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 \input{preamble} % OK, start here. % \begin{document} \title{Cohomology of Algebraic Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent 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 \cite{Kn}. \medskip\noindent An important missing ingredient in this chapter is the {\it induction principle}, i.e., the analogue for quasi-compact and quasi-separated algebraic spaces of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. This is formulated precisely and proved in detail in Derived Categories of Spaces, Section \ref{spaces-perfect-section-induction}. Instead of the induction principle, in this chapter we use the alternating {\v C}ech complex, see Section \ref{section-alternating-cech}. It is designed to prove vanishing statements such as Proposition \ref{proposition-vanishing}, 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. \section{Conventions} \label{section-conventions} \noindent 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. \medskip\noindent 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 $X \times X$. \section{Higher direct images} \label{section-higher-direct-image} \noindent 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 \ref{spaces-properties-section-quasi-coherent}). By Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent} 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 $X$. \medskip\noindent 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 \ref{descent-lemma-higher-direct-images-small-etale} 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 $Y$. \medskip\noindent 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 \'etale 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 \begin{equation} \label{equation-representable-higher-direct-image} R^if'_*(\mathcal{F}|_U) = (R^if_*\mathcal{F})|_V, \end{equation} see Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}. 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. \medskip\noindent 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 {\v C}ech complex, as discussed in Sheaves on Stacks, Sections \ref{stacks-sheaves-section-cech} and \ref{stacks-sheaves-section-sheaf-cech-complex} ff. \begin{lemma} \label{lemma-higher-direct-image} 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. \end{lemma} \begin{proof} Let $V \to Y$ be an \'etale 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 \ref{spaces-properties-lemma-pushforward-etale-base-change-modules} 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 \'etale topology on $Y$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-coherent}) we may replace $Y$ by $V$, i.e., we may assume $Y$ is an affine scheme. \medskip\noindent 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 \'etale morphism $g : U \to X$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. 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). \medskip\noindent 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 \ref{spaces-morphisms-lemma-pushforward}. Assume $IH_n$. In the rest of the proof we show that $IH_{n + 1}$ holds. \medskip\noindent 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 \ref{sites-cohomology-lemma-relative-Leray}. 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 \ref{spaces-properties-lemma-properties-quasi-coherent}) it follows that $R^{n + 1}f_*g_*\mathcal{H}$ is quasi-coherent (details omitted). \medskip\noindent 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 \'etale. 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. \end{proof} \begin{lemma} \label{lemma-quasi-coherence-higher-direct-images-application} 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 $q \in \mathbf{Z}$. \end{lemma} \begin{proof} Since formation of $Rf_*$ commutes with \'etale localization (Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}) 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 \ref{sites-cohomology-lemma-Leray}. By Lemma \ref{lemma-higher-direct-image} we see that the sheaves $R^qf_*\mathcal{F}$ are quasi-coherent. By Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} we see that $E_2^{p, q} = 0$ when $p > 0$. Hence the spectral sequence degenerates at $E_2$ and we win. \end{proof} \noindent Here is a result which holds for all abelian sheaves (in particular also quasi-coherent modules). \begin{lemma} \label{lemma-finite-higher-direct-image-zero} 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 $p > 0$. \end{lemma} \begin{proof} By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change} we may compute the higher direct images on an \'etale cover of $Y$. Hence we may assume $Y$ is a scheme. This implies that $X$ is a scheme (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-local}). In this case we may apply \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-what-integral}. For the finite case the reader may wish to consult the less technical \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-finite-higher-direct-image-zero}. \end{proof} \section{Colimits and cohomology} \label{section-colimits} \noindent The following lemma in particular applies to diagrams of quasi-coherent sheaves. \begin{lemma} \label{lemma-colimits} 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 $X_\etale$. \end{lemma} \begin{proof} This follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-colim-works-over-collection}. Namely, let $\mathcal{B} \subset \Ob(X_{spaces, \etale})$ be the set of quasi-compact and quasi-separated spaces \'etale 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 $X_\etale$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence} the set $\mathcal{B}$ satisfies all the assumptions of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-colim-works-over-collection}. Since $X \in \mathcal{B}$ we win. \end{proof} \begin{lemma} \label{lemma-colimit-cohomology} \begin{slogan} Higher direct images of qcqs morphisms commute with filtered colimits of sheaves. \end{slogan} 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 $$R^pf_*\mathcal{F} = \colim R^pf_*\mathcal{F}_i.$$ \end{lemma} \begin{proof} Recall that $R^pf_*\mathcal{F}$ is the sheaf on $Y_{spaces, \etale}$ associated to $V \mapsto H^p(V \times_Y X, \mathcal{F})$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images} and Properties of Spaces, Lemma \ref{spaces-properties-lemma-functoriality-etale-site}. Recall that the colimit is the sheaf associated to the presheaf colimit. Hence we can apply Lemma \ref{lemma-colimits} to $H^p(V \times_Y X, -)$ where $V$ is affine to conclude (because when $V$ is affine, then $V \times_Y X$ is quasi-compact and quasi-separated). Strictly speaking this also uses Properties of Spaces, Lemma \ref{spaces-properties-lemma-alternative} to see that there exist enough affine objects. \end{proof} \noindent The following lemma tells us that finitely presented modules behave as expected in quasi-compact and quasi-separated algebraic spaces. \begin{lemma} \label{lemma-finite-presentation-quasi-compact-colimit} 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 quasi-coherent $\mathcal{O}_X$-modules. Let $\mathcal{G}$ be an $\mathcal{O}_X$-module of finite presentation. Then we have $$\colim_i \Hom_X(\mathcal{G}, \mathcal{F}_i) = \Hom_X(\mathcal{G}, \colim_i \mathcal{F}_i).$$ \end{lemma} \begin{proof} Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$. Set $R = U \times_X U$. Note that $R$ is a quasi-compact (as $X$ is quasi-separated and $U$ quasi-compact) and separated (as $U$ is separated) scheme. Hence we have $$\colim_i \Hom_U(\mathcal{G}|_U, \mathcal{F}_i|_U) = \Hom_U(\mathcal{G}|_U, \colim_i \mathcal{F}_i|_U).$$ by Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-compact-colimit} (and the material on restriction to schemes \'etale over $X$, see Properties of Spaces, Sections \ref{spaces-properties-section-quasi-coherent} and \ref{spaces-properties-section-properties-modules}). Similarly for $R$. Since $\QCoh(\mathcal{O}_X) = \QCoh(U, R, s, t, c)$ (see Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent}) the result follows formally. \end{proof} \section{The alternating {\v C}ech complex} \label{section-alternating-cech} \noindent Let $S$ be a scheme. Let $f : U \to X$ be an \'etale 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 \ref{spaces-properties-section-localize}. 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 \ref{sites-modules-section-localize}. 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. \medskip\noindent 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 $f_!$ 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 \ref{sites-modules-lemma-extension-by-zero}. 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 \ref{sites-modules-lemma-stalk-j-shriek} and Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-small-etale-site}. 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 $U_\etale$. \begin{lemma} \label{lemma-product-is-tensor-product} Let $S$ be a scheme. Let $f_i : U_i \to X$ be \'etale 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 $$(f_1 \amalg f_2)_! \underline{\mathbf{Z}} \longrightarrow f_{1, !}\underline{\mathbf{Z}} \oplus f_{2, !}\underline{\mathbf{Z}}$$ \end{lemma} \begin{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. \end{proof} \noindent 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 \'etale then $\text{Tr}_f$ is surjective. \medskip\noindent Assume that $f : U \to X$ is a surjective \'etale morphism of algebraic spaces. Consider the {\it 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 \ref{more-algebra-lemma-homotopy-koszul-abstract}. 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 \begin{equation} \label{equation-quasi-isomorphism} K^\bullet \longrightarrow \underline{\mathbf{Z}}[0] \end{equation} We use the complex $K^\bullet$ to define what we call the alternating {\v C}ech complex associated to $f : U \to X$. \begin{definition} \label{definition-alternating-cech-complex} Let $S$ be a scheme. Let $f : U \to X$ be a surjective \'etale morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be an object of $\textit{Ab}(X_\etale)$. The {\it alternating {\v C}ech complex}\footnote{This may be nonstandard notation} $\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F})$ 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 $\textit{Ab}(X_\etale)$. \end{definition} \noindent 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 \ref{cohomology-section-alternating-cech} 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 {\v C}ech complex to the cohomology. \begin{lemma} \label{lemma-alternating-cech-to-cohomology} Let $S$ be a scheme. Let $f : U \to X$ be a surjective \'etale 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 $$E_1^{p, q} = \text{Ext}_{\textit{Ab}(X_\etale)}^q(K^p, \mathcal{F})$$ converging to $H^{p + q}(X, \mathcal{F})$ where $K^p = \wedge^{p + 1}f_!\underline{\mathbf{Z}}$. \end{lemma} \begin{proof} Recall that we have the quasi-isomorphism $K^\bullet \to \underline{\mathbf{Z}}[0]$, see (\ref{equation-quasi-isomorphism}). Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ in $\textit{Ab}(X_\etale)$. Consider the double complex $A^{\bullet, \bullet}$ with terms $$A^{p, q} = \Hom(K^p, \mathcal{I}^q)$$ where 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 $sA^\bullet$ the total complex associated to the double complex $A^{\bullet, \bullet}$. We will use the two spectral sequences $({}'E_r, {}'d_r)$ and $({}''E_r, {}''d_r)$ associated to this double complex, see Homology, Section \ref{homology-section-double-complex}. \medskip\noindent Because $K^\bullet$ is a resolution of $\underline{\mathbf{Z}}$ we see that the complexes $$A^{\bullet, q} : \Hom(K^0, \mathcal{I}^q) \to \Hom(K^1, \mathcal{I}^q) \to \Hom(K^2, \mathcal{I}^q) \to \ldots$$ are acyclic in positive degrees and have $H^0$ equal to $\Gamma(X, \mathcal{I}^q)$. Hence by Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution} and its proof the spectral sequence $({}''E_r, {}''d_r)$ degenerates, and the natural map $$\mathcal{I}^\bullet(X) \longrightarrow sA^\bullet$$ is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that $H^n(sA^\bullet) = H^n(X, \mathcal{F})$. \medskip\noindent The map $\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F}) \to R\Gamma(X, \mathcal{F})$ of the lemma is the composition of $\check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F}) \to SA^\bullet$ with the inverse of the displayed quasi-isomorphism. \medskip\noindent 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. \end{proof} \noindent It follows from the lemma that it is important to understand the ext groups $\text{Ext}_{\textit{Ab}(X_\etale)}(K^p, \mathcal{F})$, i.e., the right derived functors of $\mathcal{F} \mapsto \Hom(K^p, \mathcal{F})$. \begin{lemma} \label{lemma-compute} Let $S$ be a scheme. Let $f : U \to X$ be a surjective, \'etale, 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 $K^p = \wedge^{p + 1}f_!\underline{\mathbf{Z}}$. \end{lemma} \begin{proof} Because $U \to X$ is separated the diagonal $U \to U \times_X U$ is a closed immersion. Since $U \to X$ is \'etale the diagonal $U \to U \times_X U$ is an open immersion, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-unramified} and \ref{spaces-morphisms-lemma-diagonal-unramified-morphism}. 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 \ref{lemma-product-is-tensor-product} 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. \end{proof} \begin{lemma} \label{lemma-twist} 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 \ref{spaces-properties-lemma-quotient}. 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 $$H^0(W, \mathcal{F}|_W)^\chi = H^0(U, \mathcal{F} \otimes_{\mathbf{Z}} \underline{\mathbf{Z}}(\chi))$$ \end{lemma} \begin{proof} The quotient morphism $q : W \to U$ is a $G$-torsor, i.e., there exists a surjective \'etale 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. \end{proof} \noindent Now we can put everything together and obtain the following pleasing result. \begin{lemma} \label{lemma-alternating-spectral-sequence} Let $S$ be a scheme. Let $f : U \to X$ be a surjective, \'etale, 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 \ref{lemma-compute}. 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 \ref{lemma-twist}. Then the spectral sequence of Lemma \ref{lemma-alternating-cech-to-cohomology} 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 $H^{p + q}(X, \mathcal{F})$. \end{lemma} \begin{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 \'etale and since $W_p \to U_p$ is surjective \'etale, it follows that also $U_p \to X$ is \'etale, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-local}. 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 \ref{sites-cohomology-lemma-cohomology-of-open}. Moreover, the functor $\mathcal{G} \mapsto \mathcal{G} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))$ is an auto-equivalence of $\textit{Ab}(U_p)$, 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 \ref{lemma-twist} 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 \ref{lemma-compute} 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 $\text{Ext}^*_{\textit{Ab}(X_\etale)}(K^p, \mathcal{F})$. Putting everything together we win. \end{proof} \section{Higher vanishing for quasi-coherent sheaves} \label{section-higher-vanishing} \noindent 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 $n$. \begin{lemma} \label{lemma-quasi-coherent-twist} With $S$, $W$, $G$, $U$, $\chi$ as in Lemma \ref{lemma-twist}. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_U$-module, then so is $\mathcal{F} \otimes_{\mathbf{Z}} \underline{\mathbf{Z}}(\chi)$. \end{lemma} \begin{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 \'etale locally. Hence the lemma follows as $\underline{\mathbf{Z}}(\chi)$ is finite locally free as a $\underline{\mathbf{Z}}$-module. \end{proof} \noindent The following proposition is interesting even if $X$ is a scheme. It is the natural generalization of Cohomology of Schemes, Lemma \ref{coherent-lemma-vanishing-nr-affines}. Before we state it, observe that given an \'etale 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 \ref{decent-spaces-lemma-bounded-fibres}. \begin{proposition} \label{proposition-vanishing} 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 \'etale 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 $q \geq d$. \end{proposition} \begin{proof} We will use the spectral sequence of Lemma \ref{lemma-alternating-spectral-sequence}. The lemma applies since $f$ is separated as $U$ is separated, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-compose-after-separated}. 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 \ref{groupoids-proposition-finite-flat-equivalence}. The discussion in Section \ref{section-higher-direct-image} 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 \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. By Lemma \ref{lemma-quasi-coherent-twist} 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. \end{proof} \noindent 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. \begin{lemma} \label{lemma-vanishing-quasi-separated} 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 \begin{enumerate} \item an affine scheme $U$, \item a surjective \'etale morphism $f : U \to X$, \item an integer $d$ bounding the degrees of the fibres of $U \to X$, \item for every $p = 0, 1, \ldots, d$ a surjective \'etale morphism $V_p \to U_p$ from an affine scheme $V_p$ where $U_p$ is as in Lemma \ref{lemma-alternating-spectral-sequence}, and \item an integer $d_p$ bounding the degree of the fibres of $V_p \to U_p$. \end{enumerate} 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 $q \geq \max(d_p + p)$. \end{lemma} \begin{proof} Since $X$ is quasi-compact we can find a surjective \'etale morphism $U \to X$ with $U$ affine, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. By Decent Spaces, Lemma \ref{decent-spaces-lemma-bounded-fibres} 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 \ref{spaces-lemma-quotient-finite-separated}. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover} we can find the morphisms $V_p \to W_p$. By Decent Spaces, Lemma \ref{decent-spaces-lemma-bounded-fibres} we can find the integers $d_p$. \medskip\noindent 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 \ref{lemma-alternating-spectral-sequence}. By definition of the integer $d$ we see that $U_p = 0$ for $p \geq d$. By Proposition \ref{proposition-vanishing} and Lemma \ref{lemma-quasi-coherent-twist} 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. \end{proof} \section{Vanishing for higher direct images} \label{section-vanishing-higher-direct-images} \noindent We apply the results of Section \ref{section-higher-vanishing} 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. \begin{lemma} \label{lemma-vanishing-higher-direct-images} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $f$ is quasi-compact and quasi-separated, and \item $Y$ is quasi-compact. \end{enumerate} 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 $i \geq n(X \to Y)$. \end{lemma} \begin{proof} Let $V \to Y$ be a surjective \'etale morphism where $V$ is an affine scheme, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. 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 \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}. Thus we may assume that $Y$ is an affine scheme. \medskip\noindent 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 \ref{lemma-higher-direct-image}. 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 \ref{sites-cohomology-lemma-apply-Leray} and the vanishing of higher cohomology of quasi-coherent sheaves on affine algebraic spaces (Proposition \ref{proposition-vanishing}). \medskip\noindent Choose $U \to X$, $d$, $V_p \to U_p$ and $d_p$ as in Lemma \ref{lemma-vanishing-quasi-separated}. 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 \ref{lemma-vanishing-quasi-separated} 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. \end{proof} \begin{lemma} \label{lemma-affine-vanishing-higher-direct-images} 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 $\mathcal{F}$. \end{lemma} \begin{proof} Recall that an affine morphism of algebraic spaces is representable. Hence this follows from (\ref{equation-representable-higher-direct-image}) and Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}. \end{proof} \begin{lemma} \label{lemma-relative-affine-cohomology} 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 $H^i(X, \mathcal{F}) = H^i(S, f_*\mathcal{F})$ for all $i \geq 0$. \end{lemma} \begin{proof} Follows from Lemma \ref{lemma-affine-vanishing-higher-direct-images} and the Leray spectral sequence. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}. \end{proof} \section{Cohomology with support in a closed subspace} \label{section-cohomology-support} \noindent This section is the analogue of Cohomology, Section \ref{cohomology-section-cohomology-support} and \'Etale Cohomology, Section \ref{etale-cohomology-section-cohomology-support} for abelian sheaves on algebraic spaces. \medskip\noindent 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 \ref{spaces-properties-definition-support}). 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 $H^q_Z(X, \mathcal{F}) = R^q\Gamma_Z(X, \mathcal{F})$. \medskip\noindent 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 \ref{sites-cohomology-lemma-restriction-along-monomorphism-surjective}) 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 $D(X_\etale)$. \medskip\noindent For an abelian sheaf $\mathcal{F}$ on $X_\etale$ we can consider the {\it 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 \ref{spaces-properties-definition-support}. Using the equivalence of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-push-pull} 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. \begin{lemma} \label{lemma-sections-with-support-acyclic} 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 $Z_\etale$. \end{lemma} \begin{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 \ref{lemma-finite-higher-direct-image-zero}) and as $\mathcal{I}$ is injective on $X_\etale$ we conclude that $\mathcal{H}_Z(\mathcal{I})$ is injective on $Z_\etale$. \end{proof} \noindent 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 $$E_2^{p, q} = H^p(Z, \mathcal{H}^q_Z(\mathcal{F})) \Rightarrow H^{p + q}_Z(X, \mathcal{F})$$ \begin{lemma} \label{lemma-cohomology-with-support-sheaf-on-support} 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 $p > 0$. \end{lemma} \begin{proof} This is true because the functor $i_*$ is exact (Lemma \ref{lemma-finite-higher-direct-image-zero}) and transforms injective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pushforward-injective-flat}). \end{proof} \begin{lemma} \label{lemma-etale-localization-sheaf-with-support} Let $S$ be a scheme. Let $f : X \to Y$ be an \'etale 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 $H^q_Z(Y, \mathcal{F}) = H^q_{f^{-1}(Z)}(X, f^{-1}\mathcal{F})$. \end{lemma} \begin{proof} Because $f$ is \'etale 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. \end{proof} \noindent 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 $T$. \begin{lemma} \label{lemma-complexes-with-support-on-closed} 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 $$i^{-1}|_{D_Z(X_\etale)} = R\mathcal{H}_Z|_{D_{|Z|}(X_\etale)}$$ \end{lemma} \begin{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 \ref{spaces-properties-lemma-stalk-pullback} and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-push-pull}. 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. \medskip\noindent 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 \ref{derived-lemma-adjoint-preserve-K-injectives}). We see that $R\mathcal{H}_Z(K)$ is computed by $\mathcal{H}_Z(i_*\mathcal{I}^\bullet) = \mathcal{I}^\bullet$ as desired. \end{proof} \section{Vanishing above the dimension} \label{section-vanishing-above-dimension} \noindent 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 \ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral}. Moreover, the dimension of $X$ (as defined in Properties of Spaces, Definition \ref{spaces-properties-definition-dimension}) is equal to the Krull dimension of $|X|$, see Decent Spaces, Lemma \ref{decent-spaces-lemma-dimension-decent-space}. 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. \begin{lemma} \label{lemma-vanishing-above-dimension} 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 $X$. \begin{enumerate} \item $H^q(X, \mathcal{F}) = 0$ for $q > d$, \item $H^d(X, \mathcal{F}) \to H^d(U, \mathcal{F})$ is surjective for any quasi-compact open $U \subset X$, \item $H^q_Z(X, \mathcal{F}) = 0$ for $q > d$ for any closed subspace $Z \subset X$ whose complement is quasi-compact. \end{enumerate} \end{lemma} \begin{proof} By Properties of Spaces, Lemma \ref{spaces-properties-lemma-dimension-decent-invariant-under-etale} every algebraic space $Y$ \'etale 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 \ref{decent-spaces-lemma-dimension-decent-space}. Also, if $Y$ is a scheme, then \'etale cohomology of $\mathcal{F}$ over $Y$, resp.\ \'etale 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 \ref{descent-proposition-same-cohomology-quasi-coherent} and \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cohomology-with-support-quasi-coherent}. We will use these facts without further mention. \medskip\noindent By Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated} 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 \'etale morphism $f_p : V_p \to U_p$ such that $f_p^{-1}(T_p) \to T_p$ is an isomorphism. \medskip\noindent 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 \ref{cohomology-proposition-cohomological-dimension-spectral}. 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 \ref{lemma-etale-localization-sheaf-with-support} and as $V_p$ is a scheme we obtain the desired vanishing from Cohomology, Proposition \ref{cohomology-proposition-cohomological-dimension-spectral}. In this way we conclude that (1) is true. \medskip\noindent 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 \'etale morphism such that $g^{-1}(Z) \to Z$ is an isomorphism. Hence by Lemma \ref{lemma-etale-localization-sheaf-with-support} 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 \ref{cohomology-proposition-cohomological-dimension-spectral}. 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 \'etale 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. \medskip\noindent A formal argument shows that (2) implies (3). \end{proof} \section{Cohomology and base change, I} \label{section-cohomology-and-base-change} \noindent 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: \begin{equation} \label{equation-base-change-diagram} \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 basic result for a flat base change. \begin{lemma} \label{lemma-flat-base-change-cohomology} In the situation above, assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. Then we have $$R^pf'_*\mathcal{F}' = g^*R^pf_*\mathcal{F}$$ for all $p \geq 0$ with notation as in (\ref{equation-base-change-diagram}). \end{lemma} \begin{proof} The morphism $g'$ is flat by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-flat}. Note that flatness of $g$ and $g'$ is equivalent to flatness of the morphisms of small \'etale ringed sites, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-morphism-sites}. Hence we can apply Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-base-change-map-flat-case} 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 \'etale 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 $X_B$. \medskip\noindent 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 \ref{lemma-affine-vanishing-higher-direct-images}. Thus $H^p(X_B, \mathcal{F}_B) = H^p(X, g'_*\mathcal{F}_B)$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}. 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 $U$ \'etale 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 \ref{algebra-theorem-lazard}. 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 \ref{lemma-colimits}). 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. \end{proof} \begin{lemma} \label{lemma-affine-base-change} 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 (\ref{equation-base-change-diagram}) we have $g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}$. \end{lemma} \begin{proof} By the discussion surrounding (\ref{equation-representable-higher-direct-image}) this reduces to the case of an affine morphism of schemes which is treated in Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}. \end{proof} \section{Coherent modules on locally Noetherian algebraic spaces} \label{section-coherent} \noindent This section is the analogue of Cohomology of Schemes, Section \ref{coherent-section-coherent-sheaves}. In Modules on Sites, Definition \ref{sites-modules-definition-site-local} 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. \begin{definition} \label{definition-coherent} 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 {\it coherent} if $\mathcal{F}$ is a coherent $\mathcal{O}_X$-module on the site $X_\etale$ in the sense of Modules on Sites, Definition \ref{sites-modules-definition-site-local}. \end{definition} \noindent Of course this definition is a bit hard to work with. We usually use the characterization given in the lemma below. \begin{lemma} \label{lemma-coherent-Noetherian} 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 \begin{enumerate} \item $\mathcal{F}$ is coherent, \item $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_X$-module, \item $\mathcal{F}$ is a finitely presented $\mathcal{O}_X$-module, \item for any \'etale morphism $\varphi : U \to X$ where $U$ is a scheme the pullback $\varphi^*\mathcal{F}$ is a coherent module on $U$, and \item there exists a surjective \'etale morphism $\varphi : U \to X$ where $U$ is a scheme such that the pullback $\varphi^*\mathcal{F}$ is a coherent module on $U$. \end{enumerate} 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. \end{lemma} \begin{proof} To be sure, if $X$ is a locally Noetherian algebraic space and $U \to X$ is an \'etale morphism, then $U$ is locally Noetherian, see Properties of Spaces, Section \ref{spaces-properties-section-types-properties}. The lemma then follows from the points (1) -- (5) made in Properties of Spaces, Section \ref{spaces-properties-section-properties-modules} and the corresponding result for coherent modules on locally Noetherian schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}. \end{proof} \begin{lemma} \label{lemma-coherent-abelian-Noetherian} 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. \end{lemma} \begin{proof} Choose a scheme $U$ and a surjective \'etale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (\ref{spaces-properties-equation-restrict-modules}). By Lemma \ref{lemma-coherent-Noetherian} 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 \ref{coherent-lemma-coherent-abelian-Noetherian}. \end{proof} \noindent 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. \begin{lemma} \label{lemma-coherent-Noetherian-quasi-coherent-sub-quotient} 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. \end{lemma} \begin{proof} Choose a scheme $U$ and a surjective \'etale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (\ref{spaces-properties-equation-restrict-modules}). By Lemma \ref{lemma-coherent-Noetherian} 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 \ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}. \end{proof} \begin{lemma} \label{lemma-tensor-hom-coherent} 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. \end{lemma} \begin{proof} Via Lemma \ref{lemma-coherent-Noetherian} this follows from the result for schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-tensor-hom-coherent}. \end{proof} \begin{lemma} \label{lemma-local-isomorphism} 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 $x \in |X|$. \begin{enumerate} \item If $\mathcal{F}_{\overline{x}} = 0$ then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\mathcal{F}|_{X'} = 0$. \item 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. \item 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. \item 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. \end{enumerate} \end{lemma} \begin{proof} Let $\varphi : U \to X$ be an \'etale morphism where $U$ is a scheme and let $u \in U$ be a point mapping to $x$. By Properties of Spaces, Lemmas \ref{spaces-properties-lemma-stalk-quasi-coherent} and \ref{spaces-properties-lemma-describe-etale-local-ring} as well as More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization} 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 \ref{spaces-properties-lemma-open-subspaces}. Since $\{U \to X'\}$ is a covering for the \'etale 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 $\mathcal{F} \to 0$. \end{proof} \begin{lemma} \label{lemma-coherent-support-closed} 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 \ref{spaces-morphisms-definition-scheme-theoretic-support}. Then $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module. \end{lemma} \begin{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 \ref{spaces-morphisms-lemma-immersion-locally-finite-type} and \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}. Finally, as $\mathcal{G}$ is of finite type it is a coherent $\mathcal{O}_Z$-module by Lemma \ref{lemma-coherent-Noetherian} \end{proof} \begin{lemma} \label{lemma-i-star-equivalence} 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. \end{lemma} \begin{proof} The functor is fully faithful by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence}. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module annihilated by $\mathcal{I}$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence} 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 \'etale locally (Lemma \ref{lemma-coherent-Noetherian}). Choosing an \'etale covering by a scheme we conclude that $\mathcal{G}$ is coherent by the case of schemes (Cohomology of Schemes, Lemma \ref{coherent-lemma-i-star-equivalence}). Hence the functor is fully faithful and the proof is done. \end{proof} \begin{lemma} \label{lemma-finite-pushforward-coherent} 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. \end{lemma} \begin{proof} Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Then $V \times_Y X \to V$ is a finite morphism of locally Noetherian schemes. By (\ref{equation-representable-higher-direct-image}) we reduce to the case of schemes which is Cohomology of Schemes, Lemma \ref{coherent-lemma-finite-pushforward-coherent}. \end{proof} \section{Coherent sheaves on Noetherian spaces} \label{section-coherent-quasi-compact} \noindent In this section we mention some properties of coherent sheaves on Noetherian algebraic spaces. \begin{lemma} \label{lemma-acc-coherent} 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 $n \geq 0$. \end{lemma} \begin{proof} Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}). 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 \ref{coherent-lemma-acc-coherent}. \end{proof} \begin{lemma} \label{lemma-power-ideal-kills-sheaf} 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). \end{lemma} \begin{proof} Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}). 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 \ref{coherent-lemma-power-ideal-kills-sheaf}. \end{proof} \begin{lemma}[Artin-Rees] \label{lemma-Artin-Rees} 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 $$\mathcal{I}^{n - c}(\mathcal{I}^c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^n\mathcal{F}$$ \end{lemma} \begin{proof} Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}). 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 \ref{coherent-lemma-Artin-Rees}. \end{proof} \begin{lemma} \label{lemma-homs-over-open} 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 $$\colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n, \mathcal{F}) \longrightarrow \Gamma(U, \mathcal{F}).$$ \end{lemma} \begin{proof} Let $W$ be an affine scheme and let $W \to X$ be a surjective \'etale morphism (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Set $R = W \times_X W$. Then $W$ and $R$ are Noetherian schemes, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}. 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 \ref{coherent-lemma-homs-over-open}. It follows formally that the result holds over $X$. \end{proof} \section{Devissage of coherent sheaves} \label{section-devissage} \noindent This section is the analogue of Cohomology of Schemes, Section \ref{coherent-section-devissage}. \begin{lemma} \label{lemma-prepare-filter-support} 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 $\text{Supp}(\mathcal{G}) \subset Z$. \end{lemma} \begin{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 \ref{spaces-properties-lemma-reduced-closed-subspace}. 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 \ref{lemma-power-ideal-kills-sheaf} 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. \end{proof} \noindent In the following we will freely use the scheme theoretic support of finite type modules as defined in Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support}. \begin{lemma} \label{lemma-prepare-filter-irreducible} 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 $Z$. \end{lemma} \begin{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 \ref{lemma-coherent-support-closed}. Hence it suffices to prove the lemma for the case $Z = X$ and $i = \text{id}$. \medskip\noindent By Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme} 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 \ref{coherent-lemma-prepare-filter-irreducible} 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 \ref{spaces-properties-section-reduced}). By Lemma \ref{lemma-homs-over-open} 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 $W$ \'etale 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. \end{proof} \begin{lemma} \label{lemma-coherent-filter} 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 $$\mathcal{F}_j/\mathcal{F}_{j - 1} \cong (Z_j \to X)_* \mathcal{I}_j$$ \end{lemma} \begin{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 \ref{spaces-properties-lemma-Noetherian-topology}) 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 $m = 0$). \medskip\noindent 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 \ref{lemma-prepare-filter-support} 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. \medskip\noindent 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 \ref{spaces-properties-lemma-reduced-closed-subspace}. By Lemma \ref{lemma-power-ideal-kills-sheaf} 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 $\mathcal{F}$. \medskip\noindent 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 \ref{spaces-morphisms-lemma-i-star-equivalence}. Hence we can apply Lemma \ref{lemma-prepare-filter-irreducible}. 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. \end{proof} \begin{lemma} \label{lemma-property-initial} 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 \begin{enumerate} \item 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 $\mathcal{F}$. \item 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 $i_*\mathcal{I}$. \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$. \end{lemma} \begin{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 \ref{lemma-coherent-filter} we can filter any $\mathcal{F}$ with successive subquotients as in (2). Hence the lemma follows. \end{proof} \noindent Here is a more useful variant of the lemma above. \begin{lemma} \label{lemma-property-higher-rank-cohomological} 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 \begin{enumerate} \item 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 $\mathcal{F}$. \item If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both. \item For every reduced closed subspace $i : Z \to X$ with $|Z|$ irreducible there exists a coherent sheaf $\mathcal{G}$ on $Z$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item 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 $i_*\mathcal{G}'$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$. \end{lemma} \begin{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 \ref{spaces-properties-lemma-Noetherian-topology}) 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 $T \not = \emptyset$ because the only sheaf with support in $\emptyset$ for which $\mathcal{P}$ does hold (by property (2)). \medskip\noindent 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 \ref{lemma-prepare-filter-support} 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. \medskip\noindent 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 \ref{spaces-properties-lemma-reduced-closed-subspace}. By Lemma \ref{lemma-power-ideal-kills-sheaf} 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 $\mathcal{F}$. \medskip\noindent 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 \ref{spaces-morphisms-lemma-i-star-equivalence} and Lemma \ref{lemma-coherent-support-closed}. Let $\mathcal{G}$ be the coherent sheaf on $Z$ satisfying (3)(a) and (3)(b). We apply Lemma \ref{lemma-prepare-filter-irreducible} 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 \ref{lemma-homs-over-open}) 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 \ref{lemma-power-ideal-kills-sheaf}. Thus after increasing $n$ we may assume the displayed map is injective, see Lemma \ref{lemma-Artin-Rees}. 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. \end{proof} \begin{lemma} \label{lemma-property-higher-rank-cohomological-variant} 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 \begin{enumerate} \item For any short exact sequence of coherent sheaves on $X$ if two out of three have property $\mathcal{P}$ so does the third. \item If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both. \item 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 $\mathcal{G}$. \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$. \end{lemma} \begin{proof} We will show that conditions (1) and (2) of Lemma \ref{lemma-property-initial} 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. \medskip\noindent 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 \ref{lemma-prepare-filter-irreducible}. Let $$0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \Coker(\varphi)$$ be a filtration as in Lemma \ref{lemma-coherent-filter}. 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 $\mathcal{P}$. \medskip\noindent 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. \end{proof} \section{Limits of coherent modules} \label{section-limits} \noindent 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 \ref{spaces-properties-lemma-properties-quasi-coherent}. Conversely, if the algebraic space is Noetherian, then every quasi-coherent module is a filtered colimit of coherent modules. \begin{lemma} \label{lemma-directed-colimit-coherent} 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. \end{lemma} \begin{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 \ref{lemma-coherent-abelian-Noetherian} and \ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}). 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. \medskip\noindent Let $U$ be an affine scheme and $U \to X$ a surjective \'etale morphism. Set $R = U \times_X U$ so that $X = U/R$ as usual. By Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent} 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 \ref{groupoids-lemma-colimit-coherent}. \end{proof} \begin{lemma} \label{lemma-direct-colimit-finite-presentation} 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. \end{lemma} \begin{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 \ref{lemma-directed-colimit-coherent}. By Lemma \ref{lemma-coherent-Noetherian} 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 \ref{spaces-properties-section-properties-modules}. 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 \'etale 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 \'etale. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules} 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. \medskip\noindent 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 \ref{lemma-finite-presentation-quasi-compact-colimit}. 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. \end{proof} \section{Vanishing of cohomology} \label{section-vanishing} \noindent 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 \ref{proposition-vanishing-affine}. \begin{situation} \label{situation-vanishing} 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 $A = \Gamma(X, \mathcal{O}_X)$. \end{situation} \noindent We would like to show that the canonical morphism $$p : X \longrightarrow \Spec(A)$$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-morphism-to-affine-scheme}) is an isomorphism. If $M$ is an $A$-module we denote $M \otimes_A \mathcal{O}_X$ the quasi-coherent module $p^*\tilde M$. \begin{lemma} \label{lemma-vanishing-compute} In Situation \ref{situation-vanishing} for an $A$-module $M$ we have $p_*(M \otimes_A \mathcal{O}_X) = \widetilde{M}$ and $\Gamma(X, M \otimes_A \mathcal{O}_X) = M$. \end{lemma} \begin{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 \ref{spaces-morphisms-lemma-pushforward}. Observe that $\Gamma(X, \bigoplus_{i \in I} \mathcal{O}_X) = \bigoplus_{i \in I} A$ by Lemma \ref{lemma-colimits}. 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. \end{proof} \noindent The following lemma shows that Situation \ref{situation-vanishing} is preserved by base change of $X \to \Spec(A)$ by $\Spec(A') \to \Spec(A)$. \begin{lemma} \label{lemma-vanishing-base-change} In Situation \ref{situation-vanishing}. \begin{enumerate} \item 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 $\mathcal{F}'$. \item Given an $A$-algebra $A'$ setting $X' = X \times_{\Spec(A)} \Spec(A')$ the morphism $X' \to X$ is affine and $\Gamma(X', \mathcal{O}_{X'}) = A'$. \end{enumerate} \end{lemma} \begin{proof} Part (1) follows from Lemma \ref{lemma-affine-vanishing-higher-direct-images} and the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray}). 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 \ref{spaces-morphisms-lemma-base-change-affine}) 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 \ref{lemma-affine-base-change} 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 \ref{lemma-vanishing-compute}. \end{proof} \begin{lemma} \label{lemma-vanishing-separate-closed} In Situation \ref{situation-vanishing}. 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 $Z_1 \subset V(a - 1)$. \end{lemma} \begin{proof} We may and do endow $Z_0$, $Z_1$ with the reduced induced subspace structure (Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}) 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. \end{proof} \begin{lemma} \label{lemma-vanishing-injective} In Situation \ref{situation-vanishing} the morphism $p : X \to \Spec(A)$ is universally injective. \end{lemma} \begin{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 \ref{spaces-morphisms-lemma-universally-injective-local}). Using Lemma \ref{lemma-vanishing-base-change} we may assume that $A$ is a field and we have to show that $|X|$ has at most one point. \medskip\noindent 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 $k \subset K$, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-points-fields}. If $k \subset K$ 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 \ref{spaces-morphisms-lemma-large-enough}. Hence, after replacing $k$ by $K$, we see that it suffices to prove that $X(k)$ is a singleton (in the case $A = k)$. \medskip\noindent Let $x, x' \in X(k)$. By Decent Spaces, Lemma \ref{decent-spaces-lemma-algebraic-residue-field-extension-closed-point} 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 \ref{lemma-vanishing-separate-closed}. We conclude that $x = x'$ as desired. \end{proof} \begin{lemma} \label{lemma-vanishing-separated} In Situation \ref{situation-vanishing} the morphism $p : X \to \Spec(A)$ is separated. \end{lemma} \begin{proof} By Decent Spaces, Lemma \ref{decent-spaces-lemma-there-is-a-scheme-integral-over} we can find a scheme $Y$ and a surjective integral morphism $Y \to X$. Since an integral morphism is affine, we can apply Lemma \ref{lemma-vanishing-base-change} 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 \ref{coherent-lemma-quasi-compact-h1-zero-covering} 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 \ref{spaces-morphisms-lemma-integral-universally-closed}. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-universally-closed-separated} we see that $X$ is a separated algebraic space. \end{proof} \begin{proposition} \label{proposition-vanishing-affine} \begin{slogan} Serre's criterion for affineness in the setting of algebraic spaces. \end{slogan} 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 \ref{situation-vanishing} is an affine scheme. \end{proposition} \begin{proof} Choose an affine scheme $U = \Spec(B)$ and a surjective \'etale morphism $\varphi : U \to X$. Set $R = U \times_X U$. As $p$ is separated (Lemma \ref{lemma-vanishing-separated}) 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 \ref{lemma-flat-base-change-cohomology}) 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 \ref{lemma-vanishing-injective} and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite} it is also locally quasi-finite. Hence $X \to \Spec(A)$ is representable by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable} and $X$ is a scheme. Finally $X$ is affine, hence equal to $\Spec(A)$, by an application of Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-compact-h1-zero-covering}. \end{proof} \begin{lemma} \label{lemma-Noetherian-h1-zero} 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. \end{lemma} \begin{proof} The assumption implies that $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ by Lemmas \ref{lemma-directed-colimit-coherent} and \ref{lemma-colimits}. Then $X$ is affine by Proposition \ref{proposition-vanishing-affine}. \end{proof} \begin{lemma} \label{lemma-Noetherian-h1-zero-invertible} 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 $X$. \end{lemma} \begin{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. \medskip\noindent 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 \ref{proposition-vanishing-affine}. Denote $j : U \to X$ the inclusion morphism. Since \'etale locally the morphism $j$ is affine (by Morphisms, Lemma \ref{morphisms-lemma-affine-s-open}) we see that $j$ is affine (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-local}). Hence we have $$H^1(U, \mathcal{F}) = H^1(X, j_*\mathcal{F})$$ by Lemma \ref{lemma-affine-vanishing-higher-direct-images} (and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}). Write $j_*\mathcal{F} = \colim \mathcal{F}_i$ as a filtered colimit of coherent $\mathcal{O}_X$-modules, see Lemma \ref{lemma-directed-colimit-coherent}. Then $$H^1(X, j_*\mathcal{F}) = \colim H^1(X, \mathcal{F}_i)$$ by Lemma \ref{lemma-colimits}. 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. \medskip\noindent Let $x \in |X|$ be a closed point. By Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-space-closed-point} 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 \ref{decent-spaces-section-reasonable-decent}). 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 \ref{spaces-properties-lemma-subscheme}) of $X$ by our claim. \medskip\noindent 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 \ref{spaces-properties-lemma-Noetherian-sober}. Finally, if $X$ is a scheme, then we can apply Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-compact-h1-zero-invertible} to conclude that $\mathcal{L}$ is ample. \end{proof} \section{Finite morphisms and affines} \label{section-finite-affine} \noindent This section is the analogue of Cohomology of Schemes, Section \ref{coherent-section-finite-affine}. \begin{lemma} \label{lemma-image-affine-finite-morphism-affine-Noetherian} Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ finite, \item $f$ surjective, \item $Y$ affine, and \item $X$ Noetherian. \end{enumerate} Then $X$ is affine. \end{lemma} \begin{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 \ref{lemma-directed-colimit-coherent} and \ref{lemma-colimits}. Then it follows that $X$ is affine from Proposition \ref{proposition-vanishing-affine}. \medskip\noindent 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 \ref{lemma-property-higher-rank-cohomological}. 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 $W = Z \times_X Y$ and denote $i' : W \to Y$ the corresponding closed immersion. Denote $f' : W \to Z$ the other projection which is a finite morphism of algebraic spaces. Since $W$ is a closed subscheme of $Y$, it is affine. We claim that $\mathcal{G} = f_*i'_*\mathcal{O}_W = i_*f'_*\mathcal{O}_W$ satisfies properties (3)(a) and (3)(b) of Lemma \ref{lemma-property-higher-rank-cohomological} which will finish the proof. Property (3)(a) is clear as $W \to Z$ is surjective (because $f$ is surjective). To see (3)(b) let $\mathcal{I}$ be a nonzero quasi-coherent sheaf of ideals on $Z$. We simply take $\mathcal{G}' = \mathcal{I} \mathcal{G}$. Namely, we have $$\mathcal{I} \mathcal{G} = f'_*(\mathcal{I}')$$ where $\mathcal{I}' = \Im((f')^*\mathcal{I} \to \mathcal{O}_W)$. This is true because $f'$ is a (representable) affine morphism of algebraic spaces and hence the result can be checked on an \'etale covering of $Z$ by a scheme in which case the result is Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-morphism-projection-ideal}. Finally, $f'$ is affine, hence $R^1f'_*\mathcal{I}' = 0$ by Lemma \ref{lemma-affine-vanishing-higher-direct-images}. As $W$ is affine we have $H^1(W, \mathcal{I}') = 0$ hence the Leray spectral sequence (in the form Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}) 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 and we win. \end{proof} \section{A weak version of Chow's lemma} \label{section-weak-chow} \noindent 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. \begin{lemma} \label{lemma-weak-chow} 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 $\Spec(A)$. \end{lemma} \begin{proof} Let $W$ be an affine scheme and let $f : W \to X$ be a surjective \'etale 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 \ref{decent-spaces-lemma-bounded-fibres}). Picking $d$ minimal we get a nonempty open $U \subset X$ such that $f^{-1}(U) \to U$ is finite \'etale of degree $d$, see Decent Spaces, Lemma \ref{decent-spaces-lemma-quasi-compact-reasonable-stratification}. 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 \'etale the diagonal $W \to W \times_X W$ is an open immersion, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-unramified} and \ref{spaces-morphisms-lemma-diagonal-unramified-morphism}. 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 \ref{morphisms-lemma-quasi-affine-finite-type-over-S} and \ref{morphisms-lemma-H-quasi-projective-open-H-projective}). 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 \ref{morphisms-lemma-quasi-compact-immersion}. 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 \ref{morphisms-section-projective}. 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 \ref{spaces-morphisms-lemma-equality-of-morphisms}. Claim: the morphism $X' \to X$ is proper. \medskip\noindent 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 \'etale 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 $X' \amalg T' \to X$. \medskip\noindent 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 \ref{spaces-morphisms-lemma-refined-valuative-criterion-universally-closed} 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 \ref{spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism}. 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 \ref{morphisms-lemma-locally-projective-proper} and Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}. The image of $z$ is in $g_i^{-1}(W) \subset X'$ and the proof is complete. \end{proof} \section{Noetherian valuative criterion} \label{section-Noetherian-valuative-criterion} \noindent 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 \ref{spaces-limits-section-Noetherian-valuative-criterion}. \begin{lemma} \label{lemma-check-separated-dvr} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $Y$ is locally Noetherian, \item $f$ is locally of finite type and quasi-separated, \item 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. \end{enumerate} Then $f$ is separated. \end{lemma} \begin{proof} To prove $f$ is separated, we may work \'etale locally on $Y$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-separated-local}). Choose an affine scheme $U$ and an \'etale morphism $U \to X \times_Y X$. Set $V = X \times_{\Delta, X \times_Y X} U$ which is quasi-compact because $f$ is quasi-separated. Consider a commutative diagram $$\xymatrix{ \Spec(K) \ar[r] \ar[d] & V \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & U }$$ 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 \ref{limits-lemma-Noetherian-dvr-valuative-proper} 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 (\'Etale Morphisms, Lemma \ref{etale-lemma-characterize-closed-immersion}) as desired. \end{proof} \begin{lemma} \label{lemma-check-proper-dvr} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $Y$ is locally Noetherian, \item $f$ is of finite type and quasi-separated, \item 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. \end{enumerate} Then $f$ is proper. \end{lemma} \begin{proof} It suffices to prove $f$ is universally closed because $f$ is separated by Lemma \ref{lemma-check-separated-dvr}. To do this we may work \'etale locally on $Y$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-closed-local}). 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 \ref{lemma-weak-chow}). We claim that $X' \to \Spec(A)$ is universally closed. The claim implies the lemma by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}. To prove this, according to Limits, Lemma \ref{limits-lemma-check-universally-closed-Noetherian} 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 $a$. \end{proof} \begin{remark}[Variant for complete discrete valuation rings] \label{remark-variant} In Lemmas \ref{lemma-check-separated-dvr} and \ref{lemma-check-proper-dvr} it suffices to consider complete discrete valuation rings. To be precise in Lemma \ref{lemma-check-separated-dvr} 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 \ref{lemma-check-separated-dvr} (3) the completion $A^\wedge$ is a discrete valuation ring (More on Algebra, Lemma \ref{more-algebra-lemma-completion-dvr}) 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 \ref{spaces-properties-proposition-sheaf-fpqc}. Similarly in Lemma \ref{lemma-check-proper-dvr} 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 \ref{lemma-check-proper-dvr} 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 {\it 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 \ref{spaces-morphisms-lemma-push-down-solution}. 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. \end{remark} \section{Higher direct images of coherent sheaves} \label{section-proper-pushforward} \noindent 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. \begin{lemma} \label{lemma-kill-by-twisting} 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 $p > 0$. \end{lemma} \begin{proof} Checking whether $R^pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes d})$ is zero can be done \'etale locally on $Y$, see Equation (\ref{equation-representable-higher-direct-image}). 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 \ref{coherent-lemma-kill-by-twisting}. \end{proof} \begin{lemma} \label{lemma-proper-pushforward-coherent} 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 $i \geq 0$. \end{lemma} \begin{proof} We first remark that $X$ is a locally Noetherian algebraic space by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}. Hence the statement of the lemma makes sense. Moreover, computing $R^if_*\mathcal{F}$ commutes with \'etale localization on $Y$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}) and checking whether $R^if_*\mathcal{F}$ coherent can be done \'etale locally on $Y$ (Lemma \ref{lemma-coherent-Noetherian}). Hence we may assume that $Y = \Spec(A)$ is a Noetherian affine scheme. \medskip\noindent Assume $Y = \Spec(A)$ is an affine scheme. Note that $f$ is locally of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}). Thus it is of finite presentation, hence $X$ is Noetherian (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-noetherian}). Thus Lemma \ref{lemma-property-higher-rank-cohomological-variant} 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 \ref{lemma-property-higher-rank-cohomological-variant} hold for this property thereby finishing the proof of the lemma. \medskip\noindent 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 \ref{lemma-coherent-abelian-Noetherian} and \ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}. Hence property $\mathcal{P}$ holds for the third as well. \medskip\noindent 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). \medskip\noindent 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 support 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 \ref{sites-cohomology-lemma-relative-Leray}) and the fact that $R^qi_*\mathcal{G} = 0$ for $q > 0$ by Lemma \ref{lemma-affine-vanishing-higher-direct-images} and the fact that a closed immersion is affine. (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-affine}). 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 $p$. \medskip\noindent We apply Lemma \ref{lemma-weak-chow} to the morphism $Z \to \Spec(A)$. Thus we get a diagram $$\xymatrix{ Z \ar[rd]_g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_A \ar[dl] \\ & \Spec(A) & }$$