \(X\) is Noetherian, we can find an \(i : Z \to X\) which is minimal in \(\mathcal{T}\). We will show that this leads to a contradiction.

Let \(\mathcal{G}\) be the sheaf whose scheme theoretic support is \(Z\) whose existence is assumed in assumption (3). Let \(\varphi : i_*\mathcal{I}^{\oplus r} \to \mathcal{G}\) be as in Lemma 07UQ. Let \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \Coker(\varphi)\] be a filtration as in Lemma 07UR. By minimality of \(Z\) and assumption (1) we see that \(\Coker(\varphi)\) has property \(\mathcal{P}\). As \(\varphi\) is injective we conclude using assumption (1) once more that \(i_*\mathcal{I}^{\oplus r}\) has property \(\mathcal{P}\). Using assumption (2) we conclude that \(i_*\mathcal{I}\) has property \(\mathcal{P}\).

Finally, if \(\mathcal{J} \subset \mathcal{O}_Z\) is a second quasi-coherent sheaf of ideals, set \(\mathcal{K} = \mathcal{I} \cap \mathcal{J}\) and consider the short exact sequences \[0 \to \mathcal{K} \to \mathcal{I} \to \mathcal{I}/\mathcal{K} \to 0 \quad \text{and} \quad 0 \to \mathcal{K} \to \mathcal{J} \to \mathcal{J}/\mathcal{K} \to 0\] Arguing as above, using the minimality of \(Z\), we see that \(i_*\mathcal{I}/\mathcal{K}\) and \(i_*\mathcal{J}/\mathcal{K}\) satisfy \(\mathcal{P}\). Hence by assumption (1) we conclude that \(i_*\mathcal{K}\) and then \(i_*\mathcal{J}\) satisfy \(\mathcal{P}\). In other words, \(Z\) is not an element of \(\mathcal{T}\) which is the desired contradiction.

Limits of coherent modules

A colimit of coherent modules (on a locally Noetherian algebraic space) is typically not coherent. But it is quasi-coherent as any colimit of quasi-coherent modules on an algebraic space is quasi-coherent, see Properties of Spaces, Lemma 03M1. Conversely, if the algebraic space is Noetherian, then every quasi-coherent module is a filtered colimit of coherent modules.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Every quasi-coherent \(\mathcal{O}_X\)-module is the filtered colimit of its coherent submodules.

Proof

Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(\mathcal{G}, \mathcal{H} \subset \mathcal{F}\) are coherent \(\mathcal{O}_X\)-submodules then the image of \(\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}\) is another coherent \(\mathcal{O}_X\)-submodule which contains both of them (see Lemmas 07UC and 07UD). In this way we see that the system is directed. Hence it now suffices to show that \(\mathcal{F}\) can be written as a filtered colimit of coherent modules, as then we can take the images of these modules in \(\mathcal{F}\) to conclude there are enough of them.

Let \(U\) be an affine scheme and \(U \to X\) a surjective étale morphism. Set \(R = U \times_X U\) so that \(X = U/R\) as usual. By Properties of Spaces, Proposition 03M3 we see that \(\QCoh(\mathcal{O}_X) = \QCoh(U, R, s, t, c)\). Hence we reduce to showing the corresponding thing for \(\QCoh(U, R, s, t, c)\). Thus the result follows from the more general Groupoids, Lemma 07TU.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\) with \(Y\) Noetherian. Then every quasi-coherent \(\mathcal{O}_X\)-module is a filtered colimit of finitely presented \(\mathcal{O}_X\)-modules.

Proof

Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Write \(f_*\mathcal{F} = \colim \mathcal{H}_i\) with \(\mathcal{H}_i\) a coherent \(\mathcal{O}_Y\)-module, see Lemma 07UV. By Lemma 07UB the modules \(\mathcal{H}_i\) are \(\mathcal{O}_Y\)-modules of finite presentation. Hence \(f^*\mathcal{H}_i\) is an \(\mathcal{O}_X\)-module of finite presentation, see Properties of Spaces, Section 05VR. We claim the map \[\colim f^*\mathcal{H}_i = f^*f_*\mathcal{F} \to \mathcal{F}\] is surjective as \(f\) is assumed affine, Namely, choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Set \(U = X \times_Y V\). Then \(U\) is a scheme, \(f' : U \to V\) is affine, and \(U \to X\) is surjective étale. By Properties of Spaces, Lemma 03LX we see that \(f'_*(\mathcal{F}|_U) = f_*\mathcal{F}|_V\) and similarly for pullbacks. Thus the restriction of \(f^*f_*\mathcal{F} \to \mathcal{F}\) to \(U\) is the map \[f^*f_*\mathcal{F}|_U = (f')^*(f_*\mathcal{F})|_V) = (f')^*f'_*(\mathcal{F}|_U) \to \mathcal{F}|_U\] which is surjective as \(f'\) is an affine morphism of schemes. Hence the claim holds.

We conclude that every quasi-coherent module on \(X\) is a quotient of a filtered colimit of finitely presented modules. In particular, we see that \(\mathcal{F}\) is a cokernel of a map \[\colim_{j \in J} \mathcal{G}_j \longrightarrow \colim_{i \in I} \mathcal{H}_i\] with \(\mathcal{G}_j\) and \(\mathcal{H}_i\) finitely presented. Note that for every \(j \in I\) there exist \(i \in I\) and a morphism \(\alpha : \mathcal{G}_j \to \mathcal{H}_i\) such that \[\xymatrix{ \mathcal{G}_j \ar[r]_\alpha \ar[d] & \mathcal{H}_i \ar[d] \\ \colim_{j \in J} \mathcal{G}_j \ar[r] & \colim_{i \in I} \mathcal{H}_i }\] commutes, see Lemma 07U7. In this situation \(\Coker(\alpha)\) is a finitely presented \(\mathcal{O}_X\)-module which comes endowed with a map \(\Coker(\alpha) \to \mathcal{F}\). Consider the set \(K\) of triples \((i, j, \alpha)\) as above. We say that \((i, j, \alpha) \leq (i', j', \alpha')\) if and only if \(i \leq i'\), \(j \leq j'\), and the diagram \[\xymatrix{ \mathcal{G}_j \ar[r]_\alpha \ar[d] & \mathcal{H}_i \ar[d] \\ \mathcal{G}_{j'} \ar[r]^{\alpha'} & \mathcal{H}_{i'} }\] commutes. It follows from the above that \(K\) is a directed partially ordered set, \[\mathcal{F} = \colim_{(i, j, \alpha) \in K} \Coker(\alpha),\] and we win.

Vanishing of cohomology

In this section we show that a quasi-compact and quasi-separated algebraic space is affine if it has vanishing higher cohomology for all quasi-coherent sheaves. We do this in a sequence of lemmas all of which will become obsolete once we prove Proposition 07V6.

Situation

Here \(S\) is a scheme and \(X\) is a quasi-compact and quasi-separated algebraic space over \(S\) with the following property: For every quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(H^1(X, \mathcal{F}) = 0\). We set \(A = \Gamma(X, \mathcal{O}_X)\).

We would like to show that the canonical morphism \[p : X \longrightarrow \Spec(A)\] (see Properties of Spaces, Lemma 05Z1) is an isomorphism. If \(M\) is an \(A\)-module we denote \(M \otimes_A \mathcal{O}_X\) the quasi-coherent module \(p^*\tilde M\).

Lemma

In Situation 07UY for an \(A\)-module \(M\) we have \(p_*(M \otimes_A \mathcal{O}_X) = \widetilde{M}\) and \(\Gamma(X, M \otimes_A \mathcal{O}_X) = M\).

Proof

The equality \(p_*(M \otimes_A \mathcal{O}_X) = \widetilde{M}\) follows from the equality \(\Gamma(X, M \otimes_A \mathcal{O}_X) = M\) as \(p_*(M \otimes_A \mathcal{O}_X)\) is a quasi-coherent module on \(\Spec(A)\) by Morphisms of Spaces, Lemma 03M9. Observe that \(\Gamma(X, \bigoplus_{i \in I} \mathcal{O}_X) = \bigoplus_{i \in I} A\) by Lemma 073E. Hence the lemma holds for free modules. Choose a short exact sequence \(F_1 \to F_0 \to M\) where \(F_0, F_1\) are free \(A\)-modules. Since \(H^1(X, -)\) is zero the global sections functor is right exact. Moreover the pullback \(p^*\) is right exact as well. Hence we see that \[\Gamma(X, F_1 \otimes_A \mathcal{O}_X) \to \Gamma(X, F_0 \otimes_A \mathcal{O}_X) \to \Gamma(X, M \otimes_A \mathcal{O}_X) \to 0\] is exact. The result follows.

The following lemma shows that Situation 07UY is preserved by base change of \(X \to \Spec(A)\) by \(\Spec(A') \to \Spec(A)\).

Lemma

In Situation 07UY.

1. Given an affine morphism \(X' \to X\) of algebraic spaces, we have \(H^1(X', \mathcal{F}') = 0\) for every quasi-coherent \(\mathcal{O}_{X'}\)-module \(\mathcal{F}'\).

2. Given an \(A\)-algebra \(A'\) setting \(X' = X \times_{\Spec(A)} \Spec(A')\) the morphism \(X' \to X\) is affine and \(\Gamma(X', \mathcal{O}_{X'}) = A'\).

Proof

Part (1) follows from Lemma 073H and the Leray spectral sequence (Cohomology on Sites, Lemma 0732). Let \(A \to A'\) be as in (2). Then \(X' \to X\) is affine because affine morphisms are preserved under base change (Morphisms of Spaces, Lemma 03WI) and the fact that a morphism of affine schemes is affine. The equality \(\Gamma(X', \mathcal{O}_{X'}) = A'\) follows as \((X' \to X)_*\mathcal{O}_{X'} = A' \otimes_A \mathcal{O}_X\) by Lemma 07U8 and thus \[\Gamma(X', \mathcal{O}_{X'}) = \Gamma(X, (X' \to X)_*\mathcal{O}_{X'}) = \Gamma(X, A' \otimes_A \mathcal{O}_X) = A'\] by Lemma 07UZ.

Lemma

In Situation 07UY. Let \(Z_0, Z_1 \subset |X|\) be disjoint closed subsets. Then there exists an \(a \in A\) such that \(Z_0 \subset V(a)\) and \(Z_1 \subset V(a - 1)\).

Proof

We may and do endow \(Z_0\), \(Z_1\) with the reduced induced subspace structure (Properties of Spaces, Definition 047X) and we denote \(i_0 : Z_0 \to X\) and \(i_1 : Z_1 \to X\) the corresponding closed immersions. Since \(Z_0 \cap Z_1 = \emptyset\) we see that the canonical map of quasi-coherent \(\mathcal{O}_X\)-modules \[\mathcal{O}_X \longrightarrow i_{0, *}\mathcal{O}_{Z_0} \oplus i_{1, *}\mathcal{O}_{Z_1}\] is surjective (look at stalks at geometric points). Since \(H^1(X, -)\) is zero on the kernel of this map the induced map of global sections is surjective. Thus we can find \(a \in A\) which maps to the global section \((0, 1)\) of the right hand side.

Lemma

In Situation 07UY the morphism \(p : X \to \Spec(A)\) is universally injective.

Proof

Let \(A \to k\) be a ring homomorphism where \(k\) is a field. It suffices to show that \(\Spec(k) \times_{\Spec(A)} X\) has at most one point (see Morphisms of Spaces, Lemma 03MX). Using Lemma 07V0 we may assume that \(A\) is a field and we have to show that \(|X|\) has at most one point.

Let’s think of \(X\) as an algebraic space over \(\Spec(k)\) and let’s use the notation \(X(K)\) to denote \(K\)-valued points of \(X\) for any extension \(K/k\), see Morphisms of Spaces, Section 0485. If \(K/k\) is an algebraically closed field extension of large transcendence degree, then we see that \(X(K) \to |X|\) is surjective, see Morphisms of Spaces, Lemma 0488. Hence, after replacing \(k\) by \(K\), we see that it suffices to prove that \(X(k)\) is a singleton (in the case \(A = k)\).

Let \(x, x' \in X(k)\). By Decent Spaces, Lemma 07U5 we see that \(x\) and \(x'\) are closed points of \(|X|\). Hence \(x\) and \(x'\) map to distinct points of \(\Spec(k)\) if \(x \not = x'\) by Lemma 07V1. We conclude that \(x = x'\) as desired.

Lemma

In Situation 07UY the morphism \(p : X \to \Spec(A)\) is separated.

Proof

By Decent Spaces, Lemma 09YB we can find a scheme \(Y\) and a surjective integral morphism \(Y \to X\). Since an integral morphism is affine, we can apply Lemma 07V0 to see that \(H^1(Y, \mathcal{G}) = 0\) for every quasi-coherent \(\mathcal{O}_Y\)-module \(\mathcal{G}\). Since \(Y \to X\) is quasi-compact and \(X\) is quasi-compact, we see that \(Y\) is quasi-compact. Since \(Y\) is a scheme, we may apply Cohomology of Schemes, Lemma 01XF to see that \(Y\) is affine. Hence \(Y\) is separated. Note that an integral morphism is affine and universally closed, see Morphisms of Spaces, Lemma 0415. By Morphisms of Spaces, Lemma 05Z2 we see that \(X\) is a separated algebraic space.

Proposition

A quasi-compact and quasi-separated algebraic space is affine if and only if all higher cohomology groups of quasi-coherent sheaves vanish. More precisely, any algebraic space as in Situation 07UY is an affine scheme.

Proof

Choose an affine scheme \(U = \Spec(B)\) and a surjective étale morphism \(\varphi : U \to X\). Set \(R = U \times_X U\). As \(p\) is separated (Lemma 07V5) we see that \(R\) is a closed subscheme of \(U \times_{\Spec(A)} U = \Spec(B \otimes_A B)\). Hence \(R = \Spec(C)\) is affine too and the ring map \[B \otimes_A B \longrightarrow C\] is surjective. Let us denote the two maps \(s, t : B \to C\) as usual. Pick \(g_1, \ldots, g_m \in B\) such that \(s(g_1), \ldots, s(g_m)\) generate \(C\) over \(t : B \to C\) (which is possible as \(t : B \to C\) is of finite presentation and the displayed map is surjective). Then \(g_1, \ldots, g_m\) give global sections of \(\varphi_*\mathcal{O}_U\) and the map \[\mathcal{O}_X[z_1, \ldots, z_n] \longrightarrow \varphi_*\mathcal{O}_U, \quad z_j \longmapsto g_j\] is surjective: you can check this by restricting to \(U\). Namely, \(\varphi^*\varphi_*\mathcal{O}_U = t_*\mathcal{O}_R\) (by Lemma 073K) hence you get exactly the condition that \(s(g_i)\) generate \(C\) over \(t : B \to C\). By the vanishing of \(H^1\) of the kernel we see that \[\Gamma(X, \mathcal{O}_X[x_1, \ldots, x_n]) = A[x_1, \ldots, x_n] \longrightarrow \Gamma(X, \varphi_*\mathcal{O}_U) = \Gamma(U, \mathcal{O}_U) = B\] is surjective. Thus we conclude that \(B\) is a finite type \(A\)-algebra. Hence \(X \to \Spec(A)\) is of finite type and separated. By Lemma 07V4 and Morphisms of Spaces, Lemma 06RW it is also locally quasi-finite. Hence \(X \to \Spec(A)\) is representable by Morphisms of Spaces, Lemma 0418 and \(X\) is a scheme. Finally \(X\) is affine, hence equal to \(\Spec(A)\), by an application of Cohomology of Schemes, Lemma 01XF.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Assume that for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(H^1(X, \mathcal{F}) = 0\). Then \(X\) is an affine scheme.

Proof

The assumption implies that \(H^1(X, \mathcal{F}) = 0\) for every quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) by Lemmas 07UV and 073E. Then \(X\) is affine by Proposition 07V6.

Lemma

Let \(S\) be a scheme. Let \(X\) be a Noetherian algebraic space over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume that for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) there exists an \(n \geq 1\) such that \(H^1(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0\). Then \(X\) is a scheme and \(\mathcal{L}\) is ample on \(X\).

Proof

Let \(s \in H^0(X, \mathcal{L}^{\otimes d})\) be a global section. Let \(U \subset X\) be the open subspace over which \(s\) is a generator of \(\mathcal{L}^{\otimes d}\). In particular we have \(\mathcal{L}^{\otimes d}|_U \cong \mathcal{O}_U\). We claim that \(U\) is affine.

Proof of the claim. We will show that \(H^1(U, \mathcal{F}) = 0\) for every quasi-coherent \(\mathcal{O}_U\)-module \(\mathcal{F}\). This will prove the claim by Proposition 07V6. Denote \(j : U \to X\) the inclusion morphism. Since étale locally the morphism \(j\) is affine (by Morphisms, Lemma 01SF) we see that \(j\) is affine (Morphisms of Spaces, Lemma 03WG). Hence we have \[H^1(U, \mathcal{F}) = H^1(X, j_*\mathcal{F})\] by Lemma 073H (and Cohomology on Sites, Lemma 0733). Write \(j_*\mathcal{F} = \colim \mathcal{F}_i\) as a filtered colimit of coherent \(\mathcal{O}_X\)-modules, see Lemma 07UV. Then \[H^1(X, j_*\mathcal{F}) = \colim H^1(X, \mathcal{F}_i)\] by Lemma 073E. Thus it suffices to show that \(H^1(X, \mathcal{F}_i)\) maps to zero in \(H^1(U, j^*\mathcal{F}_i)\). By assumption there exists an \(n \geq 1\) such that \[H^1(X, \mathcal{F}_i \otimes_{\mathcal{O}_X} (\mathcal{O}_X \oplus \mathcal{L} \oplus \ldots \oplus \mathcal{L}^{\otimes d - 1}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0\] Hence there exists an \(a \geq 0\) such that \(H^1(X, \mathcal{F}_i \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes ad}) = 0\). On the other hand, the map \[s^a : \mathcal{F}_i \longrightarrow \mathcal{F}_i \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes ad}\] is an isomorphism after restriction to \(U\). Contemplating the commutative diagram \[\xymatrix{ H^1(X, \mathcal{F}_i) \ar[r] \ar[d]_{s^a} & H^1(U, j^*\mathcal{F}_i) \ar[d]^{\cong} \\ H^1(X, \mathcal{F}_i \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes ad}) \ar[r] & H^1(U, j^*(\mathcal{F}_i \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes ad})) }\] we conclude that the map \(H^1(X, \mathcal{F}_i) \to H^1(U, j^*\mathcal{F}_i)\) is zero and the claim holds.

Let \(x \in |X|\) be a closed point. By Decent Spaces, Lemma 0AHB we can represent \(x\) by a closed immersion \(i : \Spec(k) \to X\) (this also uses that a quasi-separated algebraic space is decent, see Decent Spaces, Section 03I7). Thus \(\mathcal{O}_X \to i_*\mathcal{O}_{\Spec(k)}\) is surjective. Let \(\mathcal{I} \subset \mathcal{O}_X\) be the kernel and choose \(d \geq 1\) such that \(H^1(X, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0\). Then \[H^0(X, \mathcal{L}^{\otimes d}) \to H^0(X, i_*\mathcal{O}_{\Spec(k)} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = H^0(\Spec(k), i^*\mathcal{L}^{\otimes d}) \cong k\] is surjective by the long exact cohomology sequence. Hence there exists an \(s \in H^0(X, \mathcal{L}^{\otimes d})\) such that \(x \in U\) where \(U\) is the open subspace corresponding to \(s\) as above. Thus \(x\) is in the schematic locus (see Properties of Spaces, Lemma 03JH) of \(X\) by our claim.

To conclude that \(X\) is a scheme, it suffices to show that any open subset of \(|X|\) which contains all the closed points is equal to \(|X|\). This follows from the fact that \(|X|\) is a Noetherian topological space, see Properties of Spaces, Lemma 04ZG. Finally, if \(X\) is a scheme, then we can apply Cohomology of Schemes, Lemma 0B5P to conclude that \(\mathcal{L}\) is ample.

Finite morphisms and affines

This section is the analogue of Cohomology of Schemes, Section 01YN.

Lemma

Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is finite, surjective and \(X\) locally Noetherian. Let \(i : Z \to X\) be a closed immersion. Denote \(i' : Z' \to Y\) the inverse image of \(Z\) (Morphisms of Spaces, Section 03MA) and \(f' : Z' \to Z\) the induced morphism. Then \(\mathcal{G} = f'_*\mathcal{O}_{Z'}\) is a coherent \(\mathcal{O}_Z\)-module whose support is \(Z\).

Proof

Observe that \(f'\) is the base change of \(f\) and hence is finite and surjective by Morphisms of Spaces, Lemmas 03MH and 03ZS. Note that \(Y\), \(Z\), and \(Z'\) are locally Noetherian by Morphisms of Spaces, Lemma 04ZK (and the fact that closed immersions and finite morphisms are of finite type). By Lemma 07UH we see that \(\mathcal{G}\) is a coherent \(\mathcal{O}_Z\)-module. The support of \(\mathcal{G}\) is closed in \(|Z|\), see Morphisms of Spaces, Lemma 07TZ. Hence if the support of \(\mathcal{G}\) is not equal to \(|Z|\), then after replacing \(X\) by an open subspace we may assume \(\mathcal{G} = 0\) but \(Z \not = \emptyset\). This would mean that \(f'_*\mathcal{O}_{Z'} = 0\). In particular the section \(1 \in \Gamma(Z', \mathcal{O}_{Z'}) = \Gamma(Z, f'_*\mathcal{O}_{Z'})\) would be zero which would imply \(Z' = \emptyset\) is the empty algebraic space. This is impossible as \(Z' \to Z\) is surjective.

Lemma

Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(Y\). Let \(\mathcal{I}\) be a quasi-coherent sheaf of ideals on \(X\). If \(f\) is affine then \(\mathcal{I}f_*\mathcal{F} = f_*(f^{-1}\mathcal{I}\mathcal{F})\) (with notation as explained in the proof).

Proof

The notation means the following. Since \(f^{-1}\) is an exact functor we see that \(f^{-1}\mathcal{I}\) is a sheaf of ideals of \(f^{-1}\mathcal{O}_X\). Via the map \(f^\sharp : f^{-1}\mathcal{O}_X \to \mathcal{O}_Y\) on \(Y_\etale\) this acts on \(\mathcal{F}\). Then \(f^{-1}\mathcal{I}\mathcal{F}\) is the subsheaf generated by sums of local sections of the form \(as\) where \(a\) is a local section of \(f^{-1}\mathcal{I}\) and \(s\) is a local section of \(\mathcal{F}\). It is a quasi-coherent \(\mathcal{O}_Y\)-submodule of \(\mathcal{F}\) because it is also the image of a natural map \(f^*\mathcal{I} \otimes_{\mathcal{O}_Y} \mathcal{F} \to \mathcal{F}\).

Having said this the proof is straightforward. Namely, the question is étale local on \(X\) and hence we may assume \(X\) is an affine scheme. In this case the result is a consequence of the corresponding result for schemes, see Cohomology of Schemes, Lemma 01YP.

Lemma

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

1. \(f\) finite,

2. \(f\) surjective,

3. \(Y\) affine, and

4. \(X\) Noetherian.

Then \(X\) is affine.

Proof

We will prove that under the assumptions of the lemma for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(H^1(X, \mathcal{F}) = 0\). This implies that \(H^1(X, \mathcal{F}) = 0\) for every quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) by Lemmas 07UV and 073E. Then it follows that \(X\) is affine from Proposition 07V6.

Let \(\mathcal{P}\) be the property of coherent sheaves \(\mathcal{F}\) on \(X\) defined by the rule \[\mathcal{P}(\mathcal{F}) \Leftrightarrow H^1(X, \mathcal{F}) = 0.\] We are going to apply Lemma 07UT. Thus we have to verify (1), (2) and (3) of that lemma for \(\mathcal{P}\). Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves. Property (2) follows since \(H^1(X, -)\) is an additive functor. To see (3) let \(i : Z \to X\) be a reduced closed subspace with \(|Z|\) irreducible. Let \(i' : Z' \to Y\) and \(f' : Z' \to Z\) be as in Lemma 0GF7 and set \(\mathcal{G} = f'_*\mathcal{O}_{Z'}\). We claim that \(\mathcal{G}\) satisfies properties (3)(a) and (3)(b) of Lemma 07UT which will finish the proof. Property (3)(a) we have seen in Lemma 0GF7. To see (3)(b) let \(\mathcal{I}\) be a nonzero quasi-coherent sheaf of ideals on \(Z\). Denote \(\mathcal{I}' \subset \mathcal{O}_{Z'}\) the quasi-coherent ideal \((f')^{-1}\mathcal{I} \mathcal{O}_{Z'}\), i.e., the image of \((f')^*\mathcal{I} \to \mathcal{O}_{Z'}\). By Lemma 0GF8 we have \(f_*\mathcal{I}' = \mathcal{I} \mathcal{G}\). We claim the common value \(\mathcal{G}' = \mathcal{I} \mathcal{G} = f'_*\mathcal{I}'\) satisfies the condition expressed in (3)(b). First, it is clear that the support of \(\mathcal{G}/\mathcal{G}'\) is contained in the support of \(\mathcal{O}_Z/\mathcal{I}\) which is a proper subspace of \(|Z|\) as \(\mathcal{I}\) is a nonzero ideal sheaf on the reduced and irreducible algebraic space \(Z\). The morphism \(f'\) is affine, hence \(R^1f'_*\mathcal{I}' = 0\) by Lemma 073H. As \(Z'\) is affine (as a closed subscheme of an affine scheme) we have \(H^1(Z', \mathcal{I}') = 0\). Hence the Leray spectral sequence (in the form Cohomology on Sites, Lemma 0733) implies that \(H^1(Z, f'_*\mathcal{I}') = 0\). Since \(i : Z \to X\) is affine we conclude that \(R^1i_*f'_*\mathcal{I}' = 0\) hence \(H^1(X, i_*f'_*\mathcal{I}') = 0\) by Leray again. In other words, we have \(H^1(X, i_*\mathcal{G}') = 0\) as desired.

A weak version of Chow’s lemma

In this section we quickly prove the following lemma in order to help us prove the basic results on cohomology of coherent modules on proper algebraic spaces.

Lemma

Let \(A\) be a ring. Let \(X\) be an algebraic space over \(\Spec(A)\) whose structure morphism \(X \to \Spec(A)\) is separated of finite type. Then there exists a proper surjective morphism \(X' \to X\) where \(X'\) is a scheme which is H-quasi-projective over \(\Spec(A)\).

Proof

Let \(W\) be an affine scheme and let \(f : W \to X\) be a surjective étale morphism. There exists an integer \(d\) such that all geometric fibres of f have \(\leq d\) points (because \(X\) is a separated algebraic hence reasonable, see Decent Spaces, Lemma 03JX). Picking \(d\) minimal we get a nonempty open \(U \subset X\) such that \(f^{-1}(U) \to U\) is finite étale of degree \(d\), see Decent Spaces, Lemma 07S8. Let \[V \subset W \times_X W \times_X \ldots \times_X W\] (\(d\) factors in the fibre product) be the complement of all the diagonals. Because \(W \to X\) is separated the diagonal \(W \to W \times_X W\) is a closed immersion. Since \(W \to X\) is étale the diagonal \(W \to W \times_X W\) is an open immersion, see Morphisms of Spaces, Lemmas 06CR and 05W1. Hence the diagonals are open and closed subschemes of the quasi-compact scheme \(W \times_X \ldots \times_X W\). In particular we conclude \(V\) is a quasi-compact scheme. Choose an open immersion \(W \subset Y\) with \(Y\) H-projective over \(A\) (this is possible as \(W\) is affine and of finite type over \(A\); for example we can use Morphisms, Lemmas 04II and 01WA). Let \[Z \subset Y \times_A Y \times_A \ldots \times_A Y\] be the scheme theoretic image of the composition \(V \to W \times_X \ldots \times_X W \to Y \times_A \ldots \times_A Y\). Observe that this morphism is quasi-compact since \(V\) is quasi-compact and \(Y \times_A \ldots \times_A Y\) is separated. Note that \(V \to Z\) is an open immersion as \(V \to Y \times_A \ldots \times_A Y\) is an immersion, see Morphisms, Lemma 01RG. The projection morphisms give \(d\) morphisms \(g_i : Z \to Y\). These morphisms \(g_i\) are projective as \(Y\) is projective over \(A\), see material in Morphisms, Section 01W7. We set \[X' = \bigcup g_i^{-1}(W) \subset Z\] There is a morphism \(X' \to X\) whose restriction to \(g_i^{-1}(W)\) is the composition \(g_i^{-1}(W) \to W \to X\). Namely, these morphisms agree over \(V\) hence agree over \(g_i^{-1}(W) \cap g_j^{-1}(W)\) by Morphisms of Spaces, Lemma 084N. Claim: the morphism \(X' \to X\) is proper.

If the claim holds, then the lemma follows by induction on \(d\). Namely, by construction \(X'\) is H-quasi-projective over \(\Spec(A)\). The image of \(X' \to X\) contains the open \(U\) as \(V\) surjects onto \(U\). Denote \(T\) the reduced induced algebraic space structure on \(X \setminus U\). Then \(T \times_X W\) is a closed subscheme of \(W\), hence affine. Moreover, the morphism \(T \times_X W \to T\) is étale and every geometric fibre has \(< d\) points. By induction hypothesis there exists a proper surjective morphism \(T' \to T\) where \(T'\) is a scheme H-quasi-projective over \(\Spec(A)\). Since \(T\) is a closed subspace of \(X\) we see that \(T' \to X\) is a proper morphism. Thus the lemma follows by taking the proper surjective morphism \(X' \amalg T' \to X\).

Proof of the claim. By construction the morphism \(X' \to X\) is separated and of finite type. We will check conditions (1) – (4) of Morphisms of Spaces, Lemma 089G for the morphisms \(V \to X'\) and \(X' \to X\). Conditions (1) and (2) we have seen above. Condition (3) holds as \(X' \to X\) is separated (as a morphism whose source is a separated algebraic space). Thus it suffices to check liftability to \(X'\) for diagrams \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & V \ar[d] \\ \Spec(R) \ar[r] & X }\] where \(R\) is a valuation ring with fraction field \(K\). Note that the top horizontal map is given by \(d\) pairwise distinct \(K\)-valued points \(w_1, \ldots, w_d\) of \(W\). In fact, this is a complete set of inverse images of the point \(x \in X(K)\) coming from the diagram. Since \(W \to X\) is surjective, we can, after possibly replacing \(R\) by an extension of valuation rings, lift the morphism \(\Spec(R) \to X\) to a morphism \(w : \Spec(R) \to W\), see Morphisms of Spaces, Lemma 089F. Since \(w_1, \ldots, w_d\) is a complete collection of inverse images of \(x\) we see that \(w|_{\Spec(K)}\) is equal to one of them, say \(w_i\). Thus we see that we get a commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & Z \ar[d]_{g_i}\\ \Spec(R) \ar[r]^w & Y }\] By the valuative criterion of properness for the projective morphism \(g_i\) we can lift \(w\) to \(z : \Spec(R) \to Z\), see Morphisms, Lemma 01WC and Schemes, Proposition 01KF. The image of \(z\) is in \(g_i^{-1}(W) \subset X'\) and the proof is complete.

Noetherian valuative criterion

We prove a version of the valuative criterion for properness using discrete valuation rings. More precise (and therefore more technical) versions can be found in Limits of Spaces, Section 0CMB.

Lemma

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

1. \(Y\) is locally Noetherian,

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

3. for every commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y }\] where \(A\) is a discrete valuation ring and \(K\) its fraction field, there is at most one dotted arrow making the diagram commute.

Then \(f\) is separated.

Proof

We have to show that the diagonal \(\Delta : X \to X \times_Y X\) is a closed immersion. We already know \(\Delta\) is representable, separated, a monomorphism, and locally of finite type, see Morphisms of Spaces, Lemma 03HK. Choose an affine scheme \(U\) and an étale morphism \(U \to X \times_Y X\). Set \(V = X \times_{\Delta, X \times_Y X} U\). It suffices to show that \(V \to U\) is a closed immersion (Morphisms of Spaces, Lemma 03M4). Since \(X \times_Y X\) is locally of finite type over \(Y\) we see that \(U\) is Noetherian (use Morphisms of Spaces, Lemmas 03XG, 03XH, and 04ZK). Note that \(V\) is a scheme as \(\Delta\) is representable. Also, \(V\) is quasi-compact because \(f\) is quasi-separated. Hence \(V \to U\) is of finite type. Consider a commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & V \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & U }\] of morphisms of schemes where \(A\) is a discrete valuation ring with fraction field \(K\). We can interpret the composition \(\Spec(A) \to U \to X \times_Y X\) as a pair of morphisms \(a, b : \Spec(A) \to X\) agreeing as morphisms into \(Y\) and equal when restricted to \(\Spec(K)\). Hence our assumption (3) guarantees \(a = b\) and we find the dotted arrow in the diagram. By Limits, Lemma 0208 we conclude that \(V \to U\) is proper. In other words, \(\Delta\) is proper. Since \(\Delta\) is a monomorphism, we find that \(\Delta\) is a closed immersion (Étale Morphisms, Lemma 04XV) as desired.

Lemma

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

1. \(Y\) is locally Noetherian,

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

3. for every commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y }\] where \(A\) is a discrete valuation ring and \(K\) its fraction field, there is a unique dotted arrow making the diagram commute.

Then \(f\) is proper.

Proof

It suffices to prove \(f\) is universally closed because \(f\) is separated by Lemma 0ARJ. To do this we may work étale locally on \(Y\) (Morphisms of Spaces, Lemma 03IT). Hence we may assume \(Y = \Spec(A)\) is a Noetherian affine scheme. Choose \(X' \to X\) as in the weak form of Chow’s lemma (Lemma 089J). We claim that \(X' \to \Spec(A)\) is universally closed. The claim implies the lemma by Morphisms of Spaces, Lemma 08AJ. To prove this, according to Limits, Lemma 05JY it suffices to prove that in every solid commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X' \ar[r] & X \ar[d] \\ \Spec(A) \ar[rr] \ar@{-->}[ru]^a \ar@{-->}[rru]_b & & Y }\] where \(A\) is a dvr with fraction field \(K\) we can find the dotted arrow \(a\). By assumption we can find the dotted arrow \(b\). Then the morphism \(X' \times_{X, b} \Spec(A) \to \Spec(A)\) is a proper morphism of schemes and by the valuative criterion for morphisms of schemes we can lift \(b\) to the desired morphism \(a\).

Remark

In Lemmas 0ARJ and 0ARK it suffices to consider complete discrete valuation rings. To be precise in Lemma 0ARJ we can replace condition (3) by the following condition: Given any commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y }\] where \(A\) is a complete discrete valuation ring with fraction field \(K\) there exists at most one dotted arrow making the diagram commute. Namely, given any diagram as in Lemma 0ARJ (3) the completion \(A^\wedge\) is a discrete valuation ring (More on Algebra, Lemma 0AP1) and the uniqueness of the arrow \(\Spec(A^\wedge) \to X\) implies the uniqueness of the arrow \(\Spec(A) \to X\) for example by Properties of Spaces, Proposition 0APL. Similarly in Lemma 0ARK we can replace condition (3) by the following condition: Given any commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] & Y }\] where \(A\) is a complete discrete valuation ring with fraction field \(K\) there exists an extension \(A \subset A'\) of complete discrete valuation rings inducing a fraction field extension \(K \subset K'\) such that there exists a unique arrow \(\Spec(A') \to X\) making the diagram \[\xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y }\] commute. Namely, given any diagram as in Lemma 0ARK part (3) the existence of any commutative diagram \[\xymatrix{ \Spec(L) \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(B) \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y }\] for any extension \(A \subset B\) of discrete valuation rings will imply there exists an arrow \(\Spec(A) \to X\) fitting into the diagram. This was shown in Morphisms of Spaces, Lemma 0ARH. In fact, it follows from these considerations that it suffices to look for dotted arrows in diagrams for any class of discrete valuation rings such that, given any discrete valuation ring, there is an extension of it that is in the class. For example, we could take complete discrete valuation rings with algebraically closed residue field.

Higher direct images of coherent sheaves

In this section we prove the fundamental fact that the higher direct images of a coherent sheaf under a proper morphism are coherent. First we prove a helper lemma.

Lemma

Let \(S\) be a scheme. Consider a commutative diagram \[\xymatrix{ X \ar[r]_i \ar[rd]_f & \mathbf{P}^n_Y \ar[d] \\ & Y }\] of algebraic spaces over \(S\). Assume \(i\) is a closed immersion and \(Y\) Noetherian. Set \(\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_Y}(1)\). Let \(\mathcal{F}\) be a coherent module on \(X\). Then there exists an integer \(d_0\) such that for all \(d \geq d_0\) we have \(R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0\) for all \(p > 0\).

Proof

Checking whether \(R^pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes d})\) is zero can be done étale locally on \(Y\), see Equation (071Z). Hence we may assume \(Y\) is the spectrum of a Noetherian ring. In this case \(X\) is a scheme and the result follows from Cohomology of Schemes, Lemma 02O1.

Lemma

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

Proof

We first remark that \(X\) is a locally Noetherian algebraic space by Morphisms of Spaces, Lemma 04ZK. Hence the statement of the lemma makes sense. Moreover, computing \(R^if_*\mathcal{F}\) commutes with étale localization on \(Y\) (Properties of Spaces, Lemma 03LX) and checking whether \(R^if_*\mathcal{F}\) coherent can be done étale locally on \(Y\) (Lemma 07UB). Hence we may assume that \(Y = \Spec(A)\) is a Noetherian affine scheme.

Assume \(Y = \Spec(A)\) is an affine scheme. Note that \(f\) is locally of finite presentation (Morphisms of Spaces, Lemma 06G4). Thus it is of finite presentation, hence \(X\) is Noetherian (Morphisms of Spaces, Lemma 04ZL). Thus Lemma 08AN applies to the category of coherent modules of \(X\). For a coherent sheaf \(\mathcal{F}\) on \(X\) we say \(\mathcal{P}\) holds if and only if \(R^if_*\mathcal{F}\) is a coherent module on \(\Spec(A)\). We will show that conditions (1), (2), and (3) of Lemma 08AN hold for this property thereby finishing the proof of the lemma.

Verification of condition (1). Let \[0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\] be a short exact sequence of coherent sheaves on \(X\). Consider the long exact sequence of higher direct images \[R^{p - 1}f_*\mathcal{F}_3 \to R^pf_*\mathcal{F}_1 \to R^pf_*\mathcal{F}_2 \to R^pf_*\mathcal{F}_3 \to R^{p + 1}f_*\mathcal{F}_1\] Then it is clear that if 2-out-of-3 of the sheaves \(\mathcal{F}_i\) have property \(\mathcal{P}\), then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemmas 07UC and 07UD. Hence property \(\mathcal{P}\) holds for the third as well.

Verification of condition (2). This follows immediately from the fact that \(R^if_*(\mathcal{F}_1 \oplus \mathcal{F}_2) = R^if_*\mathcal{F}_1 \oplus R^if_*\mathcal{F}_2\) and that a summand of a coherent module is coherent (see lemmas cited above).

Verification of condition (3). Let \(i : Z \to X\) be a closed immersion with \(Z\) reduced and \(|Z|\) irreducible. Set \(g = f \circ i : Z \to \Spec(A)\). Let \(\mathcal{G}\) be a coherent module on \(Z\) whose scheme theoretic support is equal to \(Z\) such that \(R^pg_*\mathcal{G}\) is coherent for all \(p\). Then \(\mathcal{F} = i_*\mathcal{G}\) is a coherent module on \(X\) whose scheme theoretic support is \(Z\) such that \(R^pf_*\mathcal{F} = R^pg_*\mathcal{G}\). To see this use the Leray spectral sequence (Cohomology on Sites, Lemma 0734) and the fact that \(R^qi_*\mathcal{G} = 0\) for \(q > 0\) by Lemma 073H and the fact that a closed immersion is affine. (Morphisms of Spaces, Lemma 07U2). Thus we reduce to finding a coherent sheaf \(\mathcal{G}\) on \(Z\) with support equal to \(Z\) such that \(R^pg_*\mathcal{G}\) is coherent for all \(p\).

We apply Lemma 089J to the morphism \(Z \to \Spec(A)\). Thus we get a diagram \[\xymatrix{ Z \ar[rd]_g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_A \ar[dl] \\ & \Spec(A) & }\] with \(\pi : Z' \to Z\) proper surjective and \(i\) an immersion. Since \(Z \to \Spec(A)\) is proper we conclude that \(g'\) is proper (Morphisms of Spaces, Lemma 04XY). Hence \(i\) is a closed immersion (Morphisms of Spaces, Lemmas 04NX and 04CD). It follows that the morphism \(i' = (i, \pi) : \mathbf{P}^n_A \times_{\Spec(A)} Z' = \mathbf{P}^n_Z\) is a closed immersion (Morphisms of Spaces, Lemma 03KO). Set \[\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_A}(1) = (i')^*\mathcal{O}_{\mathbf{P}^n_Z}(1)\] We may apply Lemma 08AQ to \(\mathcal{L}\) and \(\pi\) as well as \(\mathcal{L}\) and \(g'\). Hence for all \(d \gg 0\) we have \(R^p\pi_*\mathcal{L}^{\otimes d} = 0\) for all \(p > 0\) and \(R^p(g')_*\mathcal{L}^{\otimes d} = 0\) for all \(p > 0\). Set \(\mathcal{G} = \pi_*\mathcal{L}^{\otimes d}\). By the Leray spectral sequence (Cohomology on Sites, Lemma 0734) we have \[E_2^{p, q} = R^pg_* R^q\pi_*\mathcal{L}^{\otimes d} \Rightarrow R^{p + q}(g')_*\mathcal{L}^{\otimes d}\] and by choice of \(d\) the only nonzero terms in \(E_2^{p, q}\) are those with \(q = 0\) and the only nonzero terms of \(R^{p + q}(g')_*\mathcal{L}^{\otimes d}\) are those with \(p = q = 0\). This implies that \(R^pg_*\mathcal{G} = 0\) for \(p > 0\) and that \(g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes d}\). Applying Cohomology of Schemes, Lemma 02O4 we see that \(g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes d}\) is coherent.

We still have to check that the support of \(\mathcal{G}\) is \(Z\). This follows from the fact that \(\mathcal{L}^{\otimes d}\) has lots of global sections. We spell it out here. Note that \(\mathcal{L}^{\otimes d}\) is globally generated for all \(d \geq 0\) because the same is true for \(\mathcal{O}_{\mathbf{P}^n}(d)\). Pick a point \(z \in Z'\) mapping to the generic point \(\xi\) of \(Z\) which we can do as \(\pi\) is surjective. (Observe that \(Z\) does indeed have a generic point as \(|Z|\) is irreducible and \(Z\) is Noetherian, hence quasi-separated, hence \(|Z|\) is a sober topological space by Properties of Spaces, Lemma 06NJ.) Pick \(s \in \Gamma(Z', \mathcal{L}^{\otimes d})\) which does not vanish at \(z\). Since \(\Gamma(Z, \mathcal{G}) = \Gamma(Z', \mathcal{L}^{\otimes d})\) we may think of \(s\) as a global section of \(\mathcal{G}\). Choose a geometric point \(\overline{z}\) of \(Z'\) lying over \(z\) and denote \(\overline{\xi} = g' \circ \overline{z}\) the corresponding geometric point of \(Z\). The adjunction map \[(g')^*\mathcal{G} = (g')^*g'_*\mathcal{L}^{\otimes d} \longrightarrow \mathcal{L}^{\otimes d}\] induces a map of stalks \(\mathcal{G}_{\overline{\xi}} \to \mathcal{L}_{\overline{z}}\), see Properties of Spaces, Lemma 05VQ. Moreover the adjunction map sends the pullback of \(s\) (viewed as a section of \(\mathcal{G}\)) to \(s\) (viewed as a section of \(\mathcal{L}^{\otimes d}\)). Thus the image of \(s\) in the vector space which is the source of the arrow \[\mathcal{G}_{\overline{\xi}} \otimes \kappa(\overline{\xi}) \longrightarrow \mathcal{L}^{\otimes d}_{\overline{z}} \otimes \kappa(\overline{z})\] isn’t zero since by choice of \(s\) the image in the target of the arrow is nonzero. Hence \(\xi\) is in the support of \(\mathcal{G}\) (Morphisms of Spaces, Lemma 07TZ). Since \(|Z|\) is irreducible and \(Z\) is reduced we conclude that the scheme theoretic support of \(\mathcal{G}\) is all of \(Z\) as desired.

Lemma

Let \(A\) be a Noetherian ring. Let \(f : X \to \Spec(A)\) be a proper morphism of algebraic spaces. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Then \(H^i(X, \mathcal{F})\) is finite \(A\)-module for all \(i \geq 0\).

Proof

This is just the affine case of Lemma 08AR. Namely, by Lemma 0720 we know that \(R^if_*\mathcal{F}\) is a quasi-coherent sheaf. Hence it is the quasi-coherent sheaf associated to the \(A\)-module \(\Gamma(\Spec(A), R^if_*\mathcal{F}) = H^i(X, \mathcal{F})\). The equality holds by Cohomology on Sites, Lemma 0733 and vanishing of higher cohomology groups of quasi-coherent modules on affine schemes (Cohomology of Schemes, Lemma 01XB). By Lemma 07UB we see \(R^if_*\mathcal{F}\) is a coherent sheaf if and only if \(H^i(X, \mathcal{F})\) is an \(A\)-module of finite type. Hence Lemma 08AR gives us the conclusion.

Lemma

Let \(A\) be a Noetherian ring. Let \(B\) be a finitely generated graded \(A\)-algebra. Let \(f : X \to \Spec(A)\) be a proper morphism of algebraic spaces. Set \(\mathcal{B} = f^*\widetilde B\). Let \(\mathcal{F}\) be a quasi-coherent graded \(\mathcal{B}\)-module of finite type. For every \(p \geq 0\) the graded \(B\)-module \(H^p(X, \mathcal{F})\) is a finite \(B\)-module.

Proof

To prove this we consider the fibre product diagram \[\xymatrix{ X' = \Spec(B) \times_{\Spec(A)} X \ar[r]_-\pi \ar[d]_{f'} & X \ar[d]^f \\ \Spec(B) \ar[r] & \Spec(A) }\] Note that \(f'\) is a proper morphism, see Morphisms of Spaces, Lemma 04WP. Also, \(B\) is a finitely generated \(A\)-algebra, and hence Noetherian (Algebra, Lemma 00FN). This implies that \(X'\) is a Noetherian algebraic space (Morphisms of Spaces, Lemma 04ZL). Note that \(X'\) is the relative spectrum of the quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{B}\) by Morphisms of Spaces, Lemma 081V. Since \(\mathcal{F}\) is a quasi-coherent \(\mathcal{B}\)-module we see that there is a unique quasi-coherent \(\mathcal{O}_{X'}\)-module \(\mathcal{F}'\) such that \(\pi_*\mathcal{F}' = \mathcal{F}\), see Morphisms of Spaces, Lemma 08AI. Since \(\mathcal{F}\) is finite type as a \(\mathcal{B}\)-module we conclude that \(\mathcal{F}'\) is a finite type \(\mathcal{O}_{X'}\)-module (details omitted). In other words, \(\mathcal{F}'\) is a coherent \(\mathcal{O}_{X'}\)-module (Lemma 07UB). Since the morphism \(\pi : X' \to X\) is affine we have \[H^p(X, \mathcal{F}) = H^p(X', \mathcal{F}')\] by Lemma 073H and Cohomology on Sites, Lemma 0733. Thus the lemma follows from Lemma 08AS.

Ample invertible sheaves and cohomology

Here is a criterion for ampleness on proper algebraic spaces over affine bases in terms of vanishing of cohomology after twisting.

Lemma

Let \(R\) be a Noetherian ring. Let \(X\) be a proper algebraic space over \(R\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. The following are equivalent

1. \(X\) is a scheme and \(\mathcal{L}\) is ample on \(X\),

2. for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) there exists an \(n_0 \geq 0\) such that \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0\) for all \(n \geq n_0\) and \(p > 0\), and

3. for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) there exists an \(n \geq 1\) such that \(H^1(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0\).

Proof

The implication (1) \(\Rightarrow\) (2) follows from Cohomology of Schemes, Lemma 0B5U. The implication (2) \(\Rightarrow\) (3) is trivial. The implication (3) \(\Rightarrow\) (1) is Lemma 0D2W.

Lemma

Let \(R\) be a Noetherian ring. Let \(f : Y \to X\) be a morphism of algebraic spaces proper over \(R\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume \(f\) is finite and surjective. The following are equivalent

1. \(X\) is a scheme and \(\mathcal{L}\) is ample, and

2. \(Y\) is a scheme and \(f^*\mathcal{L}\) is ample.

Proof

Assume (1). Then \(Y\) is a scheme as a finite morphism is representable (by schemes), see Morphisms of Spaces, Lemma 03ZQ. Hence (2) follows from Cohomology of Schemes, Lemma 0B5V.

Assume (2). Let \(P\) be the following property on coherent \(\mathcal{O}_X\)-modules \(\mathcal{F}\): there exists an \(n_0\) such that \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0\) for all \(n \geq n_0\) and \(p > 0\). We will prove that \(P\) holds for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), which implies \(\mathcal{L}\) is ample by Lemma 0GFA. We are going to apply Lemma 07UT. Thus we have to verify (1), (2) and (3) of that lemma for \(P\). Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves and the fact that tensoring with an invertible sheaf is an exact functor. Property (2) follows since \(H^p(X, -)\) is an additive functor.

To see (3) let \(i : Z \to X\) be a reduced closed subspace with \(|Z|\) irreducible. Let \(i' : Z' \to Y\) and \(f' : Z' \to Z\) be as in Lemma 0GF7 and set \(\mathcal{G} = f'_*\mathcal{O}_{Z'}\). We claim that \(\mathcal{G}\) satisfies properties (3)(a) and (3)(b) of Lemma 07UT which will finish the proof. Property (3)(a) we have seen in Lemma 0GF7. To see (3)(b) let \(\mathcal{I}\) be a nonzero quasi-coherent sheaf of ideals on \(Z\). Denote \(\mathcal{I}' \subset \mathcal{O}_{Z'}\) the quasi-coherent ideal \((f')^{-1}\mathcal{I} \mathcal{O}_{Z'}\), i.e., the image of \((f')^*\mathcal{I} \to \mathcal{O}_{Z'}\). By Lemma 0GF8 we have \(f_*\mathcal{I}' = \mathcal{I} \mathcal{G}\). We claim the common value \(\mathcal{G}' = \mathcal{I} \mathcal{G} = f'_*\mathcal{I}'\) satisfies the condition expressed in (3)(b). First, it is clear that the support of \(\mathcal{G}/\mathcal{G}'\) is contained in the support of \(\mathcal{O}_Z/\mathcal{I}\) which is a proper subspace of \(|Z|\) as \(\mathcal{I}\) is a nonzero ideal sheaf on the reduced and irreducible algebraic space \(Z\). Recall that \(f'_*\), \(i_*\), and \(i'_*\) transform coherent modules into coherent modules, see Lemmas 07UH and 08AM. As \(Y\) is a scheme and \(\mathcal{L}\) is ample we see from Lemma 0GFA that there exists an \(n_0\) such that \[H^p(Y, i'_*\mathcal{I}' \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0\] for \(n \geq n_0\) and \(p > 0\). Now we get \[\begin{align*} H^p(X, i_*\mathcal{G}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) & = H^p(Z, \mathcal{G'} \otimes_{\mathcal{O}_Z} i^*\mathcal{L}^{\otimes n}) \\ & = H^p(Z, f'_*\mathcal{I}' \otimes_{\mathcal{O}_Z} i^*\mathcal{L}^{\otimes n})) \\ & = H^p(Z, f'_*(\mathcal{I}' \otimes_{\mathcal{O}_{Z'}} (f')^*i^*\mathcal{L}^{\otimes n})) \\ & = H^p(Z, f'_*(\mathcal{I}' \otimes_{\mathcal{O}_{Z'}} (i')^*f^*\mathcal{L}^{\otimes n})) \\ & = H^p(Z', \mathcal{I}' \otimes_{\mathcal{O}_{Z'}} (i')^*f^*\mathcal{L}^{\otimes n})) \\ & = H^p(Y, i'_*\mathcal{I}' \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0 \end{align*}\] Here we have used the projection formula and the Leray spectral sequence (see Cohomology on Sites, Sections 0943 and 072X) and Lemma 0A4K. This verifies property (3)(b) of Lemma 07UT as desired.

The theorem on formal functions

This section is the analogue of Cohomology of Schemes, Section 02O7. We encourage the reader to read that section first.

Situation

Here \(A\) is a Noetherian ring and \(I \subset A\) is an ideal. Also, \(f : X \to \Spec(A)\) is a proper morphism of algebraic spaces and \(\mathcal{F}\) is a coherent sheaf on \(X\).

In this situation we denote \(I^n\mathcal{F}\) the quasi-coherent submodule of \(\mathcal{F}\) generated as an \(\mathcal{O}_X\)-module by products of local sections of \(\mathcal{F}\) and elements of \(I^n\). In other words, it is the image of the map \(f^*\widetilde{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{F}\).

Lemma

In Situation 08AV. Set \(B = \bigoplus_{n \geq 0} I^n\). Then for every \(p \geq 0\) the graded \(B\)-module \(\bigoplus_{n \geq 0} H^p(X, I^n\mathcal{F})\) is a finite \(B\)-module.

Proof

Let \(\mathcal{B} = \bigoplus I^n\mathcal{O}_X = f^*\widetilde{B}\). Then \(\bigoplus I^n\mathcal{F}\) is a finite type graded \(\mathcal{B}\)-module. Hence the result follows from Lemma 08AT.

Lemma

In Situation 08AV. For every \(p \geq 0\) there exists an integer \(c \geq 0\) such that

1. the multiplication map \(I^{n - c} \otimes H^p(X, I^c\mathcal{F}) \to H^p(X, I^n\mathcal{F})\) is surjective for all \(n \geq c\), and

2. the image of \(H^p(X, I^{n + m}\mathcal{F}) \to H^p(X, I^n\mathcal{F})\) is contained in the submodule \(I^{m - c} H^p(X, I^n\mathcal{F})\) for all \(n \geq 0\), \(m \geq c\).

Proof

By Lemma 08AW we can find \(d_1, \ldots, d_t \geq 0\), and \(x_i \in H^p(X, I^{d_i}\mathcal{F})\) such that \(\bigoplus_{n \geq 0} H^p(X, I^n\mathcal{F})\) is generated by \(x_1, \ldots, x_t\) over \(B = \bigoplus_{n \geq 0} I^n\). Take \(c = \max\{d_i\}\). It is clear that (1) holds. For (2) let \(b = \max(0, n - c)\). Consider the commutative diagram of \(A\)-modules \[\xymatrix{ I^{n + m - c - b} \otimes I^b \otimes H^p(X, I^c\mathcal{F}) \ar[r] \ar[d] & I^{n + m - c} \otimes H^p(X, I^c\mathcal{F}) \ar[r] & H^p(X, I^{n + m}\mathcal{F}) \ar[d] \\ I^{n + m - c - b} \otimes H^p(X, I^n\mathcal{F}) \ar[rr] & & H^p(X, I^n\mathcal{F}) }\] By part (1) of the lemma the composition of the horizontal arrows is surjective if \(n + m \geq c\). On the other hand, it is clear that \(n + m - c - b \geq m - c\). Hence part (2).

Lemma

In Situation 08AV. Fix \(p \geq 0\).

1. There exists a \(c_1 \geq 0\) such that for all \(n \geq c_1\) we have \[\Ker( H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}) ) \subset I^{n - c_1}H^p(X, \mathcal{F}).\]

2. The inverse system \[\left(H^p(X, \mathcal{F}/I^n\mathcal{F})\right)_{n \in \mathbf{N}}\] satisfies the Mittag-Leffler condition (see Homology, Definition 02N0).

3. In fact for any \(p\) and \(n\) there exists a \(c_2(n) \geq n\) such that \[\Im(H^p(X, \mathcal{F}/I^k\mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) = \Im(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}))\] for all \(k \geq c_2(n)\).

Proof

Let \(c_1 = \max\{c_p, c_{p + 1}\}\), where \(c_p, c_{p +1}\) are the integers found in Lemma 08AX for \(H^p\) and \(H^{p + 1}\). We will use this constant in the proofs of (1), (2) and (3).

Let us prove part (1). Consider the short exact sequence \[0 \to I^n\mathcal{F} \to \mathcal{F} \to \mathcal{F}/I^n\mathcal{F} \to 0\] From the long exact cohomology sequence we see that \[\Ker( H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}) ) = \Im( H^p(X, I^n\mathcal{F}) \to H^p(X, \mathcal{F}) )\] Hence by our choice of \(c_1\) we see that this is contained in \(I^{n - c_1}H^p(X, \mathcal{F})\) for \(n \geq c_1\).

Note that part (3) implies part (2) by definition of the Mittag-Leffler condition.

Let us prove part (3). Fix an \(n\) throughout the rest of the proof. Consider the commutative diagram \[\xymatrix{ 0 \ar[r] & I^n\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & \mathcal{F}/I^n\mathcal{F} \ar[r] & 0 \\ 0 \ar[r] & I^{n + m}\mathcal{F} \ar[r] \ar[u] & \mathcal{F} \ar[r] \ar[u] & \mathcal{F}/I^{n + m}\mathcal{F} \ar[r] \ar[u] & 0 }\] This gives rise to the following commutative diagram \[\xymatrix{ H^p(X, I^n\mathcal{F}) \ar[r] & H^p(X, \mathcal{F}) \ar[r] & H^p(X, \mathcal{F}/I^n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^n\mathcal{F}) \\ H^p(X, I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^a }\] If \(m \geq c_1\) we see that the image of \(a\) is contained in \(I^{m - c_1} H^{p + 1}(X, I^n\mathcal{F})\). By the Artin-Rees lemma (see Algebra, Lemma 00IO) there exists an integer \(c_3(n)\) such that \[I^N H^{p + 1}(X, I^n\mathcal{F}) \cap \Im(\delta) \subset \delta\left(I^{N - c_3(n)}H^p(X, \mathcal{F}/I^n\mathcal{F})\right)\] for all \(N \geq c_3(n)\). As \(H^p(X, \mathcal{F}/I^n\mathcal{F})\) is annihilated by \(I^n\), we see that if \(m \geq c_3(n) + c_1 + n\), then \[\Im(H^p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) = \Im(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}))\] In other words, part (3) holds with \(c_2(n) = c_3(n) + c_1 + n\).

Theorem

In Situation 08AV. Fix \(p \geq 0\). The system of maps \[H^p(X, \mathcal{F})/I^nH^p(X, \mathcal{F}) \longrightarrow H^p(X, \mathcal{F}/I^n\mathcal{F})\] define an isomorphism of limits \[H^p(X, \mathcal{F})^\wedge \longrightarrow \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F})\] where the left hand side is the completion of the \(A\)-module \(H^p(X, \mathcal{F})\) with respect to the ideal \(I\), see Algebra, Section 00M9. Moreover, this is in fact a homeomorphism for the limit topologies.

Proof

In fact, this follows immediately from Lemma 08AY. We spell out the details. Set \(M = H^p(X, \mathcal{F})\) and \(M_n = H^p(X, \mathcal{F}/I^n\mathcal{F})\). Denote \(N_n = \Im(M \to M_n)\). By the description of the limit in Homology, Section 02MY we have \[\lim_n M_n = \{(x_n) \in \prod M_n \mid \varphi_i(x_n) = x_{n - 1}, \ n = 2, 3, \ldots\}\] Pick an element \(x = (x_n) \in \lim_n M_n\). By Lemma 08AY part (3) we have \(x_n \in N_n\) for all \(n\) since by definition \(x_n\) is the image of some \(x_{n + m} \in M_{n + m}\) for all \(m\). By Lemma 08AY part (1) we see that there exists a factorization \[M \to N_n \to M/I^{n - c_1}M\] of the reduction map. Denote \(y_n \in M/I^{n - c_1}M\) the image of \(x_n\) for \(n \geq c_1\). Since for \(n' \geq n\) the composition \(M \to M_{n'} \to M_n\) is the given map \(M \to M_n\) we see that \(y_{n'}\) maps to \(y_n\) under the canonical map \(M/I^{n' - c_1}M \to M/I^{n - c_1}M\). Hence \(y = (y_{n + c_1})\) defines an element of \(\lim_n M/I^nM\). We omit the verification that \(y\) maps to \(x\) under the map \[M^\wedge = \lim_n M/I^nM \longrightarrow \lim_n M_n\] of the lemma. We also omit the verification on topologies.

Lemma

Let \(A\) be a ring. Let \(I \subset A\) be an ideal. Assume \(A\) is Noetherian and complete with respect to \(I\). Let \(f : X \to \Spec(A)\) be a proper morphism of algebraic spaces. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Then \[H^p(X, \mathcal{F}) = \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F})\] for all \(p \geq 0\).

Proof

This is a reformulation of the theorem on formal functions (Theorem 08AZ) in the case of a complete Noetherian base ring. Namely, in this case the \(A\)-module \(H^p(X, \mathcal{F})\) is finite (Lemma 08AS) hence \(I\)-adically complete (Algebra, Lemma 00MA) and we see that completion on the left hand side is not necessary.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\) and let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Assume

1. \(Y\) locally Noetherian,

2. \(f\) proper, and

3. \(\mathcal{F}\) coherent.

Let \(\overline{y}\) be a geometric point of \(Y\). Consider the “infinitesimal neighbourhoods” \[\xymatrix{ X_n = \Spec(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^n) \times_Y X \ar[r]_-{i_n} \ar[d]_{f_n} & X \ar[d]^f \\ \Spec(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^n) \ar[r]^-{c_n} & Y }\] of the fibre \(X_1 = X_{\overline{y}}\) and set \(\mathcal{F}_n = i_n^*\mathcal{F}\). Then we have \[\left(R^pf_*\mathcal{F}\right)_{\overline{y}}^\wedge \cong \lim_n H^p(X_n, \mathcal{F}_n)\] as \(\mathcal{O}_{Y, \overline{y}}^\wedge\)-modules.

Proof

This is just a reformulation of a special case of the theorem on formal functions, Theorem 08AZ. Let us spell it out. Note that \(\mathcal{O}_{Y, \overline{y}}\) is a Noetherian local ring, see Properties of Spaces, Lemma 08AH. Consider the canonical morphism \(c : \Spec(\mathcal{O}_{Y, \overline{y}}) \to Y\). This is a flat morphism as it identifies local rings. Denote \(f' : X' \to \Spec(\mathcal{O}_{Y, \overline{y}})\) the base change of \(f\) to this local ring. We see that \(c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'\) by Lemma 073K. Moreover, we have canonical identifications \(X_n = X'_n\) for all \(n \geq 1\).

Hence we may assume that \(Y = \Spec(A)\) is the spectrum of a strictly henselian Noetherian local ring \(A\) with maximal ideal \(\mathfrak m\) and that \(\overline{y} \to Y\) is equal to \(\Spec(A/\mathfrak m) \to Y\). It follows that \[\left(R^pf_*\mathcal{F}\right)_{\overline{y}} = \Gamma(Y, R^pf_*\mathcal{F}) = H^p(X, \mathcal{F})\] because \((Y, \overline{y})\) is an initial object in the category of étale neighbourhoods of \(\overline{y}\). The morphisms \(c_n\) are each closed immersions. Hence their base changes \(i_n\) are closed immersions as well. Note that \(i_{n, *}\mathcal{F}_n = i_{n, *}i_n^*\mathcal{F} = \mathcal{F}/\mathfrak m^n\mathcal{F}\). By the Leray spectral sequence for \(i_n\), and Lemma 07UH we see that \[H^p(X_n, \mathcal{F}_n) = H^p(X, i_{n, *}\mathcal{F}) = H^p(X, \mathcal{F}/\mathfrak m^n\mathcal{F})\] Hence we may indeed apply the theorem on formal functions to compute the limit in the statement of the lemma and we win.

Here is a lemma which we will generalize later to fibres of dimension \(> 0\), namely the next lemma.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\overline{y}\) be a geometric point of \(Y\). Assume

1. \(Y\) locally Noetherian,

2. \(f\) is proper, and

3. \(X_{\overline{y}}\) has discrete underlying topological space.

Then for any coherent sheaf \(\mathcal{F}\) on \(X\) we have \((R^pf_*\mathcal{F})_{\overline{y}} = 0\) for all \(p > 0\).

Proof

Let \(\kappa(\overline{y})\) be the residue field of the local ring of \(\mathcal{O}_{Y, \overline{y}}\). As in Lemma 08B1 we set \(X_{\overline{y}} = X_1 = \Spec(\kappa(\overline{y})) \times_Y X\). By Morphisms of Spaces, Lemma 06LS the morphism \(f : X \to Y\) is quasi-finite at each of the points of the fibre of \(X \to Y\) over \(\overline{y}\). It follows that \(X_{\overline{y}} \to \overline{y}\) is separated and quasi-finite. Hence \(X_{\overline{y}}\) is a scheme by Morphisms of Spaces, Proposition 03XX. Since it is quasi-compact its underlying topological space is a finite discrete space. Then it is an affine scheme by Schemes, Lemma 02O0. By Lemma 07VP it follows that the algebraic spaces \(X_n\) are affine schemes as well. Moreover, the underlying topological of each \(X_n\) is the same as that of \(X_1\). Hence it follows that \(H^p(X_n, \mathcal{F}_n) = 0\) for all \(p > 0\). Hence we see that \((R^pf_*\mathcal{F})_{\overline{y}}^\wedge = 0\) by Lemma 08B1. Note that \(R^pf_*\mathcal{F}\) is coherent by Lemma 08AR and hence \(R^pf_*\mathcal{F}_{\overline{y}}\) is a finite \(\mathcal{O}_{Y, \overline{y}}\)-module. By Algebra, Lemma 00MA this implies that \((R^pf_*\mathcal{F})_{\overline{y}} = 0\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\overline{y}\) be a geometric point of \(Y\). Assume

1. \(Y\) locally Noetherian,

2. \(f\) is proper, and

3. \(\dim(X_{\overline{y}}) = d\).

Then for any coherent sheaf \(\mathcal{F}\) on \(X\) we have \((R^pf_*\mathcal{F})_{\overline{y}} = 0\) for all \(p > d\).

Proof

Let \(\kappa(\overline{y})\) be the residue field of the local ring of \(\mathcal{O}_{Y, \overline{y}}\). As in Lemma 08B1 we set \(X_{\overline{y}} = X_1 = \Spec(\kappa(\overline{y})) \times_Y X\). Moreover, the underlying topological space of each infinitesimal neighbourhood \(X_n\) is the same as that of \(X_{\overline{y}}\). Hence \(H^p(X_n, \mathcal{F}_n) = 0\) for all \(p > d\) by Lemma 0A4R. Hence we see that \((R^pf_*\mathcal{F})_{\overline{y}}^\wedge = 0\) by Lemma 08B1 for \(p > d\). Note that \(R^pf_*\mathcal{F}\) is coherent by Lemma 08AR and hence \(R^pf_*\mathcal{F}_{\overline{y}}\) is a finite \(\mathcal{O}_{Y, \overline{y}}\)-module. By Algebra, Lemma 00MA this implies that \((R^pf_*\mathcal{F})_{\overline{y}} = 0\).

Applications of the theorem on formal functions

We will add more here as needed.

Lemma

(For a more general version see More on Morphisms of Spaces, Lemma 0A4X). Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(Y\) is locally Noetherian. The following are equivalent

1. \(f\) is finite, and

2. \(f\) is proper and \(|X_k|\) is a discrete space for every morphism \(\Spec(k) \to Y\) where \(k\) is a field.

Proof

A finite morphism is proper according to Morphisms of Spaces, Lemma 04NZ. A finite morphism is quasi-finite according to Morphisms of Spaces, Lemma 04NY. A quasi-finite morphism has discrete fibres \(X_k\), see Morphisms of Spaces, Lemma 06RW. Hence a finite morphism is proper and has discrete fibres \(X_k\).

Assume \(f\) is proper with discrete fibres \(X_k\). We want to show \(f\) is finite. In fact it suffices to prove \(f\) is affine. Namely, if \(f\) is affine, then it follows that \(f\) is integral by Morphisms of Spaces, Lemma 0415 whereupon it follows from Morphisms of Spaces, Lemma 0414 that \(f\) is finite.

To show that \(f\) is affine we may assume that \(Y\) is affine, and our goal is to show that \(X\) is affine too. Since \(f\) is proper we see that \(X\) is separated and quasi-compact. We will show that for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(H^1(X, \mathcal{F}) = 0\). This implies that \(H^1(X, \mathcal{F}) = 0\) for every quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) by Lemmas 07UV and 073E. Then it follows that \(X\) is affine from Proposition 07V6. By Lemma 0A4S we conclude that the stalks of \(R^1f_*\mathcal{F}\) are zero for all geometric points of \(Y\). In other words, \(R^1f_*\mathcal{F} = 0\). Hence we see from the Leray Spectral Sequence for \(f\) that \(H^1(X , \mathcal{F}) = H^1(Y, f_*\mathcal{F})\). Since \(Y\) is affine, and \(f_*\mathcal{F}\) is quasi-coherent (Morphisms of Spaces, Lemma 03M9) we conclude \(H^1(Y, f_*\mathcal{F}) = 0\) from Cohomology of Schemes, Lemma 01XB. Hence \(H^1(X, \mathcal{F}) = 0\) as desired.

As a consequence we have the following useful result.

Lemma

(For a more general version see More on Morphisms of Spaces, Lemma 0A4Y). Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\overline{y}\) be a geometric point of \(Y\). Assume

1. \(Y\) is locally Noetherian,

2. \(f\) is proper, and

3. \(|X_{\overline{y}}|\) is finite.

Then there exists an open neighbourhood \(V \subset Y\) of \(\overline{y}\) such that \(f|_{f^{-1}(V)} : f^{-1}(V) \to V\) is finite.

Proof

The morphism \(f\) is quasi-finite at all the geometric points of \(X\) lying over \(\overline{y}\) by Morphisms of Spaces, Lemma 06LS. By Morphisms of Spaces, Lemma 04NW the set of points at which \(f\) is quasi-finite is an open subspace \(U \subset X\). Let \(Z = X \setminus U\). Then \(\overline{y} \not \in f(Z)\). Since \(f\) is proper the set \(f(Z) \subset Y\) is closed. Choose any open neighbourhood \(V \subset Y\) of \(\overline{y}\) with \(Z \cap V = \emptyset\). Then \(f^{-1}(V) \to V\) is locally quasi-finite and proper. Hence \(f^{-1}(V) \to V\) has discrete fibres \(X_k\) (Morphisms of Spaces, Lemma 06RW) which are quasi-compact hence finite. Thus \(f^{-1}(V) \to V\) is finite by Lemma 0A4V.