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 \input{preamble} % OK, start here. % \begin{document} \title{Cohomology of Schemes} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we first prove a number of results on the cohomology of quasi-coherent sheaves. A fundamental reference is \cite{EGA}. Having done this we will elaborate on cohomology of coherent sheaves in the Noetherian setting. See \cite{FAC}. \section{{\v C}ech cohomology of quasi-coherent sheaves} \label{section-cech-quasi-coherent} \noindent Let $X$ be a scheme. Let $U \subset X$ be an affine open. Recall that a {\it standard open covering} of $U$ is a covering of the form $\mathcal{U} : U = \bigcup_{i = 1}^n D(f_i)$ where $f_1, \ldots, f_n \in \Gamma(U, \mathcal{O}_X)$ generate the unit ideal, see Schemes, Definition \ref{schemes-definition-standard-covering}. \begin{lemma} \label{lemma-cech-cohomology-quasi-coherent-trivial} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{U} : U = \bigcup_{i = 1}^n D(f_i)$ be a standard open covering of an affine open of $X$. Then $\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$ for all $p > 0$. \end{lemma} \begin{proof} Write $U = \Spec(A)$ for some ring $A$. In other words, $f_1, \ldots, f_n$ are elements of $A$ which generate the unit ideal of $A$. Write $\mathcal{F}|_U = \widetilde{M}$ for some $A$-module $M$. Clearly the {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is identified with the complex $$\prod\nolimits_{i_0} M_{f_{i_0}} \to \prod\nolimits_{i_0i_1} M_{f_{i_0}f_{i_1}} \to \prod\nolimits_{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \to \ldots$$ We are asked to show that the extended complex \begin{equation} \label{equation-extended} 0 \to M \to \prod\nolimits_{i_0} M_{f_{i_0}} \to \prod\nolimits_{i_0i_1} M_{f_{i_0}f_{i_1}} \to \prod\nolimits_{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \to \ldots \end{equation} (whose truncation we have studied in Algebra, Lemma \ref{algebra-lemma-cover-module}) is exact. It suffices to show that (\ref{equation-extended}) is exact after localizing at a prime $\mathfrak p$, see Algebra, Lemma \ref{algebra-lemma-characterize-zero-local}. In fact we will show that the extended complex localized at $\mathfrak p$ is homotopic to zero. \medskip\noindent There exists an index $i$ such that $f_i \not \in \mathfrak p$. Choose and fix such an element $i_{\text{fix}}$. Note that $M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}$. Similarly for a localization at a product $f_{i_0} \ldots f_{i_p}$ and $\mathfrak p$ we can drop any $f_{i_j}$ for which $i_j = i_{\text{fix}}$. Let us define a homotopy $$h : \prod\nolimits_{i_0 \ldots i_{p + 1}} M_{f_{i_0} \ldots f_{i_{p + 1}}, \mathfrak p} \longrightarrow \prod\nolimits_{i_0 \ldots i_p} M_{f_{i_0} \ldots f_{i_p}, \mathfrak p}$$ by the rule $$h(s)_{i_0 \ldots i_p} = s_{i_{\text{fix}} i_0 \ldots i_p}$$ (This is dual'' to the homotopy in the proof of Cohomology, Lemma \ref{cohomology-lemma-homology-complex}.) In other words, $h : \prod_{i_0} M_{f_{i_0}, \mathfrak p} \to M$ is projection onto the factor $M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}$ and in general the map $h$ equal projection onto the factors $M_{f_{i_{\text{fix}}} f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p} = M_{f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p}$. We compute \begin{align*} (dh + hd)(s)_{i_0 \ldots i_p} & = \sum\nolimits_{j = 0}^p (-1)^j h(s)_{i_0 \ldots \hat i_j \ldots i_p} + d(s)_{i_{\text{fix}} i_0 \ldots i_p}\\ & = \sum\nolimits_{j = 0}^p (-1)^j s_{i_{\text{fix}} i_0 \ldots \hat i_j \ldots i_p} + s_{i_0 \ldots i_p} + \sum\nolimits_{j = 0}^p (-1)^{j + 1} s_{i_{\text{fix}} i_0 \ldots \hat i_j \ldots i_p} \\ & = s_{i_0 \ldots i_p} \end{align*} This proves the identity map is homotopic to zero as desired. \end{proof} \noindent The following lemma says in particular that for any affine scheme $X$ and any quasi-coherent sheaf $\mathcal{F}$ on $X$ we have $$H^p(X, \mathcal{F}) = 0$$ for all $p > 0$. \begin{lemma} \label{lemma-quasi-coherent-affine-cohomology-zero} \begin{slogan} Serre vanishing: Higher cohomology vanishes on affine schemes for quasi-coherent modules. \end{slogan} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. For any affine open $U \subset X$ we have $H^p(U, \mathcal{F}) = 0$ for all $p > 0$. \end{lemma} \begin{proof} We are going to apply Cohomology, Lemma \ref{cohomology-lemma-cech-vanish-basis}. As our basis $\mathcal{B}$ for the topology of $X$ we are going to use the affine opens of $X$. As our set $\text{Cov}$ of open coverings we are going to use the standard open coverings of affine opens of $X$. Next we check that conditions (1), (2) and (3) of Cohomology, Lemma \ref{cohomology-lemma-cech-vanish-basis} hold. Note that the intersection of standard opens in an affine is another standard open. Hence property (1) holds. The coverings form a cofinal system of open coverings of any element of $\mathcal{B}$, see Schemes, Lemma \ref{schemes-lemma-standard-open}. Hence (2) holds. Finally, condition (3) of the lemma follows from Lemma \ref{lemma-cech-cohomology-quasi-coherent-trivial}. \end{proof} \noindent Here is a relative version of the vanishing of cohomology of quasi-coherent sheaves on affines. \begin{lemma} \label{lemma-relative-affine-vanishing} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $f$ is affine then $R^if_*\mathcal{F} = 0$ for all $i > 0$. \end{lemma} \begin{proof} According to Cohomology, Lemma \ref{cohomology-lemma-describe-higher-direct-images} the sheaf $R^if_*\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto H^i(f^{-1}(V), \mathcal{F}|_{f^{-1}(V)})$. By assumption, whenever $V$ is affine we have that $f^{-1}(V)$ is affine, see Morphisms, Definition \ref{morphisms-definition-affine}. By Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero} we conclude that $H^i(f^{-1}(V), \mathcal{F}|_{f^{-1}(V)}) = 0$ whenever $V$ is affine. Since $S$ has a basis consisting of affine opens we win. \end{proof} \begin{lemma} \label{lemma-relative-affine-cohomology} Let $f : X \to S$ be an affine morphism of schemes. 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-relative-affine-vanishing} and the Leray spectral sequence. See Cohomology, Lemma \ref{cohomology-lemma-apply-Leray}. \end{proof} \noindent The following two lemmas explain when {\v C}ech cohomology can be used to compute cohomology of quasi-coherent modules. \begin{lemma} \label{lemma-affine-diagonal} Let $X$ be a scheme. The following are equivalent \begin{enumerate} \item $X$ has affine diagonal $\Delta : X \to X \times X$, \item for $U, V \subset X$ affine open, the intersection $U \cap V$ is affine, and \item there exists an open covering $\mathcal{U} : X = \bigcup_{i \in I} U_i$ such that $U_{i_0 \ldots i_p}$ is affine open for all $p \ge 0$ and all $i_0, \ldots, i_p \in I$. \end{enumerate} In particular this holds if $X$ is separated. \end{lemma} \begin{proof} Assume $X$ has affine diagonal. Let $U, V \subset X$ be affine opens. Then $U \cap V = \Delta^{-1}(U \times V)$ is affine. Thus (2) holds. It is immediate that (2) implies (3). Conversely, if there is a covering of $X$ as in (3), then $X \times X = \bigcup U_i \times U_{i'}$ is an affine open covering, and we see that $\Delta^{-1}(U_i \times U_{i'}) = U_i \cap U_{i'}$ is affine. Then $\Delta$ is an affine morphism by Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}. The final assertion follows from Schemes, Lemma \ref{schemes-lemma-characterize-separated}. \end{proof} \begin{lemma} \label{lemma-cech-cohomology-quasi-coherent} Let $X$ be a scheme. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering such that $U_{i_0 \ldots i_p}$ is affine open for all $p \ge 0$ and all $i_0, \ldots, i_p \in I$. In this case for any quasi-coherent sheaf $\mathcal{F}$ we have $$\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(X, \mathcal{F})$$ as $\Gamma(X, \mathcal{O}_X)$-modules for all $p$. \end{lemma} \begin{proof} In view of Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero} this is a special case of Cohomology, Lemma \ref{cohomology-lemma-cech-spectral-sequence-application}. \end{proof} \section{Vanishing of cohomology} \label{section-vanishing} \noindent We have seen that on an affine scheme the higher cohomology groups of any quasi-coherent sheaf vanish (Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero}). It turns out that this also characterizes affine schemes. We give two versions. \begin{lemma} \label{lemma-quasi-compact-h1-zero-covering} \begin{reference} \cite{Serre-criterion}, \cite[II, Theorem 5.2.1 (d') and IV (1.7.17)]{EGA} \end{reference} \begin{slogan} Serre's criterion for affineness. \end{slogan} Let $X$ be a scheme. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ we have $H^1(X, \mathcal{I}) = 0$. \end{enumerate} Then $X$ is affine. \end{lemma} \begin{proof} Let $x \in X$ be a closed point. Let $U \subset X$ be an affine open neighbourhood of $x$. Write $U = \Spec(A)$ and let $\mathfrak m \subset A$ be the maximal ideal corresponding to $x$. Set $Z = X \setminus U$ and $Z' = Z \cup \{x\}$. By Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme} there are quasi-coherent sheaves of ideals $\mathcal{I}$, resp.\ $\mathcal{I}'$ cutting out the reduced closed subschemes $Z$, resp.\ $Z'$. Consider the short exact sequence $$0 \to \mathcal{I}' \to \mathcal{I} \to \mathcal{I}/\mathcal{I}' \to 0.$$ Since $x$ is a closed point of $X$ and $x \not \in Z$ we see that $\mathcal{I}/\mathcal{I}'$ is supported at $x$. In fact, the restriction of $\mathcal{I}/\mathcal{I'}$ to $U$ corresponds to the $A$-module $A/\mathfrak m$. Hence we see that $\Gamma(X, \mathcal{I}/\mathcal{I'}) = A/\mathfrak m$. Since by assumption $H^1(X, \mathcal{I}') = 0$ we see there exists a global section $f \in \Gamma(X, \mathcal{I})$ which maps to the element $1 \in A/\mathfrak m$ as a section of $\mathcal{I}/\mathcal{I'}$. Clearly we have $x \in X_f \subset U$. This implies that $X_f = D(f_A)$ where $f_A$ is the image of $f$ in $A = \Gamma(U, \mathcal{O}_X)$. In particular $X_f$ is affine. \medskip\noindent Consider the union $W = \bigcup X_f$ over all $f \in \Gamma(X, \mathcal{O}_X)$ such that $X_f$ is affine. Obviously $W$ is open in $X$. By the arguments above every closed point of $X$ is contained in $W$. The closed subset $X \setminus W$ of $X$ is also quasi-compact (see Topology, Lemma \ref{topology-lemma-closed-in-quasi-compact}). Hence it has a closed point if it is nonempty (see Topology, Lemma \ref{topology-lemma-quasi-compact-closed-point}). This would contradict the fact that all closed points are in $W$. Hence we conclude $X = W$. \medskip\noindent Choose finitely many $f_1, \ldots, f_n \in \Gamma(X, \mathcal{O}_X)$ such that $X = X_{f_1} \cup \ldots \cup X_{f_n}$ and such that each $X_{f_i}$ is affine. This is possible as we've seen above. By Properties, Lemma \ref{properties-lemma-characterize-affine} to finish the proof it suffices to show that $f_1, \ldots, f_n$ generate the unit ideal in $\Gamma(X, \mathcal{O}_X)$. Consider the short exact sequence $$\xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{O}_X^{\oplus n} \ar[rr]^{f_1, \ldots, f_n} & & \mathcal{O}_X \ar[r] & 0 }$$ The arrow defined by $f_1, \ldots, f_n$ is surjective since the opens $X_{f_i}$ cover $X$. We let $\mathcal{F}$ be the kernel of this surjective map. Observe that $\mathcal{F}$ has a filtration $$0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_n = \mathcal{F}$$ so that each subquotient $\mathcal{F}_i/\mathcal{F}_{i - 1}$ is isomorphic to a quasi-coherent sheaf of ideals. Namely we can take $\mathcal{F}_i$ to be the intersection of $\mathcal{F}$ with the first $i$ direct summands of $\mathcal{O}_X^{\oplus n}$. The assumption of the lemma implies that $H^1(X, \mathcal{F}_i/\mathcal{F}_{i - 1}) = 0$ for all $i$. This implies that $H^1(X, \mathcal{F}_2) = 0$ because it is sandwiched between $H^1(X, \mathcal{F}_1)$ and $H^1(X, \mathcal{F}_2/\mathcal{F}_1)$. Continuing like this we deduce that $H^1(X, \mathcal{F}) = 0$. Therefore we conclude that the map $$\xymatrix{ \bigoplus\nolimits_{i = 1, \ldots, n} \Gamma(X, \mathcal{O}_X) \ar[rr]^{f_1, \ldots, f_n} & & \Gamma(X, \mathcal{O}_X) }$$ is surjective as desired. \end{proof} \noindent Note that if $X$ is a Noetherian scheme then every quasi-coherent sheaf of ideals is automatically a coherent sheaf of ideals and a finite type quasi-coherent sheaf of ideals. Hence the preceding lemma and the next lemma both apply in this case. \begin{lemma} \label{lemma-quasi-separated-h1-zero-covering} \begin{reference} \cite{Serre-criterion}, \cite[II, Theorem 5.2.1]{EGA} \end{reference} \begin{slogan} Serre's criterion for affineness. \end{slogan} Let $X$ be a scheme. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item $X$ is quasi-separated, and \item $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals $\mathcal{I}$ of finite type. \end{enumerate} Then $X$ is affine. \end{lemma} \begin{proof} By Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} every quasi-coherent sheaf of ideals is a directed colimit of quasi-coherent sheaves of ideals of finite type. By Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit} taking cohomology on $X$ commutes with directed colimits. Hence we see that $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals on $X$. In other words we see that Lemma \ref{lemma-quasi-compact-h1-zero-covering} applies. \end{proof} \noindent We can use the arguments given above to find a sufficient condition to see when an invertible sheaf is ample. However, we warn the reader that this condition is not necessary. \begin{lemma} \label{lemma-quasi-compact-h1-zero-invertible} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ there exists an $n \geq 1$ such that $H^1(X, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0$. \end{enumerate} Then $\mathcal{L}$ is ample. \end{lemma} \begin{proof} This is proved in exactly the same way as Lemma \ref{lemma-quasi-compact-h1-zero-covering}. Let $x \in X$ be a closed point. Let $U \subset X$ be an affine open neighbourhood of $x$ such that $\mathcal{L}|_U \cong \mathcal{O}_U$. Write $U = \Spec(A)$ and let $\mathfrak m \subset A$ be the maximal ideal corresponding to $x$. Set $Z = X \setminus U$ and $Z' = Z \cup \{x\}$. By Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme} there are quasi-coherent sheaves of ideals $\mathcal{I}$, resp.\ $\mathcal{I}'$ cutting out the reduced closed subschemes $Z$, resp.\ $Z'$. Consider the short exact sequence $$0 \to \mathcal{I}' \to \mathcal{I} \to \mathcal{I}/\mathcal{I}' \to 0.$$ For every $n \geq 1$ we obtain a short exact sequence $$0 \to \mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} \to \mathcal{I}/\mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} \to 0.$$ By our assumption we may pick $n$ such that $H^1(X, \mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0$. Since $x$ is a closed point of $X$ and $x \not \in Z$ we see that $\mathcal{I}/\mathcal{I}'$ is supported at $x$. In fact, the restriction of $\mathcal{I}/\mathcal{I'}$ to $U$ corresponds to the $A$-module $A/\mathfrak m$. Since $\mathcal{L}$ is trivial on $U$ we see that the restriction of $\mathcal{I}/\mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}$ to $U$ also corresponds to the $A$-module $A/\mathfrak m$. Hence we see that $\Gamma(X, \mathcal{I}/\mathcal{I'} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = A/\mathfrak m$. By our choice of $n$ we see there exists a global section $s \in \Gamma(X, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})$ which maps to the element $1 \in A/\mathfrak m$. Clearly we have $x \in X_s \subset U$ because $s$ vanishes at points of $Z$. This implies that $X_s = D(f)$ where $f \in A$ is the image of $s$ in $A \cong \Gamma(U, \mathcal{L}^{\otimes n})$. In particular $X_s$ is affine. \medskip\noindent Consider the union $W = \bigcup X_s$ over all $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ for $n \geq 1$ such that $X_s$ is affine. Obviously $W$ is open in $X$. By the arguments above every closed point of $X$ is contained in $W$. The closed subset $X \setminus W$ of $X$ is also quasi-compact (see Topology, Lemma \ref{topology-lemma-closed-in-quasi-compact}). Hence it has a closed point if it is nonempty (see Topology, Lemma \ref{topology-lemma-quasi-compact-closed-point}). This would contradict the fact that all closed points are in $W$. Hence we conclude $X = W$. This means that $\mathcal{L}$ is ample by Properties, Definition \ref{properties-definition-ample}. \end{proof} \noindent There is a variant of Lemma \ref{lemma-quasi-compact-h1-zero-invertible} with finite type ideal sheaves which we will formulate and prove here if we ever need it. \section{Quasi-coherence of higher direct images} \label{section-quasi-coherence} \noindent We have seen that the higher cohomology groups of a quasi-coherent module on an affine is zero. For (quasi-)separated quasi-compact schemes $X$ this implies vanishing of cohomology groups of quasi-coherent sheaves beyond a certain degree. However, it may not be the case that $X$ has finite cohomological dimension, because that is defined in terms of vanishing of cohomology of {\it all} $\mathcal{O}_X$-modules. \begin{lemma} \label{lemma-induction-principle} Let $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property of the quasi-compact opens of $X$. Assume that \begin{enumerate} \item $P$ holds for every affine open of $X$, \item if $U$ is quasi-compact open, $V$ affine open, $P$ holds for $U$, $V$, and $U \cap V$, then $P$ holds for $U \cup V$. \end{enumerate} Then $P$ holds for every quasi-compact open of $X$ and in particular for $X$. \end{lemma} \begin{proof} First we argue by induction that $P$ holds for {\it separated} quasi-compact opens $W \subset X$. Namely, such an open can be written as $W = U_1 \cup \ldots \cup U_n$ and we can do induction on $n$ using property (2) with $U = U_1 \cup \ldots \cup U_{n - 1}$ and $V = U_n$. This is allowed because $U \cap V = (U_1 \cap U_n) \cup \ldots \cup (U_{n - 1} \cap U_n)$ is also a union of $n - 1$ affine open subschemes by Schemes, Lemma \ref{schemes-lemma-characterize-separated} applied to the affine opens $U_i$ and $U_n$ of $W$. Having said this, for any quasi-compact open $W \subset X$ we can do induction on the number of affine opens needed to cover $W$ using the same trick as before and using that the quasi-compact open $U_i \cap U_n$ is separated as an open subscheme of the affine scheme $U_n$. \end{proof} \begin{lemma} \label{lemma-vanishing-nr-affines} \begin{slogan} For schemes with affine diagonal, the cohomology of quasi-coherent modules vanishes in degrees bigger than the number of affine opens needed in a covering. \end{slogan} Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Let $t = t(X)$ be the minimal number of affine opens needed to cover $X$. Then $H^n(X, \mathcal{F}) = 0$ for all $n \geq t$ and all quasi-coherent sheaves $\mathcal{F}$. \end{lemma} \begin{proof} First proof. By induction on $t$. If $t = 1$ the result follows from Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero}. If $t > 1$ write $X = U \cup V$ with $V$ affine open and $U = U_1 \cup \ldots \cup U_{t - 1}$ a union of $t - 1$ open affines. Note that in this case $U \cap V = (U_1 \cap V) \cup \ldots (U_{t - 1} \cap V)$ is also a union of $t - 1$ affine open subschemes. Namely, since the diagonal is affine, the intersection of two affine opens is affine, see Lemma \ref{lemma-affine-diagonal}. We apply the Mayer-Vietoris long exact sequence $$0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots$$ see Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris}. By induction we see that the groups $H^i(U, \mathcal{F})$, $H^i(V, \mathcal{F})$, $H^i(U \cap V, \mathcal{F})$ are zero for $i \geq t - 1$. It follows immediately that $H^i(X, \mathcal{F})$ is zero for $i \geq t$. \medskip\noindent Second proof. Let $\mathcal{U} : X = \bigcup_{i = 1}^t U_i$ be a finite affine open covering. Since $X$ is has affine diagonal the multiple intersections $U_{i_0 \ldots i_p}$ are all affine, see Lemma \ref{lemma-affine-diagonal}. By Lemma \ref{lemma-cech-cohomology-quasi-coherent} the {\v C}ech cohomology groups $\check{H}^p(\mathcal{U}, \mathcal{F})$ agree with the cohomology groups. By Cohomology, Lemma \ref{cohomology-lemma-alternating-usual} the {\v C}ech cohomology groups may be computed using the alternating {\v C}ech complex $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$. As the covering consists of $t$ elements we see immediately that $\check{\mathcal{C}}_{alt}^p(\mathcal{U}, \mathcal{F}) = 0$ for all $p \geq t$. Hence the result follows. \end{proof} \begin{lemma} \label{lemma-affine-diagonal-universal-delta-functor} Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Then \begin{enumerate} \item given a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an embedding $\mathcal{F} \to \mathcal{F}'$ of quasi-coherent $\mathcal{O}_X$-modules such that $H^p(X, \mathcal{F}') = 0$ for all $p \geq 1$, and \item $\{H^n(X, -)\}_{n \geq 0}$ is a universal $\delta$-functor from $\QCoh(\mathcal{O}_X)$ to $\textit{Ab}$. \end{enumerate} \end{lemma} \begin{proof} Let $X = \bigcup U_i$ be an affine open covering. Set $U = \coprod U_i$ and denote $f : U \to X$ the morphism inducing the given open immersions $U_i \to X$. For every $\mathcal{O}_X$-module $\mathcal{F}$ there is a canonical map $\mathcal{F} \to j_*j^*\mathcal{F}$. This map is injective as can be seen by checking on stalks: if $x \in U_i$, then we have a factorization $$\mathcal{F}_x \to (j_*j^*\mathcal{F})_x \to (j^*\mathcal{F})_{x'} = \mathcal{F}_x$$ where $x' \in U$ is the point $x$ viewed as a point of $U_i \subset U$. Now if $\mathcal{F}$ is quasi-coherent, then $j^*\mathcal{F}$ is quasi-coherent on the affine scheme $U$ hence has vanishing higher cohomology. Then $H^p(X, j_*j^*\mathcal{F}) = 0$ for $p > 0$ by Lemma \ref{lemma-relative-affine-cohomology} as $j$ is affine by Morphisms, Lemma \ref{morphisms-lemma-affine-permanence}. This proves (1). Then $H^p(X, \mathcal{F}) \to H^p(X, j_*j^*\mathcal{F})$ is zero and part (2) follows from Homology, Lemma \ref{homology-lemma-efface-implies-universal}. \end{proof} \begin{lemma} \label{lemma-vanishing-nr-affines-quasi-separated} Let $X$ be a quasi-compact quasi-separated scheme. Let $X = U_1 \cup \ldots \cup U_t$ be an affine open covering. Set $$d = \max\nolimits_{I \subset \{1, \ldots, t\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_i)\right)$$ where $t(U)$ is the minimal number of affines needed to cover the scheme $U$. Then $H^n(X, \mathcal{F}) = 0$ for all $n \geq d$ and all quasi-coherent sheaves $\mathcal{F}$. \end{lemma} \begin{proof} Note that since $X$ is quasi-separated the numbers $t(\bigcap_{i \in I} U_i)$ are finite. Let $\mathcal{U} : X = \bigcup_{i = 1}^t U_i$. By Cohomology, Lemma \ref{cohomology-lemma-cech-spectral-sequence} there is a spectral sequence $$E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))$$ converging to $H^{p + q}(U, \mathcal{F})$. By Cohomology, Lemma \ref{cohomology-lemma-alternating-usual} we have $$E_2^{p, q} = H^p(\check{\mathcal{C}}_{alt}^\bullet( \mathcal{U}, \underline{H}^q(\mathcal{F}))$$ The alternating {\v C}ech complex with values in the presheaf $\underline{H}^q(\mathcal{F})$ vanishes in high degrees by Lemma \ref{lemma-vanishing-nr-affines}, more precisely $E_2^{p, q} = 0$ for $p + q \geq d$. Hence the result follows. \end{proof} \begin{lemma} \label{lemma-quasi-coherence-higher-direct-images} Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. \begin{enumerate} \item For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the higher direct images $R^pf_*\mathcal{F}$ are quasi-coherent on $S$. \item If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$ such that $R^pf_*\mathcal{F} = 0$ for all $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}$ on $X$. \item In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$ such that for every morphism of schemes $S' \to S$ we have $R^p(f')_*\mathcal{F}' = 0$ for $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}'$ on $X'$. Here $f' : X' = S' \times_S X \to S'$ is the base change of $f$. \end{enumerate} \end{lemma} \begin{proof} We first prove (1). Note that under the hypotheses of the lemma the sheaf $R^0f_*\mathcal{F} = f_*\mathcal{F}$ is quasi-coherent by Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Using Cohomology, Lemma \ref{cohomology-lemma-localize-higher-direct-images} we see that forming higher direct images commutes with restriction to open subschemes. Since being quasi-coherent is local on $S$ we may assume $S$ is affine. \medskip\noindent Assume $S$ is affine and $f$ quasi-compact and separated. Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$. We will prove this case of (1) by induction on $t$. If $t = 1$ then the morphism $f$ is affine by Morphisms, Lemma \ref{morphisms-lemma-morphism-affines-affine} and (1) follows from Lemma \ref{lemma-relative-affine-vanishing}. If $t > 1$ write $X = U \cup V$ with $V$ affine open and $U = U_1 \cup \ldots \cup U_{t - 1}$ a union of $t - 1$ open affines. Note that in this case $U \cap V = (U_1 \cap V) \cup \ldots (U_{t - 1} \cap V)$ is also a union of $t - 1$ affine open subschemes, see Schemes, Lemma \ref{schemes-lemma-characterize-separated}. We will apply the relative Mayer-Vietoris sequence $$0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots$$ see Cohomology, Lemma \ref{cohomology-lemma-relative-mayer-vietoris}. By induction we see that $R^pa_*\mathcal{F}$, $R^pb_*\mathcal{F}$ and $R^pc_*\mathcal{F}$ are all quasi-coherent. This implies that each of the sheaves $R^pf_*\mathcal{F}$ is quasi-coherent since it sits in the middle of a short exact sequence with a cokernel of a map between quasi-coherent sheaves on the left and a kernel of a map between quasi-coherent sheaves on the right. Using the results on quasi-coherent sheaves in Schemes, Section \ref{schemes-section-quasi-coherent} we see conclude $R^pf_*\mathcal{F}$ is quasi-coherent. \medskip\noindent Assume $S$ is affine and $f$ quasi-compact and quasi-separated. Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$. We will prove (1) by induction on $t$. In case $t = 1$ the morphism $f$ is separated and we are back in the previous case (see previous paragraph). If $t > 1$ write $X = U \cup V$ with $V$ affine open and $U$ a union of $t - 1$ open affines. Note that in this case $U \cap V$ is an open subscheme of an affine scheme and hence separated (see Schemes, Lemma \ref{schemes-lemma-affine-separated}). We will apply the relative Mayer-Vietoris sequence $$0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots$$ see Cohomology, Lemma \ref{cohomology-lemma-relative-mayer-vietoris}. By induction and the result of the previous paragraph we see that $R^pa_*\mathcal{F}$, $R^pb_*\mathcal{F}$ and $R^pc_*\mathcal{F}$ are quasi-coherent. As in the previous paragraph this implies each of sheaves $R^pf_*\mathcal{F}$ is quasi-coherent. \medskip\noindent Next, we prove (3) and a fortiori (2). Choose a finite affine open covering $S = \bigcup_{j = 1, \ldots m} S_j$. For each $i$ choose a finite affine open covering $f^{-1}(S_j) = \bigcup_{i = 1, \ldots t_j} U_{ji}$. Let $$d_j = \max\nolimits_{I \subset \{1, \ldots, t_j\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_{ji})\right)$$ be the integer found in Lemma \ref{lemma-vanishing-nr-affines-quasi-separated}. We claim that $n(X, S, f) = \max d_j$ works. \medskip\noindent Namely, let $S' \to S$ be a morphism of schemes and let $\mathcal{F}'$ be a quasi-coherent sheaf on $X' = S' \times_S X$. We want to show that $R^pf'_*\mathcal{F}' = 0$ for $p \geq n(X, S, f)$. Since this question is local on $S'$ we may assume that $S'$ is affine and maps into $S_j$ for some $j$. Then $X' = S' \times_{S_j} f^{-1}(S_j)$ is covered by the open affines $S' \times_{S_j} U_{ji}$, $i = 1, \ldots t_j$ and the intersections $$\bigcap\nolimits_{i \in I} S' \times_{S_j} U_{ji} = S' \times_{S_j} \bigcap\nolimits_{i \in I} U_{ji}$$ are covered by the same number of affines as before the base change. Applying Lemma \ref{lemma-vanishing-nr-affines-quasi-separated} we get $H^p(X', \mathcal{F}') = 0$. By the first part of the proof we already know that each $R^qf'_*\mathcal{F}'$ is quasi-coherent hence has vanishing higher cohomology groups on our affine scheme $S'$, thus we see that $H^0(S', R^pf'_*\mathcal{F}') = H^p(X', \mathcal{F}') = 0$ by Cohomology, Lemma \ref{cohomology-lemma-apply-Leray}. Since $R^pf'_*\mathcal{F}'$ is quasi-coherent we conclude that $R^pf'_*\mathcal{F}' = 0$. \end{proof} \begin{lemma} \label{lemma-quasi-coherence-higher-direct-images-application} Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. Assume $S$ is affine. For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $$H^q(X, \mathcal{F}) = H^0(S, R^qf_*\mathcal{F})$$ for all $q \in \mathbf{Z}$. \end{lemma} \begin{proof} Consider the Leray spectral sequence $E_2^{p, q} = H^p(S, R^qf_*\mathcal{F})$ converging to $H^{p + q}(X, \mathcal{F})$, see Cohomology, Lemma \ref{cohomology-lemma-Leray}. By Lemma \ref{lemma-quasi-coherence-higher-direct-images} we see that the sheaves $R^qf_*\mathcal{F}$ are quasi-coherent. By Lemma \ref{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. See also Cohomology, Lemma \ref{cohomology-lemma-apply-Leray} (2) for the general principle. \end{proof} \section{Cohomology and base change, I} \label{section-cohomology-and-base-change} \noindent Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Suppose further that $g : S' \to S$ is any morphism of schemes. Denote $X' = X_{S'} = S' \times_S X$ the base change of $X$ and denote $f' : X' \to S'$ 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}' & S' \ar[r]^g & S & Rf_*\mathcal{F} } } \end{equation} Here is the simplest case of the base change property we have in mind. \begin{lemma} \label{lemma-affine-base-change} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ is affine. In this case $f_*\mathcal{F} \cong Rf_*\mathcal{F}$ is a quasi-coherent sheaf, and for every base change diagram (\ref{equation-base-change-diagram}) we have $$g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}.$$ \end{lemma} \begin{proof} The vanishing of higher direct images is Lemma \ref{lemma-relative-affine-vanishing}. The statement is local on $S$ and $S'$. Hence we may assume $X = \Spec(A)$, $S = \Spec(R)$, $S' = \Spec(R')$ and $\mathcal{F} = \widetilde{M}$ for some $A$-module $M$. We use Schemes, Lemma \ref{schemes-lemma-widetilde-pullback} to describe pullbacks and pushforwards of $\mathcal{F}$. Namely, $X' = \Spec(R' \otimes_R A)$ and $\mathcal{F}'$ is the quasi-coherent sheaf associated to $(R' \otimes_R A) \otimes_A M$. Thus we see that the lemma boils down to the equality $$(R' \otimes_R A) \otimes_A M = R' \otimes_R M$$ as $R'$-modules. \end{proof} \noindent In many situations it is sufficient to know about the following special case of cohomology and base change. It follows immediately from the stronger results in Section \ref{section-cohomology-and-base-change-derived}, but since it is so important it deserves its own proof. \begin{lemma}[Flat base change] \label{lemma-flat-base-change-cohomology} Consider a cartesian diagram of schemes $$\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S }$$ Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module with pullback $\mathcal{F}' = (g')^*\mathcal{F}$. Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. For any $i \geq 0$ \begin{enumerate} \item the base change map of Cohomology, Lemma \ref{cohomology-lemma-base-change-map-flat-case} is an isomorphism $$g^*R^if_*\mathcal{F} \longrightarrow R^if'_*\mathcal{F}',$$ \item if $S = \Spec(A)$ and $S' = \Spec(B)$, then $H^i(X, \mathcal{F}) \otimes_A B = H^i(X', \mathcal{F}')$. \end{enumerate} \end{lemma} \begin{proof} We claim that part (1) follows from part (2). Namely, part (1) is local on $S'$ and hence we may assume $S$ and $S'$ are affine. In other words, we have $S = \Spec(A)$ and $S' = \Spec(B)$ as in (2). Then since $R^if_*\mathcal{F}$ is quasi-coherent (Lemma \ref{lemma-quasi-coherence-higher-direct-images}), it is the quasi-coherent $\mathcal{O}_S$-module associated to the $A$-module $H^0(S, R^if_*\mathcal{F}) = H^i(X, \mathcal{F})$ (equality by Lemma \ref{lemma-quasi-coherence-higher-direct-images-application}). Similarly, $R^if'_*\mathcal{F}'$ is the quasi-coherent $\mathcal{O}_{S'}$-module associated to the $B$-module $H^i(X', \mathcal{F}')$. Since pullback by $g$ corresponds to $- \otimes_A B$ on modules (Schemes, Lemma \ref{schemes-lemma-widetilde-pullback}) we see that it suffices to prove (2). \medskip\noindent Let $A \to B$ be a flat ring homomorphism. Let $X$ be a quasi-compact and quasi-separated scheme over $A$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Set $X_B = X \times_{\Spec(A)} \Spec(B)$ and denote $\mathcal{F}_B$ the pullback of $\mathcal{F}$. We are trying to show that the map $$H^i(X, \mathcal{F}) \otimes_A B \longrightarrow H^i(X_B, \mathcal{F}_B)$$ (given by the reference in the statement of the lemma) 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 In case $X$ is separated, choose an affine open covering $\mathcal{U} : X = U_1 \cup \ldots \cup U_t$ and recall that $$\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(X, \mathcal{F}),$$ see Lemma \ref{lemma-cech-cohomology-quasi-coherent}. If $\mathcal{U}_B : X_B = (U_1)_B \cup \ldots \cup (U_t)_B$ we obtain by base change, then it is still the case that each $(U_i)_B$ is affine and that $X_B$ is separated. Thus we obtain $$\check{H}^p(\mathcal{U}_B, \mathcal{F}_B) = H^p(X_B, \mathcal{F}_B).$$ We have the following relation between the {\v C}ech complexes $$\check{\mathcal{C}}^\bullet(\mathcal{U}_B, \mathcal{F}_B) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A B$$ as follows from Lemma \ref{lemma-affine-base-change}. Since $A \to B$ is flat, the same thing remains true on taking cohomology. \medskip\noindent In case $X$ is quasi-separated, choose an affine open covering $\mathcal{U} : X = U_1 \cup \ldots \cup U_t$. We will use the {\v C}ech-to-cohomology spectral sequence Cohomology, Lemma \ref{cohomology-lemma-cech-spectral-sequence}. The reader who wishes to avoid this spectral sequence can use Mayer-Vietoris and induction on $t$ as in the proof of Lemma \ref{lemma-quasi-coherence-higher-direct-images}. The spectral sequence has $E_2$-page $E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))$ and converges to $H^{p + q}(X, \mathcal{F})$. Similarly, we have a spectral sequence with $E_2$-page $E_2^{p, q} = \check{H}^p(\mathcal{U}_B, \underline{H}^q(\mathcal{F}_B))$ which converges to $H^{p + q}(X_B, \mathcal{F}_B)$. Since the intersections $U_{i_0 \ldots i_p}$ are quasi-compact and separated, the result of the second paragraph of the proof gives $\check{H}^p(\mathcal{U}_B, \underline{H}^q(\mathcal{F}_B)) = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) \otimes_A B$. Using that $A \to B$ is flat we conclude that $H^i(X, \mathcal{F}) \otimes_A B \to H^i(X_B, \mathcal{F}_B)$ is an isomorphism for all $i$ and we win. \end{proof} \begin{lemma}[Finite locally free base change] \label{lemma-finite-locally-free-base-change-cohomology} Consider a cartesian diagram of schemes $$\xymatrix{ Y \ar[d]_{g} \ar[r]_h & X \ar[d]^f \\ \Spec(B) \ar[r] & \Spec(A) }$$ Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module with pullback $\mathcal{G} = h^*\mathcal{F}$. If $B$ is a finite locally free $A$-module, then $H^i(X, \mathcal{F}) \otimes_A B = H^i(Y, \mathcal{G})$. \end{lemma} \noindent {\bf Warning}: Do not use this lemma unless you understand the difference between this and Lemma \ref{lemma-flat-base-change-cohomology}. \begin{proof} In case $X$ is separated, choose an affine open covering $\mathcal{U} : X = \bigcup_{i \in I} U_i$ and recall that $$\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(X, \mathcal{F}),$$ see Lemma \ref{lemma-cech-cohomology-quasi-coherent}. Let $\mathcal{V} : Y = \bigcup_{i \in I} g^{-1}(U_i)$ be the corresponding affine open covering of $Y$. The opens $V_i = g^{-1}(U_i) = U_i \times_{\Spec(A)} \Spec(B)$ are affine and $Y$ is separated. Thus we obtain $$\check{H}^p(\mathcal{V}, \mathcal{G}) = H^p(Y, \mathcal{G}).$$ We claim the map of {\v C}ech complexes $$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A B \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})$$ is an isomorphism. Namely, as $B$ is finitely presented as an $A$-module we see that tensoring with $B$ over $A$ commutes with products, see Algebra, Proposition \ref{algebra-proposition-fp-tensor}. Thus it suffices to show that the maps $\Gamma(U_{i_0 \ldots i_p}, \mathcal{F}) \otimes_A B \to \Gamma(V_{i_0 \ldots i_p}, \mathcal{G})$ are isomorphisms which follows from Lemma \ref{lemma-affine-base-change}. Since $A \to B$ is flat, the same thing remains true on taking cohomology. \medskip\noindent In the general case we argue in exactly the same way using affine open covering $\mathcal{U} : X = \bigcup_{i \in I} U_i$ and the corresponding covering $\mathcal{V} : Y = \bigcup_{i \in I} V_i$ with $V_i = g^{-1}(U_i)$ as above. We will use the {\v C}ech-to-cohomology spectral sequence Cohomology, Lemma \ref{cohomology-lemma-cech-spectral-sequence}. The spectral sequence has $E_2$-page $E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))$ and converges to $H^{p + q}(X, \mathcal{F})$. Similarly, we have a spectral sequence with $E_2$-page $E_2^{p, q} = \check{H}^p(\mathcal{V}, \underline{H}^q(\mathcal{G}))$ which converges to $H^{p + q}(Y, \mathcal{G})$. Since the intersections $U_{i_0 \ldots i_p}$ are separated, the result of the previous paragraph gives isomorphisms $\Gamma(U_{i_0 \ldots i_p}, \underline{H}^q(\mathcal{F})) \otimes_A B \to \Gamma(V_{i_0 \ldots i_p}, \underline{H}^q(\mathcal{G}))$. Using that $- \otimes_A B$ commutes with products and is exact, we conclude that $\check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) \otimes_A B \to \check{H}^p(\mathcal{V}, \underline{H}^q(\mathcal{G}))$ is an isomorphism. Using that $A \to B$ is flat we conclude that $H^i(X, \mathcal{F}) \otimes_A B \to H^i(Y, \mathcal{G})$ is an isomorphism for all $i$ and we win. \end{proof} \section{Colimits and higher direct images} \label{section-colimits} \noindent General results of this nature can be found in Cohomology, Section \ref{cohomology-section-limits}, Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections}, and Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-compact-colimit}. \begin{lemma} \label{lemma-colimit-cohomology} Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F} = \colim \mathcal{F}_i$ be a filtered colimit of quasi-coherent sheaves on $X$. 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 associated to $U \mapsto H^p(f^{-1}U, \mathcal{F})$, see Cohomology, Lemma \ref{cohomology-lemma-describe-higher-direct-images}. Recall that the colimit is the sheaf associated to the presheaf colimit (taking colimits over opens). Hence we can apply Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit} to $H^p(f^{-1}U, -)$ where $U$ is affine to conclude. (Because the basis of affine opens in $f^{-1}U$ satisfies the assumptions of that lemma.) \end{proof} \section{Cohomology and base change, II} \label{section-cohomology-and-base-change-derived} \noindent Let $f : X \to S$ be a morphism of schemes and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $f$ is quasi-compact and quasi-separated we would like to represent $Rf_*\mathcal{F}$ by a complex of quasi-coherent sheaves on $S$. This follows from the fact that the sheaves $R^if_*\mathcal{F}$ are quasi-coherent if $S$ is quasi-compact and has affine diagonal, using that $D_\QCoh(S)$ is equivalent to $D(\QCoh(\mathcal{O}_S))$, see Derived Categories of Schemes, Proposition \ref{perfect-proposition-quasi-compact-affine-diagonal}. \medskip\noindent In this section we will use a different approach which produces an explicit complex having a good base change property. The construction is particularly easy if $f$ and $S$ are separated, or more generally have affine diagonal. Since this is the case which by far the most often used we treat it separately. \begin{lemma} \label{lemma-separated-case-relative-cech} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $X$ is quasi-compact and $X$ and $S$ have affine diagonal (e.g., if $X$ and $S$ are separated). In this case we can compute $Rf_*\mathcal{F}$ as follows: \begin{enumerate} \item Choose a finite affine open covering $\mathcal{U} : X = \bigcup_{i = 1, \ldots, n} U_i$. \item For $i_0, \ldots, i_p \in \{1, \ldots, n\}$ denote $f_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to S$ the restriction of $f$ to the intersection $U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$. \item Set $\mathcal{F}_{i_0 \ldots i_p}$ equal to the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_p}$. \item Set $$\check{\mathcal{C}}^p(\mathcal{U}, f, \mathcal{F}) = \bigoplus\nolimits_{i_0 \ldots i_p} f_{i_0 \ldots i_p *} \mathcal{F}_{i_0 \ldots i_p}$$ and define differentials $d : \check{\mathcal{C}}^p(\mathcal{U}, f, \mathcal{F}) \to \check{\mathcal{C}}^{p + 1}(\mathcal{U}, f, \mathcal{F})$ as in Cohomology, Equation (\ref{cohomology-equation-d-cech}). \end{enumerate} Then the complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})$ is a complex of quasi-coherent sheaves on $S$ which comes equipped with an isomorphism $$\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf_*\mathcal{F}$$ in $D^{+}(S)$. This isomorphism is functorial in the quasi-coherent sheaf $\mathcal{F}$. \end{lemma} \begin{proof} Consider the resolution $\mathcal{F} \to {\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})$ of Cohomology, Lemma \ref{cohomology-lemma-covering-resolution}. We have an equality of complexes $\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) = f_*{\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})$ of quasi-coherent $\mathcal{O}_S$-modules. The morphisms $j_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to X$ and the morphisms $f_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to S$ are affine by Morphisms, Lemma \ref{morphisms-lemma-affine-permanence} and Lemma \ref{lemma-affine-diagonal}. Hence $R^qj_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}$ as well as $R^qf_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}$ are zero for $q > 0$ (Lemma \ref{lemma-relative-affine-vanishing}). Using $f \circ j_{i_0 \ldots i_p} = f_{i_0 \ldots i_p}$ and the spectral sequence of Cohomology, Lemma \ref{cohomology-lemma-relative-Leray} we conclude that $R^qf_*(j_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}) = 0$ for $q > 0$. Since the terms of the complex ${\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})$ are finite direct sums of the sheaves $j_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}$ we conclude using Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) that $$Rf_* \mathcal{F} = f_*{\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})$$ as desired. \end{proof} \noindent Next, we are going to consider what happens if we do a base change. \begin{lemma} \label{lemma-base-change-complex} With notation as in diagram (\ref{equation-base-change-diagram}). Assume $f : X \to S$ and $\mathcal{F}$ satisfy the hypotheses of Lemma \ref{lemma-separated-case-relative-cech}. Choose a finite affine open covering $\mathcal{U} : X = \bigcup U_i$ of $X$. There is a canonical isomorphism $$g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf'_*\mathcal{F}'$$ in $D^{+}(S')$. Moreover, if $S' \to S$ is affine, then in fact $$g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}')$$ with $\mathcal{U}' : X' = \bigcup U_i'$ where $U_i' = (g')^{-1}(U_i) = U_{i, S'}$ is also affine. \end{lemma} \begin{proof} In fact we may define $U_i' = (g')^{-1}(U_i) = U_{i, S'}$ no matter whether $S'$ is affine over $S$ or not. Let $\mathcal{U}' : X' = \bigcup U_i'$ be the induced covering of $X'$. In this case we claim that $$g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}')$$ with $\check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}')$ defined in exactly the same manner as in Lemma \ref{lemma-separated-case-relative-cech}. This is clear from the case of affine morphisms (Lemma \ref{lemma-affine-base-change}) by working locally on $S'$. Moreover, exactly as in the proof of Lemma \ref{lemma-separated-case-relative-cech} one sees that there is an isomorphism $$\check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}') \longrightarrow Rf'_*\mathcal{F}'$$ in $D^{+}(S')$ since the morphisms $U_i' \to X'$ and $U_i' \to S'$ are still affine (being base changes of affine morphisms). Details omitted. \end{proof} \noindent The lemma above says that the complex $$\mathcal{K}^\bullet = \check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})$$ is a bounded below complex of quasi-coherent sheaves on $S$ which {\it universally} computes the higher direct images of $f : X \to S$. This is something about this particular complex and it is not preserved by replacing $\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})$ by a quasi-isomorphic complex in general! In other words, this is not a statement that makes sense in the derived category. The reason is that the pullback $g^*\mathcal{K}^\bullet$ is {\it not} equal to the derived pullback $Lg^*\mathcal{K}^\bullet$ of $\mathcal{K}^\bullet$ in general! \medskip\noindent Here is a more general case where we can prove this statement. We remark that the condition of $S$ being separated is harmless in most applications, since this is usually used to prove some local property of the total derived image. The proof is significantly more involved and uses hypercoverings; it is a nice example of how you can use them sometimes. \begin{lemma} \label{lemma-hypercoverings} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume that $f$ is quasi-compact and quasi-separated and that $S$ is quasi-compact and separated. There exists a bounded below complex $\mathcal{K}^\bullet$ of quasi-coherent $\mathcal{O}_S$-modules with the following property: For every morphism $g : S' \to S$ the complex $g^*\mathcal{K}^\bullet$ is a representative for $Rf'_*\mathcal{F}'$ with notation as in diagram (\ref{equation-base-change-diagram}). \end{lemma} \begin{proof} (If $f$ is separated as well, please see Lemma \ref{lemma-base-change-complex}.) The assumptions imply in particular that $X$ is quasi-compact and quasi-separated as a scheme. Let $\mathcal{B}$ be the set of affine opens of $X$. By Hypercoverings, Lemma \ref{hypercovering-lemma-quasi-separated-quasi-compact-hypercovering} we can find a hypercovering $K = (I, \{U_i\})$ such that each $I_n$ is finite and each $U_i$ is an affine open of $X$. By Hypercoverings, Lemma \ref{hypercovering-lemma-cech-spectral-sequence} there is a spectral sequence with $E_2$-page $$E_2^{p, q} = \check{H}^p(K, \underline{H}^q(\mathcal{F}))$$ converging to $H^{p + q}(X, \mathcal{F})$. Note that $\check{H}^p(K, \underline{H}^q(\mathcal{F}))$ is the $p$th cohomology group of the complex $$\prod\nolimits_{i \in I_0} H^q(U_i, \mathcal{F}) \to \prod\nolimits_{i \in I_1} H^q(U_i, \mathcal{F}) \to \prod\nolimits_{i \in I_2} H^q(U_i, \mathcal{F}) \to \ldots$$ Since each $U_i$ is affine we see that this is zero unless $q = 0$ in which case we obtain $$\prod\nolimits_{i \in I_0} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_1} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_2} \mathcal{F}(U_i) \to \ldots$$ Thus we conclude that $R\Gamma(X, \mathcal{F})$ is computed by this complex. \medskip\noindent For any $n$ and $i \in I_n$ denote $f_i : U_i \to S$ the restriction of $f$ to $U_i$. As $S$ is separated and $U_i$ is affine this morphism is affine. Consider the complex of quasi-coherent sheaves $$\mathcal{K}^\bullet = ( \prod\nolimits_{i \in I_0} f_{i, *}\mathcal{F}|_{U_i} \to \prod\nolimits_{i \in I_1} f_{i, *}\mathcal{F}|_{U_i} \to \prod\nolimits_{i \in I_2} f_{i, *}\mathcal{F}|_{U_i} \to \ldots )$$ on $S$. As in Hypercoverings, Lemma \ref{hypercovering-lemma-cech-spectral-sequence} we obtain a map $\mathcal{K}^\bullet \to Rf_*\mathcal{F}$ in $D(\mathcal{O}_S)$ by choosing an injective resolution of $\mathcal{F}$ (details omitted). Consider any affine scheme $V$ and a morphism $g : V \to S$. Then the base change $X_V$ has a hypercovering $K_V = (I, \{U_{i, V}\})$ obtained by base change. Moreover, $g^*f_{i, *}\mathcal{F} = f_{i, V, *}(g')^*\mathcal{F}|_{U_{i, V}}$. Thus the arguments above prove that $\Gamma(V, g^*\mathcal{K}^\bullet)$ computes $R\Gamma(X_V, (g')^*\mathcal{F})$. This finishes the proof of the lemma as it suffices to prove the equality of complexes Zariski locally on $S'$. \end{proof} \section{Cohomology of projective space} \label{section-cohomology-projective-space} \noindent In this section we compute the cohomology of the twists of the structure sheaf on $\mathbf{P}^n_S$ over a scheme $S$. Recall that $\mathbf{P}^n_S$ was defined as the fibre product $\mathbf{P}^n_S = S \times_{\Spec(\mathbf{Z})} \mathbf{P}^n_{\mathbf{Z}}$ in Constructions, Definition \ref{constructions-definition-projective-space}. It was shown to be equal to $$\mathbf{P}^n_S = \underline{\text{Proj}}_S(\mathcal{O}_S[T_0, \ldots, T_n])$$ in Constructions, Lemma \ref{constructions-lemma-projective-space-bundle}. In particular, projective space is a particular case of a projective bundle. If $S = \Spec(R)$ is affine then we have $$\mathbf{P}^n_S = \mathbf{P}^n_R = \text{Proj}(R[T_0, \ldots, T_n]).$$ All these identifications are compatible and compatible with the constructions of the twisted structure sheaves $\mathcal{O}_{\mathbf{P}^n_S}(d)$. \medskip\noindent Before we state the result we need some notation. Let $R$ be a ring. Recall that $R[T_0, \ldots, T_n]$ is a graded $R$-algebra where each $T_i$ is homogeneous of degree $1$. Denote $(R[T_0, \ldots, T_n])_d$ the degree $d$ summand. It is a finite free $R$-module of rank $\binom{n + d}{d}$ when $d \geq 0$ and zero else. It has a basis consisting of monomials $T_0^{e_0} \ldots T_n^{e_n}$ with $\sum e_i = d$. We will also use the following notation: $R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]$ denotes the $\mathbf{Z}$-graded ring with $\frac{1}{T_i}$ in degree $-1$. In particular the $\mathbf{Z}$-graded $R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]$ module $$\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]$$ which shows up in the statement below is zero in degrees $\geq -n$, is free on the generator $\frac{1}{T_0 \ldots T_n}$ in degree $-n - 1$ and is free of rank $(-1)^n\binom{n + d}{d}$ for $d \leq -n - 1$. \begin{lemma} \label{lemma-cohomology-projective-space-over-ring} \begin{reference} \cite[III Proposition 2.1.12]{EGA} \end{reference} Let $R$ be a ring. Let $n \geq 0$ be an integer. We have $$H^q(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n_R}(d)) = \left\{ \begin{matrix} (R[T_0, \ldots, T_n])_d & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \left(\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\right)_d & \text{if} & q = n \end{matrix} \right.$$ as $R$-modules. \end{lemma} \begin{proof} We will use the standard affine open covering $$\mathcal{U} : \mathbf{P}^n_R = \bigcup\nolimits_{i = 0}^n D_{+}(T_i)$$ to compute the cohomology using the {\v C}ech complex. This is permissible by Lemma \ref{lemma-cech-cohomology-quasi-coherent} since any intersection of finitely many affine $D_{+}(T_i)$ is also a standard affine open (see Constructions, Section \ref{constructions-section-proj}). In fact, we can use the alternating or ordered {\v C}ech complex according to Cohomology, Lemmas \ref{cohomology-lemma-ordered-alternating} and \ref{cohomology-lemma-alternating-usual}. \medskip\noindent The ordering we will use on $\{0, \ldots, n\}$ is the usual one. Hence the complex we are looking at has terms $$\check{\mathcal{C}}_{ord}^p(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) = \bigoplus\nolimits_{i_0 < \ldots < i_p} (R[T_0, \ldots, T_n, \frac{1}{T_{i_0} \ldots T_{i_p}}])_d$$ Moreover, the maps are given by the usual formula $$d(s)_{i_0 \ldots i_{p + 1}} = \sum\nolimits_{j = 0}^{p + 1} (-1)^j s_{i_0 \ldots \hat i_j \ldots i_{p + 1}}$$ see Cohomology, Section \ref{cohomology-section-alternating-cech}. Note that each term of this complex has a natural $\mathbf{Z}^{n + 1}$-grading. Namely, we get this by declaring a monomial $T_0^{e_0} \ldots T_n^{e_n}$ to be homogeneous with weight $(e_0, \ldots, e_n) \in \mathbf{Z}^{n + 1}$. It is clear that the differential given above respects the grading. In a formula we have $$\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) = \bigoplus\nolimits_{\vec{e} \in \mathbf{Z}^{n + 1}} \check{\mathcal{C}}^\bullet(\vec{e})$$ where not all summands on the right hand side occur (see below). Hence in order to compute the cohomology modules of the complex it suffices to compute the cohomology of the graded pieces and take the direct sum at the end. \medskip\noindent Fix $\vec{e} = (e_0, \ldots, e_n) \in \mathbf{Z}^{n + 1}$. In order for this weight to occur in the complex above we need to assume $e_0 + \ldots + e_n = d$ (if not then it occurs for a different twist of the structure sheaf of course). Assuming this, set $$NEG(\vec{e}) = \{i \in \{0, \ldots, n\} \mid e_i < 0\}.$$ With this notation the weight $\vec{e}$ summand $\check{\mathcal{C}}^\bullet(\vec{e})$ of the {\v C}ech complex above has the following terms $$\check{\mathcal{C}}^p(\vec{e}) = \bigoplus\nolimits_{i_0 < \ldots < i_p, \ NEG(\vec{e}) \subset \{i_0, \ldots, i_p\}} R \cdot T_0^{e_0} \ldots T_n^{e_n}$$ In other words, the terms corresponding to $i_0 < \ldots < i_p$ such that $NEG(\vec{e})$ is not contained in $\{i_0 \ldots i_p\}$ are zero. The differential of the complex $\check{\mathcal{C}}^\bullet(\vec{e})$ is still given by the exact same formula as above. \medskip\noindent Suppose that $NEG(\vec{e}) = \{0, \ldots, n\}$, i.e., that all exponents $e_i$ are negative. In this case the complex $\check{\mathcal{C}}^\bullet(\vec{e})$ has only one term, namely $\check{\mathcal{C}}^n(\vec{e}) = R \cdot \frac{1}{T_0^{-e_0} \ldots T_n^{-e_n}}$. Hence in this case $$H^q(\check{\mathcal{C}}^\bullet(\vec{e})) = \left\{ \begin{matrix} R \cdot \frac{1}{T_0^{-e_0} \ldots T_n^{-e_n}} & \text{if} & q = n \\ 0 & \text{if} & \text{else} \end{matrix} \right.$$ The direct sum of all of these terms clearly gives the value $$\left(\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\right)_d$$ in degree $n$ as stated in the lemma. Moreover these terms do not contribute to cohomology in other degrees (also in accordance with the statement of the lemma). \medskip\noindent Assume $NEG(\vec{e}) = \emptyset$. In this case the complex $\check{\mathcal{C}}^\bullet(\vec{e})$ has a summand $R$ corresponding to all $i_0 < \ldots < i_p$. Let us compare the complex $\check{\mathcal{C}}^\bullet(\vec{e})$ to another complex. Namely, consider the affine open covering $$\mathcal{V} : \Spec(R) = \bigcup\nolimits_{i \in \{0, \ldots, n\}} V_i$$ where $V_i = \Spec(R)$ for all $i$. Consider the alternating {\v C}ech complex $$\check{\mathcal{C}}_{ord}^\bullet(\mathcal{V}, \mathcal{O}_{\Spec(R)})$$ By the same reasoning as above this computes the cohomology of the structure sheaf on $\Spec(R)$. Hence we see that $H^p( \check{\mathcal{C}}_{ord}^\bullet(\mathcal{V}, \mathcal{O}_{\Spec(R)}) ) = R$ if $p = 0$ and is $0$ whenever $p > 0$. For these facts, see Lemma \ref{lemma-cech-cohomology-quasi-coherent-trivial} and its proof. Note that also $\check{\mathcal{C}}_{ord}^\bullet(\mathcal{V}, \mathcal{O}_{\Spec(R)})$ has a summand $R$ for every $i_0 < \ldots < i_p$ and has exactly the same differential as $\check{\mathcal{C}}^\bullet(\vec{e})$. In other words these complexes are isomorphic complexes and hence have the same cohomology. We conclude that $$H^q(\check{\mathcal{C}}^\bullet(\vec{e})) = \left\{ \begin{matrix} R \cdot T_0^{e_0} \ldots T_n^{e_n} & \text{if} & q = 0 \\ 0 & \text{if} & \text{else} \end{matrix} \right.$$ in the case that $NEG(\vec{e}) = \emptyset$. The direct sum of all of these terms clearly gives the value $$(R[T_0, \ldots, T_n])_d$$ in degree $0$ as stated in the lemma. Moreover these terms do not contribute to cohomology in other degrees (also in accordance with the statement of the lemma). \medskip\noindent To finish the proof of the lemma we have to show that the complexes $\check{\mathcal{C}}^\bullet(\vec{e})$ are acyclic when $NEG(\vec{e})$ is neither empty nor equal to $\{0, \ldots, n\}$. Pick an index $i_{\text{fix}} \not \in NEG(\vec{e})$ (such an index exists). Consider the map $$h : \check{\mathcal{C}}^{p + 1}(\vec{e}) \to \check{\mathcal{C}}^p(\vec{e})$$ given by the rule $$h(s)_{i_0 \ldots i_p} = s_{i_{\text{fix}} i_0 \ldots i_p}$$ (compare with the proof of Lemma \ref{lemma-cech-cohomology-quasi-coherent-trivial}). It is clear that this is well defined since $$NEG(\vec{e}) \subset \{i_0, \ldots, i_p\} \Leftrightarrow NEG(\vec{e}) \subset \{i_{\text{fix}}, i_0, \ldots, i_p\}$$ Also $\check{\mathcal{C}}^0(\vec{e}) = 0$ so that this formula does work for all $p$ including $p = - 1$. The exact same (combinatorial) computation as in the proof of Lemma \ref{lemma-cech-cohomology-quasi-coherent-trivial} shows that $$(hd + dh)(s)_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}$$ Hence we see that the identity map of the complex $\check{\mathcal{C}}^\bullet(\vec{e})$ is homotopic to zero which implies that it is acyclic. \end{proof} \noindent In the following lemma we are going to use the pairing of free $R$-modules $$R[T_0, \ldots, T_n] \times \frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}] \longrightarrow R$$ which is defined by the rule $$(f, g) \longmapsto \text{coefficient of } \frac{1}{T_0 \ldots T_n} \text{ in }fg.$$ In other words, the basis element $T_0^{e_0} \ldots T_n^{e_n}$ pairs with the basis element $T_0^{d_0} \ldots T_n^{d_n}$ to give $1$ if and only if $e_i + d_i = -1$ for all $i$, and pairs to zero in all other cases. Using this pairing we get an identification $$\left(\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\right)_d = \Hom_R((R[T_0, \ldots, T_n])_{-n - 1 - d}, R)$$ Thus we can reformulate the result of Lemma \ref{lemma-cohomology-projective-space-over-ring} as saying that \begin{equation} \label{equation-identify} H^q(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n_R}(d)) = \left\{ \begin{matrix} (R[T_0, \ldots, T_n])_d & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \Hom_R((R[T_0, \ldots, T_n])_{-n - 1 - d}, R) & \text{if} & q = n \end{matrix} \right. \end{equation} \begin{lemma} \label{lemma-identify-functorially} The identifications of Equation (\ref{equation-identify}) are compatible with base change w.r.t.\ ring maps $R \to R'$. Moreover, for any $f \in R[T_0, \ldots, T_n]$ homogeneous of degree $m$ the map multiplication by $f$ $$\mathcal{O}_{\mathbf{P}^n_R}(d) \longrightarrow \mathcal{O}_{\mathbf{P}^n_R}(d + m)$$ induces the map on the cohomology group via the identifications of Equation (\ref{equation-identify}) which is multiplication by $f$ for $H^0$ and the contragredient of multiplication by $f$ $$(R[T_0, \ldots, T_n])_{-n - 1 - (d + m)} \longrightarrow (R[T_0, \ldots, T_n])_{-n - 1 - d}$$ on $H^n$. \end{lemma} \begin{proof} Suppose that $R \to R'$ is a ring map. Let $\mathcal{U}$ be the standard affine open covering of $\mathbf{P}^n_R$, and let $\mathcal{U}'$ be the standard affine open covering of $\mathbf{P}^n_{R'}$. Note that $\mathcal{U}'$ is the pullback of the covering $\mathcal{U}$ under the canonical morphism $\mathbf{P}^n_{R'} \to \mathbf{P}^n_R$. Hence there is a map of {\v C}ech complexes $$\gamma : \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) \longrightarrow \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}', \mathcal{O}_{\mathbf{P}_{R'}}(d))$$ which is compatible with the map on cohomology by Cohomology, Lemma \ref{cohomology-lemma-functoriality-cech}. It is clear from the computations in the proof of Lemma \ref{lemma-cohomology-projective-space-over-ring} that this map of {\v C}ech complexes is compatible with the identifications of the cohomology groups in question. (Namely the basis elements for the {\v C}ech complex over $R$ simply map to the corresponding basis elements for the {\v C}ech complex over $R'$.) Whence the first statement of the lemma. \medskip\noindent Now fix the ring $R$ and consider two homogeneous polynomials $f, g \in R[T_0, \ldots, T_n]$ both of the same degree $m$. Since cohomology is an additive functor, it is clear that the map induced by multiplication by $f + g$ is the same as the sum of the maps induced by multiplication by $f$ and the map induced by multiplication by $g$. Moreover, since cohomology is a functor, a similar result holds for multiplication by a product $fg$ where $f, g$ are both homogeneous (but not necessarily of the same degree). Hence to verify the second statement of the lemma it suffices to prove this when $f = x \in R$ or when $f = T_i$. In the case of multiplication by an element $x \in R$ the result follows since every cohomology groups or complex in sight has the structure of an $R$-module or complex of $R$-modules. Finally, we consider the case of multiplication by $T_i$ as a $\mathcal{O}_{\mathbf{P}^n_R}$-linear map $$\mathcal{O}_{\mathbf{P}^n_R}(d) \longrightarrow \mathcal{O}_{\mathbf{P}^n_R}(d + 1)$$ The statement on $H^0$ is clear. For the statement on $H^n$ consider multiplication by $T_i$ as a map on {\v C}ech complexes $$\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) \longrightarrow \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d + 1))$$ We are going to use the notation introduced in the proof of Lemma \ref{lemma-cohomology-projective-space-over-ring}. We consider the effect of multiplication by $T_i$ in terms of the decompositions $$\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) = \bigoplus\nolimits_{\vec{e} \in \mathbf{Z}^{n + 1}, \ \sum e_i = d} \check{\mathcal{C}}^\bullet(\vec{e})$$ and $$\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d + 1)) = \bigoplus\nolimits_{\vec{e} \in \mathbf{Z}^{n + 1}, \ \sum e_i = d + 1} \check{\mathcal{C}}^\bullet(\vec{e})$$ It is clear that it maps the subcomplex $\check{\mathcal{C}}^\bullet(\vec{e})$ to the subcomplex $\check{\mathcal{C}}^\bullet(\vec{e} + \vec{b}_i)$ where $\vec{b}_i = (0, \ldots, 0, 1, 0, \ldots, 0))$ the $i$th basis vector. In other words, it maps the summand of $H^n$ corresponding to $\vec{e}$ with $e_i < 0$ and $\sum e_i = d$ to the summand of $H^n$ corresponding to $\vec{e} + \vec{b}_i$ (which is zero if $e_i + b_i \geq 0$). It is easy to see that this corresponds exactly to the action of the contragredient of multiplication by $T_i$ as a map $$(R[T_0, \ldots, T_n])_{-n - 1 - (d + 1)} \longrightarrow (R[T_0, \ldots, T_n])_{-n - 1 - d}$$ This proves the lemma. \end{proof} \noindent Before we state the relative version we need some notation. Namely, recall that $\mathcal{O}_S[T_0, \ldots, T_n]$ is a graded $\mathcal{O}_S$-module where each $T_i$ is homogeneous of degree $1$. Denote $(\mathcal{O}_S[T_0, \ldots, T_n])_d$ the degree $d$ summand. It is a finite locally free sheaf of rank $\binom{n + d}{d}$ on $S$. \begin{lemma} \label{lemma-cohomology-projective-space-over-base} Let $S$ be a scheme. Let $n \geq 0$ be an integer. Consider the structure morphism $$f : \mathbf{P}^n_S \longrightarrow S.$$ We have $$R^qf_*(\mathcal{O}_{\mathbf{P}^n_S}(d)) = \left\{ \begin{matrix} (\mathcal{O}_S[T_0, \ldots, T_n])_d & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \SheafHom_{\mathcal{O}_S}( (\mathcal{O}_S[T_0, \ldots, T_n])_{- n - 1 - d}, \mathcal{O}_S) & \text{if} & q = n \end{matrix} \right.$$ \end{lemma} \begin{proof} Omitted. Hint: This follows since the identifications in (\ref{equation-identify}) are compatible with affine base change by Lemma \ref{lemma-identify-functorially}. \end{proof} \noindent Next we state the version for projective bundles associated to finite locally free sheaves. Let $S$ be a scheme. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_S$-module of constant rank $n + 1$, see Modules, Section \ref{modules-section-locally-free}. In this case we think of $\text{Sym}(\mathcal{E})$ as a graded $\mathcal{O}_S$-module where $\mathcal{E}$ is the graded part of degree $1$. And $\text{Sym}^d(\mathcal{E})$ is the degree $d$ summand. It is a finite locally free sheaf of rank $\binom{n + d}{d}$ on $S$. Recall that our normalization is that $$\pi : \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_S(\text{Sym}(\mathcal{E})) \longrightarrow S$$ and that there are natural maps $\text{Sym}^d(\mathcal{E}) \to \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)$. \begin{lemma} \label{lemma-cohomology-projective-bundle} Let $S$ be a scheme. Let $n \geq 1$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_S$-module of constant rank $n + 1$. Consider the structure morphism $$\pi : \mathbf{P}(\mathcal{E}) \longrightarrow S.$$ We have $$R^q\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) = \left\{ \begin{matrix} \text{Sym}^d(\mathcal{E}) & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \SheafHom_{\mathcal{O}_S}( \text{Sym}^{- n - 1 - d}(\mathcal{E}) \otimes_{\mathcal{O}_S} \wedge^{n + 1}\mathcal{E}, \mathcal{O}_S) & \text{if} & q = n \end{matrix} \right.$$ These identifications are compatible with base change and isomorphism between locally free sheaves. \end{lemma} \begin{proof} Consider the canonical map $$\pi^*\mathcal{E} \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$$ and twist down by $1$ to get $$\pi^*(\mathcal{E})(-1) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}$$ This is a surjective map from a locally free rank $n + 1$ sheaf onto the structure sheaf. Hence the corresponding Koszul complex is exact (More on Algebra, Lemma \ref{more-algebra-lemma-homotopy-koszul-abstract}). In other words there is an exact complex $$0 \to \pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1) \to \ldots \to \pi^*(\wedge^i\mathcal{E})(-i) \to \ldots \to \pi^*\mathcal{E}(-1) \to \mathcal{O}_{\mathbf{P}(\mathcal{E})} \to 0$$ We will think of the term $\pi^*(\wedge^i\mathcal{E})(-i)$ as being in degree $-i$. We are going to compute the higher direct images of this acyclic complex using the first spectral sequence of Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Namely, we see that there is a spectral sequence with terms $$E_1^{p, q} = R^q\pi_*\left(\pi^*(\wedge^{-p}\mathcal{E})(p)\right)$$ converging to zero! By the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) we have $$E_1^{p, q} = \wedge^{-p} \mathcal{E} \otimes_{\mathcal{O}_S} R^q\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(p)\right).$$ Note that locally on $S$ the sheaf $\mathcal{E}$ is trivial, i.e., isomorphic to $\mathcal{O}_S^{\oplus n + 1}$, hence locally on $S$ the morphism $\mathbf{P}(\mathcal{E}) \to S$ can be identified with $\mathbf{P}^n_S \to S$. Hence locally on $S$ we can use the result of Lemmas \ref{lemma-cohomology-projective-space-over-ring}, \ref{lemma-identify-functorially}, or \ref{lemma-cohomology-projective-space-over-base}. It follows that $E_1^{p, q} = 0$ unless $(p, q)$ is $(0, 0)$ or $(-n - 1, n)$. The nonzero terms are \begin{align*} E_1^{0, 0} & = \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})} = \mathcal{O}_S \\ E_1^{-n - 1, n} & = R^n\pi_*\left(\pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1)\right) = \wedge^{n + 1}\mathcal{E} \otimes_{\mathcal{O}_S} R^n\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) \end{align*} Hence there can only be one nonzero differential in the spectral sequence namely the map $d_{n + 1}^{-n - 1, n} : E_{n + 1}^{-n - 1, n} \to E_{n + 1}^{0, 0}$ which has to be an isomorphism (because the spectral sequence converges to the $0$ sheaf). Thus $E_1^{p, q} = E_{n + 1}^{p, q}$ and we obtain a canonical isomorphism $$\wedge^{n + 1}\mathcal{E} \otimes_{\mathcal{O}_S} R^n\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) = R^n\pi_*\left(\pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1)\right) \xrightarrow{d_{n + 1}^{-n - 1, n}} \mathcal{O}_S$$ Since $\wedge^{n + 1}\mathcal{E}$ is an invertible sheaf, this implies that $R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)$ is invertible as well and canonically isomorphic to the inverse of $\wedge^{n + 1}\mathcal{E}$. In other words we have proved the case $d = - n - 1$ of the lemma. \medskip\noindent Working locally on $S$ we see immediately from the computation of cohomology in Lemmas \ref{lemma-cohomology-projective-space-over-ring}, \ref{lemma-identify-functorially}, or \ref{lemma-cohomology-projective-space-over-base} the statements on vanishing of the lemma. Moreover the result on $R^0\pi_*$ is clear as well, since there are canonical maps $\text{Sym}^d(\mathcal{E}) \to \pi_* \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)$ for all $d$. It remains to show that the description of $R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)$ is correct for $d < -n - 1$. In order to do this we consider the map $$\pi^*(\text{Sym}^{-d - n - 1}(\mathcal{E})) \otimes_{\mathcal{O}_{\mathbf{P}(\mathcal{E})}} \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)$$ Applying $R^n\pi_*$ and the projection formula (see above) we get a map $$\text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes_{\mathcal{O}_S} R^n\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) \longrightarrow R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1) = (\wedge^{n + 1}\mathcal{E})^{\otimes -1}$$ (the last equality we have shown above). Again by the local calculations of Lemmas \ref{lemma-cohomology-projective-space-over-ring}, \ref{lemma-identify-functorially}, or \ref{lemma-cohomology-projective-space-over-base} it follows that this map induces a perfect pairing between $R^n\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d))$ and $\text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes \wedge^{n + 1}(\mathcal{E})$ as desired. \end{proof} \section{Coherent sheaves on locally Noetherian schemes} \label{section-coherent-sheaves} \noindent We have defined the notion of a coherent module on any ringed space in Modules, Section \ref{modules-section-coherent}. Although it is possible to consider coherent sheaves on non-Noetherian schemes we will always assume the base scheme is locally Noetherian when we consider coherent sheaves. Here is a characterization of coherent sheaves on locally Noetherian schemes. \begin{lemma} \label{lemma-coherent-Noetherian} Let $X$ be a locally Noetherian scheme. 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 affine open $\Spec(A) = U \subset X$ we have $\mathcal{F}|_U = \widetilde M$ with $M$ a finite $A$-module, and \item there exists an affine open covering $X = \bigcup U_i$, $U_i = \Spec(A_i)$ such that each $\mathcal{F}|_{U_i} = \widetilde M_i$ with $M_i$ a finite $A_i$-module. \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} The implications (1) $\Rightarrow$ (2) and (1) $\Rightarrow$ (3) hold in general, see Modules, Lemma \ref{modules-lemma-coherent-finite-presentation}. If $\mathcal{F}$ is finitely presented then $\mathcal{F}$ is quasi-coherent, see Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-coherent}. Hence also (3) $\Rightarrow$ (2). \medskip\noindent Assume $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_X$-module. By Properties, Lemma \ref{properties-lemma-finite-type-module} we see that on any affine open $\Spec(A) = U \subset X$ we have $\mathcal{F}|_U = \widetilde M$ with $M$ a finite $A$-module. Since $A$ is Noetherian we see that $M$ has a finite resolution $$A^{\oplus m} \to A^{\oplus n} \to M \to 0.$$ Hence $\mathcal{F}$ is of finite presentation by Properties, Lemma \ref{properties-lemma-finite-presentation-module}. In other words (2) $\Rightarrow$ (3). \medskip\noindent By Modules, Lemma \ref{modules-lemma-coherent-structure-sheaf} it suffices to show that $\mathcal{O}_X$ is coherent in order to show that (3) implies (1). Thus we have to show: given any open $U \subset X$ and any finite collection of sections $f_i \in \mathcal{O}_X(U)$, $i = 1, \ldots, n$ the kernel of the map $\bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{O}_U$ is of finite type. Since being of finite type is a local property it suffices to check this in a neighbourhood of any $x \in U$. Thus we may assume $U = \Spec(A)$ is affine. In this case $f_1, \ldots, f_n \in A$ are elements of $A$. Since $A$ is Noetherian, see Properties, Lemma \ref{properties-lemma-locally-Noetherian} the kernel $K$ of the map $\bigoplus_{i = 1, \ldots, n} A \to A$ is a finite $A$-module. See for example Algebra, Lemma \ref{algebra-lemma-Noetherian-basic}. As the functor\ $\widetilde{ }$\ is exact, see Schemes, Lemma \ref{schemes-lemma-spec-sheaves} we get an exact sequence $$\widetilde K \to \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{O}_U$$ and by Properties, Lemma \ref{properties-lemma-finite-type-module} again we see that $\widetilde K$ is of finite type. We conclude that (1), (2) and (3) are all equivalent. \medskip\noindent It follows from Properties, Lemma \ref{properties-lemma-finite-type-module} that (2) implies (4). It is trivial that (4) implies (5). The discussion in Schemes, Section \ref{schemes-section-quasi-coherent} show that (5) implies that $\mathcal{F}$ is quasi-coherent and it is clear that (5) implies that $\mathcal{F}$ is of finite type. Hence (5) implies (2) and we win. \end{proof} \begin{lemma} \label{lemma-coherent-abelian-Noetherian} Let $X$ be a locally Noetherian scheme. 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} This is a restatement of Modules, Lemma \ref{modules-lemma-coherent-abelian} in a particular case. \end{proof} \noindent The following lemma does not always hold for the category of coherent $\mathcal{O}_X$-modules on a general ringed space $X$. \begin{lemma} \label{lemma-coherent-Noetherian-quasi-coherent-sub-quotient} Let $X$ be a locally Noetherian scheme. 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} We may assume that $X$ is affine, say $X = \Spec(A)$. Properties, Lemma \ref{properties-lemma-locally-Noetherian} implies that $A$ is Noetherian. Lemma \ref{lemma-coherent-Noetherian} turns this into algebra. The algebraic counter part of the lemma is that a quotient, or a submodule of a finite $A$-module is a finite $A$-module, see for example Algebra, Lemma \ref{algebra-lemma-Noetherian-basic}. \end{proof} \begin{lemma} \label{lemma-tensor-hom-coherent} Let $X$ be a locally Noetherian scheme. 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} It is shown in Modules, Lemma \ref{modules-lemma-internal-hom-locally-kernel-direct-sum} that $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is coherent. The result for tensor products is Modules, Lemma \ref{modules-lemma-tensor-product-permanence} \end{proof} \begin{lemma} \label{lemma-local-isomorphism} Let $X$ be a locally Noetherian scheme. 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 $x \in X$. \begin{enumerate} \item If $\mathcal{F}_x = 0$ then there exists an open neighbourhood $U \subset X$ of $x$ such that $\mathcal{F}|_U = 0$. \item If $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ is injective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi|_U$ is injective. \item If $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ is surjective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi|_U$ is surjective. \item If $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ is bijective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi|_U$ is an isomorphism. \end{enumerate} \end{lemma} \begin{proof} See Modules, Lemmas \ref{modules-lemma-finite-type-surjective-on-stalk}, \ref{modules-lemma-finite-type-stalk-zero}, and \ref{modules-lemma-finite-type-to-coherent-injective-on-stalk}. \end{proof} \begin{lemma} \label{lemma-map-stalks-local-map} Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. Let $x \in X$. Suppose $\psi : \mathcal{G}_x \to \mathcal{F}_x$ is a map of $\mathcal{O}_{X, x}$-modules. Then there exists an open neighbourhood $U \subset X$ of $x$ and a map $\varphi : \mathcal{G}|_U \to \mathcal{F}|_U$ such that $\varphi_x = \psi$. \end{lemma} \begin{proof} In view of Lemma \ref{lemma-coherent-Noetherian} this is a reformulation of Modules, Lemma \ref{modules-lemma-stalk-internal-hom}. \end{proof} \begin{lemma} \label{lemma-coherent-support-closed} Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\text{Supp}(\mathcal{F})$ is closed, and $\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}. \end{lemma} \begin{proof} Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and let $\mathcal{G}$ be the finite type quasi-coherent sheaf on $Z$ such that $i_*\mathcal{G} \cong \mathcal{F}$. Since $Z = \text{Supp}(\mathcal{F})$ we see that the support is closed. The scheme $Z$ is locally Noetherian by Morphisms, Lemmas \ref{morphisms-lemma-immersion-locally-finite-type} and \ref{morphisms-lemma-finite-type-noetherian}. Finally, $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module by Lemma \ref{lemma-coherent-Noetherian} \end{proof} \begin{lemma} \label{lemma-i-star-equivalence} Let $i : Z \to X$ be a closed immersion of locally Noetherian schemes. 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, Lemma \ref{morphisms-lemma-i-star-equivalence}. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module annihilated by $\mathcal{I}$. By Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence} we can write $\mathcal{F} = i_*\mathcal{G}$ for some quasi-coherent sheaf $\mathcal{G}$ on $Z$. By Modules, Lemma \ref{modules-lemma-i-star-reflects-finite-type} we see that $\mathcal{G}$ is of finite type. Hence $\mathcal{G}$ is coherent by Lemma \ref{lemma-coherent-Noetherian}. Thus the functor is also essentially surjective as desired. \end{proof} \begin{lemma} \label{lemma-finite-pushforward-coherent} Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-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 if $\mathcal{F}$ is coherent. \end{lemma} \begin{proof} The higher direct images vanish by Lemma \ref{lemma-relative-affine-vanishing} and because a finite morphism is affine (by definition). Note that the assumptions imply that also $X$ is locally Noetherian (see Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}) and hence the statement makes sense. Let $\Spec(A) = V \subset Y$ be an affine open subset. By Morphisms, Definition \ref{morphisms-definition-integral} we see that $f^{-1}(V) = \Spec(B)$ with $A \to B$ finite. Lemma \ref{lemma-coherent-Noetherian} turns the statement of the lemma into the following algebra fact: If $M$ is a finite $B$-module, then $M$ is also finite viewed as a $A$-module, see Algebra, Lemma \ref{algebra-lemma-finite-module-over-finite-extension}. \end{proof} \noindent In the situation of the lemma also the higher direct images are coherent since they vanish. We will show that this is always the case for a proper morphism between locally Noetherian schemes (insert future reference here). \begin{lemma} \label{lemma-coherent-support-dimension-0} Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf with $\dim(\text{Supp}(\mathcal{F})) \leq 0$. Then $\mathcal{F}$ is generated by global sections and $H^i(X, \mathcal{F}) = 0$ for $i > 0$. \end{lemma} \begin{proof} By Lemma \ref{lemma-coherent-support-closed} we see that $\mathcal{F} = i_*\mathcal{G}$ where $i : Z \to X$ is the inclusion of the scheme theoretic support of $\mathcal{F}$ and where $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module. Since the dimension of $Z$ is $0$, we see $Z$ is a disjoint union of affines (Properties, Lemma \ref{properties-lemma-locally-Noetherian-dimension-0}). Hence $\mathcal{G}$ is globally generated and the higher cohomology groups of $\mathcal{G}$ are zero (Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero}). Hence $\mathcal{F} = i_*\mathcal{G}$ is globally generated. Since the cohomologies of $\mathcal{F}$ and $\mathcal{G}$ agree (Lemma \ref{lemma-relative-affine-cohomology} applies as a closed immersion is affine) we conclude that the higher cohomology groups of $\mathcal{F}$ are zero. \end{proof} \begin{lemma} \label{lemma-pushforward-coherent-on-open} Let $X$ be a scheme. Let $j : U \to X$ be the inclusion of an open. Let $T \subset X$ be a closed subset contained in $U$. If $\mathcal{F}$ is a coherent $\mathcal{O}_U$-module with $\text{Supp}(\mathcal{F}) \subset T$, then $j_*\mathcal{F}$ is a coherent $\mathcal{O}_X$-module. \end{lemma} \begin{proof} Consider the open covering $X = U \cup (X \setminus T)$. Then $j_*\mathcal{F}|_U = \mathcal{F}$ is coherent and $j_*\mathcal{F}|_{X \setminus T} = 0$ is also coherent. Hence $j_*\mathcal{F}$ is coherent. \end{proof} \section{Coherent sheaves on Noetherian schemes} \label{section-coherent-quasi-compact} \noindent In this section we mention some properties of coherent sheaves on Noetherian schemes. \begin{lemma} \label{lemma-acc-coherent} Let $X$ be a Noetherian scheme. 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 a finite affine open covering. On each member of the covering we get stabilization by Algebra, Lemma \ref{algebra-lemma-Noetherian-basic}. Hence the lemma follows. \end{proof} \begin{lemma} \label{lemma-power-ideal-kills-sheaf} Let $X$ be a Noetherian scheme. 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 subscheme $Z \subset X$. 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} This follows immediately from Algebra, Lemma \ref{algebra-lemma-Noetherian-power-ideal-kills-module} because $X$ has a finite covering by spectra of Noetherian rings. \end{proof} \begin{lemma}[Artin-Rees] \label{lemma-Artin-Rees} Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}$ be a quasi-coherent subsheaf. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Then there exists a $c \geq 0$ such that for all $n \geq c$ we have $$\mathcal{I}^{n - c}(\mathcal{I}^c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^n\mathcal{F} \cap \mathcal{G}$$ \end{lemma} \begin{proof} This follows immediately from Algebra, Lemma \ref{algebra-lemma-Artin-Rees} because $X$ has a finite covering by spectra of Noetherian rings. \end{proof} \begin{lemma} \label{lemma-homs-over-open} Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G}$ be 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 subscheme 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} We first prove the second map is an isomorphism. It is injective by Properties, Lemma \ref{properties-lemma-sections-over-quasi-compact-open}. Since $\mathcal{F}$ is the union of its coherent submodules, see Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} (and Lemma \ref{lemma-coherent-Noetherian}) we may and do assume that $\mathcal{F}$ is coherent to prove surjectivity. Let $\mathcal{F}_n$ denote the quasi-coherent subsheaf of $\mathcal{F}$ consisting of sections annihilated by $\mathcal{I}^n$, see Properties, Lemma \ref{properties-lemma-sections-over-quasi-compact-open}. Since $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots$ we see that $\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots$ for some $n \geq 0$ by Lemma \ref{lemma-acc-coherent}. Set $\mathcal{H} = \mathcal{F}_n$ for this $n$. By Artin-Rees (Lemma \ref{lemma-Artin-Rees}) there exists an $c \geq 0$ such that $\mathcal{I}^m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^{m - c}\mathcal{H}$. Picking $m = n + c$ we get $\mathcal{I}^m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^n\mathcal{H} = 0$. Thus if we set $\mathcal{F}' = \mathcal{I}^m\mathcal{F}$ then we see that $\mathcal{F}' \cap \mathcal{F}_n = 0$ and $\mathcal{F}'|_U = \mathcal{F}|_U$. Note in particular that the subsheaf $(\mathcal{F}')_N$ of sections annihilated by $\mathcal{I}^N$ is zero for all $N \geq 0$. Hence by Properties, Lemma \ref{properties-lemma-sections-over-quasi-compact-open} we deduce that the top horizontal arrow in the following commutative diagram is a bijection: $$\xymatrix{ \colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}') \ar[r] \ar[d] & \Gamma(U, \mathcal{F}') \ar[d] \\ \colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}) \ar[r] & \Gamma(U, \mathcal{F}) }$$ Since also the right vertical arrow is a bijection we conclude that the bottom horizontal arrow is surjective as desired. \medskip\noindent Next, we prove the first arrow of the lemma is a bijection. By Lemma \ref{lemma-coherent-Noetherian} the sheaf $\mathcal{G}$ is of finite presentation and hence the sheaf $\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})$ is quasi-coherent, see Schemes, Section \ref{schemes-section-quasi-coherent}. By definition we have $$\mathcal{H}(U) = \Hom_{\mathcal{O}_U}(\mathcal{G}|_U, \mathcal{F}|_U)$$ Pick a $\psi$ in the right hand side of the first arrow of the lemma, i.e., $\psi \in \mathcal{H}(U)$. The result just proved applies to $\mathcal{H}$ and hence there exists an $n \geq 0$ and an $\varphi : \mathcal{I}^n \to \mathcal{H}$ which recovers $\psi$ on restriction to $U$. By Modules, Lemma \ref{modules-lemma-internal-hom} $\varphi$ corresponds to a map $$\varphi : \mathcal{I}^{\otimes n} \otimes_{\mathcal{O}_X} \mathcal{G} \longrightarrow \mathcal{F}.$$ This is almost what we want except that the source of the arrow is the tensor product of $\mathcal{I}^n$ and $\mathcal{G}$ and not the product. We will show that, at the cost of increasing $n$, the difference is irrelevant. Consider the short exact sequence $$0 \to \mathcal{K} \to \mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{I}^n\mathcal{G} \to 0$$ where $\mathcal{K}$ is defined as the kernel. Note that $\mathcal{I}^n\mathcal{K} = 0$ (proof omitted). By Artin-Rees again we see that $$\mathcal{K} \cap \mathcal{I}^m(\mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G}) = 0$$ for some $m$ large enough. In other words we see that $$\mathcal{I}^m(\mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G}) \longrightarrow \mathcal{I}^{n + m}\mathcal{G}$$ is an isomorphism. Let $\varphi'$ be the restriction of $\varphi$ to this submodule thought of as a map $\mathcal{I}^{m + n}\mathcal{G} \to \mathcal{F}$. Then $\varphi'$ gives an element of the left hand side of the first arrow of the lemma which maps to $\psi$ via the arrow. In other words we have proved surjectivity of the arrow. We omit the proof of injectivity. \end{proof} \section{Depth} \label{section-depth} \noindent In this section we talk a little bit about depth and property $(S_k)$ for coherent modules on locally Noetherian schemes. Note that we have already discussed this notion for locally Noetherian schemes in Properties, Section \ref{properties-section-Rk}. \begin{definition} \label{definition-depth} Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $k \geq 0$ be an integer. \begin{enumerate} \item We say $\mathcal{F}$ has {\it depth $k$ at a point} $x$ of $X$ if $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) = k$. \item We say $X$ has {\it depth $k$ at a point} $x$ of $X$ if $\text{depth}(\mathcal{O}_{X, x}) = k$. \item We say $\mathcal{F}$ has property {\it $(S_k)$} if $$\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) \geq \min(k, \dim(\text{Supp}(\mathcal{F}_x)))$$ for all $x \in X$. \item We say $X$ has property {\it $(S_k)$} if $\mathcal{O}_X$ has property $(S_k)$. \end{enumerate} \end{definition} \noindent Any coherent sheaf satisfies condition $(S_0)$. Condition $(S_1)$ is equivalent to having no embedded associated points, see Divisors, Lemma \ref{divisors-lemma-S1-no-embedded}. \begin{lemma} \label{lemma-hom-into-depth} Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules and $x \in X$. \begin{enumerate} \item If $\mathcal{G}_x$ has depth $\geq 1$, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x$ has depth $\geq 1$. \item If $\mathcal{G}_x$ has depth $\geq 2$, then $\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x$ has depth $\geq 2$. \end{enumerate} \end{lemma} \begin{proof} Observe that $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is a coherent $\mathcal{O}_X$-module by Lemma \ref{lemma-tensor-hom-coherent}. Coherent modules are of finite presentation (Lemma \ref{lemma-coherent-Noetherian}) hence taking stalks commutes with taking $\SheafHom$ and $\Hom$, see Modules, Lemma \ref{modules-lemma-stalk-internal-hom}. Thus we reduce to the case of finite modules over local rings which is More on Algebra, Lemma \ref{more-algebra-lemma-hom-into-depth}. \end{proof} \begin{lemma} \label{lemma-hom-into-S2} Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{G}$ has property $(S_1)$, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ has property $(S_1)$. \item If $\mathcal{G}$ has property $(S_2)$, then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ has property $(S_2)$. \end{enumerate} \end{lemma} \begin{proof} Follows immediately from Lemma \ref{lemma-hom-into-depth} and the definitions. \end{proof} \noindent We have seen in Properties, Lemma \ref{properties-lemma-scheme-CM-iff-all-Sk} that a locally Noetherian scheme is Cohen-Macaulay if and only if $(S_k)$ holds for all $k$. Thus it makes sense to introduce the following definition, which is equivalent to the condition that all stalks are Cohen-Macaulay modules. \begin{definition} \label{definition-Cohen-Macaulay} Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. We say $\mathcal{F}$ is {\it Cohen-Macaulay} if and only if $(S_k)$ holds for all $k \geq 0$. \end{definition} \begin{lemma} \label{lemma-Cohen-Macaulay-over-regular} Let $X$ be a regular scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is Cohen-Macaulay and $\text{Supp}(\mathcal{F}) = X$, \item $\mathcal{F}$ is finite locally free of rank $>0$. \end{enumerate} \end{lemma} \begin{proof} Let $x \in X$. If (2) holds, then $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module of rank $> 0$. Hence $\text{depth}(\mathcal{F}_x) = \dim(\mathcal{O}_{X, x})$ because a regular local ring is Cohen-Macaulay (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}). Conversely, if (1) holds, then $\mathcal{F}_x$ is a maximal Cohen-Macaulay module over $\mathcal{O}_{X, x}$ (Algebra, Definition \ref{algebra-definition-maximal-CM}). Hence $\mathcal{F}_x$ is free by Algebra, Lemma \ref{algebra-lemma-regular-mcm-free}. \end{proof} \section{Devissage of coherent sheaves} \label{section-devissage} \noindent Let $X$ be a Noetherian scheme. Consider an integral closed subscheme $i : Z \to X$. It is often convenient to consider coherent sheaves of the form $i_*\mathcal{G}$ where $\mathcal{G}$ is a coherent sheaf on $Z$. In particular we are interested in these sheaves when $\mathcal{G}$ is a torsion free rank $1$ sheaf. For example $\mathcal{G}$ could be a nonzero sheaf of ideals on $Z$, or even more specifically $\mathcal{G} = \mathcal{O}_Z$. \medskip\noindent Throughout this section we will use that a coherent sheaf is the same thing as a finite type quasi-coherent sheaf and that a quasi-coherent subquotient of a coherent sheaf is coherent, see Section \ref{section-coherent-sheaves}. The support of a coherent sheaf is closed, see Modules, Lemma \ref{modules-lemma-support-finite-type-closed}. \begin{lemma} \label{lemma-prepare-filter-support} Let $X$ be a Noetherian scheme. 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 subscheme structure on $Z$, see Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme}. 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 point $x$ of $Z' \setminus Z$ we have $\mathcal{I}_x = \mathcal{O}_{X, x}$ and hence $\mathcal{G}_{n, x} = 0$. Thus we see that $\text{Supp}(\mathcal{G}_n) \subset Z$. Note that $X \setminus Z'$ is a Noetherian scheme. 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} \begin{lemma} \label{lemma-prepare-filter-irreducible} Let $X$ be a Noetherian scheme. Let $i : Z \to X$ be an integral closed subscheme. Let $\xi \in Z$ be the generic point. Let $\mathcal{F}$ be a coherent sheaf on $X$. Assume that $\mathcal{F}_\xi$ is annihilated by $\mathfrak m_\xi$. Then there exists an integer $r \geq 0$ and a sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ and an injective map of coherent sheaves $$i_*\left(\mathcal{I}^{\oplus r}\right) \to \mathcal{F}$$ which is an isomorphism in a neighbourhood of $\xi$. \end{lemma} \begin{proof} Let $\mathcal{J} \subset \mathcal{O}_X$ be the ideal sheaf of $Z$. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of local sections of $\mathcal{F}$ which are annihilated by $\mathcal{J}$. It is a quasi-coherent sheaf by Properties, Lemma \ref{properties-lemma-sections-annihilated-by-ideal}. Moreover, $\mathcal{F}'_\xi = \mathcal{F}_\xi$ because $\mathcal{J}_\xi = \mathfrak m_\xi$ and part (3) of Properties, Lemma \ref{properties-lemma-sections-annihilated-by-ideal}. By Lemma \ref{lemma-local-isomorphism} we see that $\mathcal{F}' \to \mathcal{F}$ induces an isomorphism in a neighbourhood of $\xi$. Hence we may replace $\mathcal{F}$ by $\mathcal{F}'$ and assume that $\mathcal{F}$ is annihilated by $\mathcal{J}$. \medskip\noindent Assume $\mathcal{J}\mathcal{F} = 0$. By Lemma \ref{lemma-i-star-equivalence} we can write $\mathcal{F} = i_*\mathcal{G}$ for some coherent sheaf $\mathcal{G}$ on $Z$. Suppose we can find a morphism $\mathcal{I}^{\oplus r} \to \mathcal{G}$ which is an isomorphism in a neighbourhood of the generic point $\xi$ of $Z$. Then applying $i_*$ (which is left exact) we get the result of the lemma. Hence we have reduced to the case $X = Z$. \medskip\noindent Suppose $Z = X$ is an integral Noetherian scheme with generic point $\xi$. Note that $\mathcal{O}_{X, \xi} = \kappa(\xi)$ is the function field of $X$ in this case. Since $\mathcal{F}_\xi$ is a finite $\mathcal{O}_\xi$-module we see that $r = \dim_{\kappa(\xi)} \mathcal{F}_\xi$ is finite. Hence the sheaves $\mathcal{O}_X^{\oplus r}$ and $\mathcal{F}$ have isomorphic stalks at $\xi$. By Lemma \ref{lemma-map-stalks-local-map} there exists a nonempty open $U \subset X$ and a morphism $\psi : \mathcal{O}_X^{\oplus r}|_U \to \mathcal{F}|_U$ which is an isomorphism at $\xi$, and hence an isomorphism in a neighbourhood of $\xi$ by Lemma \ref{lemma-local-isomorphism}. By Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme} there exists a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ whose associated closed subscheme $Z \subset X$ is the complement of $U$. 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 $\psi$ 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 because $X$ is integral and it is injective at the generic point of $X$ (easy proof omitted). \end{proof} \begin{lemma} \label{lemma-coherent-filter} Let $X$ be a Noetherian scheme. 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 an integral closed subscheme $Z_j \subset X$ 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 $$\mathcal{T} = \left\{ \begin{matrix} Z \subset X \text{ closed such that there exists a coherent sheaf } \mathcal{F} \\ \text{ with } \text{Supp}(\mathcal{F}) = Z \text{ for which the lemma is wrong} \end{matrix} \right\}$$ We are trying to show that $\mathcal{T}$ is empty. If not, then because $X$ is Noetherian we can choose a minimal element $Z \in \mathcal{T}$. This means that there exists a coherent sheaf $\mathcal{F}$ on $X$ whose support is $Z$ and for which the lemma does not hold. Clearly $Z \not = \emptyset$ since the only sheaf whose support is empty is the zero sheaf for which the lemma does hold (with $m = 0$). \medskip\noindent If $Z$ is not irreducible, then we can write $Z = Z_1 \cup Z_2$ with $Z_1, Z_2$ closed and strictly smaller than $Z$. 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 $Z$ 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 $Z$ is irreducible. \medskip\noindent Suppose $Z$ is irreducible. Let $\mathcal{J}$ be the sheaf of ideals cutting out the reduced induced closed subscheme structure of $Z$, see Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme}. 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 $$0 = \mathcal{J}^n\mathcal{F} \subset \mathcal{J}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{J}\mathcal{F} \subset \mathcal{F}$$ each of whose successive subquotients is annihilated by $\mathcal{J}$. Hence if each of these subquotients has a filtration as in the statement of the lemma then also $\mathcal{F}$ does. In other words we may assume that $\mathcal{J}$ does annihilate $\mathcal{F}$. \medskip\noindent In the case where $Z$ is irreducible and $\mathcal{J}\mathcal{F} = 0$ 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 $\mathcal{Q}$ is defined as the quotient. Since $\mathcal{Q}$ is zero in a neighbourhood of $\xi$ by the lemma just cited we see that the support of $\mathcal{Q}$ is strictly smaller than $Z$. Hence we see that $\mathcal{Q}$ has a filtration of the desired type by minimality of $Z$. But then clearly $\mathcal{F}$ does too, which is our final contradiction. \end{proof} \begin{lemma} \label{lemma-property-initial} Let $X$ be a Noetherian scheme. 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 integral closed subscheme $Z \subset X$ 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} \begin{lemma} \label{lemma-property-irreducible} Let $X$ be a Noetherian scheme. Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ with support contained in $Z_0$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third. \item For every integral closed subscheme $Z \subset Z_0 \subset X$, $Z \not = Z_0$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. \item There exists some coherent sheaf $\mathcal{G}$ on $X$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z_0$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, \item $\dim_{\kappa(\xi)} \mathcal{G}_\xi = 1$, and \item property $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$. \end{lemma} \begin{proof} First note that if $\mathcal{F}$ is a coherent sheaf with support contained in $Z_0$ 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 $\mathcal{F}$. Or, if $\mathcal{F}$ has property $\mathcal{P}$ and all but one of the $\mathcal{F}_i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$ then so does the last one. This follows from assumption (1). \medskip\noindent As a first application we conclude that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$. Namely, such a sheaf has a filtration (see Lemma \ref{lemma-coherent-filter}) whose subquotients have property $\mathcal{P}$ according to (2). \medskip\noindent Let $\mathcal{G}$ be as in (3). By Lemma \ref{lemma-prepare-filter-irreducible} there exist a sheaf of ideals $\mathcal{I}$ on $Z_0$, an integer $r \geq 1$, and a short exact sequence $$0 \to \left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0$$ where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. By (3)(c) we see that $r = 1$. Since $\mathcal{Q}$ has property $\mathcal{P}$ too we conclude that $(Z_0 \to X)_*\mathcal{I}$ has property $\mathcal{P}$. \medskip\noindent Next, suppose that $\mathcal{I}' \not = 0$ is another quasi-coherent sheaf of ideals on $Z_0$. Then we can consider the intersection $\mathcal{I}'' = \mathcal{I}' \cap \mathcal{I}$ and we get two short exact sequences $$0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I} \to \mathcal{Q} \to 0$$ and $$0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I}' \to \mathcal{Q}' \to 0.$$ Note that the support of the coherent sheaves $\mathcal{Q}$ and $\mathcal{Q}'$ are strictly contained in $Z_0$. Hence $\mathcal{Q}$ and $\mathcal{Q}'$ have property $\mathcal{P}$ (see above). Hence we conclude using (1) that $(Z_0 \to X)_*\mathcal{I}''$ and $(Z_0 \to X)_*\mathcal{I}'$ both have $\mathcal{P}$ as well. \medskip\noindent The final step of the proof is to note that any coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$ has a filtration (see Lemma \ref{lemma-coherent-filter} again) whose subquotients all have property $\mathcal{P}$ by what we just said. \end{proof} \begin{lemma} \label{lemma-property} Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third. \item For every integral closed subscheme $Z \subset X$ with generic point $\xi$ there exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, \item $\dim_{\kappa(\xi)} \mathcal{G}_\xi = 1$, and \item property $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$. \end{lemma} \begin{proof} According to Lemma \ref{lemma-property-initial} it suffices to show that for all integral closed subschemes $Z \subset X$ and all quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. If this fails, then since $X$ is Noetherian there is a minimal integral closed subscheme $Z_0 \subset X$ such that $\mathcal{P}$ fails for $(Z_0 \to X)_*\mathcal{I}_0$ for some quasi-coherent sheaf of ideals $\mathcal{I}_0 \subset \mathcal{O}_{Z_0}$, but $\mathcal{P}$ does hold for $(Z \to X)_*\mathcal{I}$ for all integral closed subschemes $Z \subset Z_0$, $Z \not = Z_0$ and quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_Z$. Since we have the existence of $\mathcal{G}$ for $Z_0$ by part (2), according to Lemma \ref{lemma-property-irreducible} this cannot happen. \end{proof} \begin{lemma} \label{lemma-property-irreducible-higher-rank-cohomological} Let $X$ be a Noetherian scheme. Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \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 $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$. \item For every integral closed subscheme $Z \subset Z_0 \subset X$, $Z \not = Z_0$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. \item There exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z_0$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and \item for every quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_X$ such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ and such that $\mathcal{P}$ holds for $\mathcal{G}'$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$. \end{lemma} \begin{proof} 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 assumption (1). \medskip\noindent As a first application we conclude that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$. Namely, such a sheaf has a filtration (see Lemma \ref{lemma-coherent-filter}) whose subquotients have property $\mathcal{P}$ according to (3). \medskip\noindent Let us denote $i : Z_0 \to X$ the closed immersion. Consider a coherent sheaf $\mathcal{G}$ as in (4). By Lemma \ref{lemma-prepare-filter-irreducible} there exists a sheaf of ideals $\mathcal{I}$ on $Z_0$ and a short exact sequence $$0 \to i_*\mathcal{I}^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0$$ where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. In particular $r > 0$ and $\mathcal{I}$ is nonzero because the support of $\mathcal{G}$ is equal to $Z_0$. Let $\mathcal{I}' \subset \mathcal{I}$ be any nonzero quasi-coherent sheaf of ideals on $Z_0$ contained in $\mathcal{I}$. Then we also get a short exact sequence $$0 \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{G} \to \mathcal{Q}' \to 0$$ where $\mathcal{Q}'$ has support properly contained in $Z_0$. Let $\mathcal{J} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals cutting out the support of $\mathcal{Q}'$ (for example the ideal corresponding to the reduced induced closed subscheme structure on the support of $\mathcal{Q}'$). Then $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$. By Lemma \ref{lemma-power-ideal-kills-sheaf} we see that $\mathcal{J}^n\mathcal{Q}' = 0$ for some $n$. Hence $\mathcal{J}^n\mathcal{G} \subset i_*(\mathcal{I}')^{\oplus r}$. By assumption (4)(c) of the lemma we see there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}^n\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ for which property $\mathcal{P}$ holds. Hence we get a short exact sequence $$0 \to \mathcal{G}' \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{Q}'' \to 0$$ where $\mathcal{Q}''$ has support properly contained in $Z_0$. Thus by our initial remarks and property (1) of the lemma we conclude that $i_*(\mathcal{I}')^{\oplus r}$ satisfies $\mathcal{P}$. Hence we see that $i_*\mathcal{I}'$ satisfies $\mathcal{P}$ by (2). Finally, for an arbitrary quasi-coherent sheaf of ideals $\mathcal{I}'' \subset \mathcal{O}_{Z_0}$ we can set $\mathcal{I}' = \mathcal{I}'' \cap \mathcal{I}$ and we get a short exact sequence $$0 \to i_*(\mathcal{I}') \to i_*(\mathcal{I}'') \to \mathcal{Q}''' \to 0$$ where $\mathcal{Q}'''$ has support properly contained in $Z_0$. Hence we conclude that property $\mathcal{P}$ holds for $i_*\mathcal{I}''$ as well. \medskip\noindent The final step of the proof is to note that any coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$ has a filtration (see Lemma \ref{lemma-coherent-filter} again) whose subquotients all have property $\mathcal{P}$ by what we just said. \end{proof} \begin{lemma} \label{lemma-property-higher-rank-cohomological} Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \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 $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$. \item For every integral closed subscheme $Z \subset X$ with generic point $\xi$ there exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and \item for every quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_X$ such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ and such that $\mathcal{P}$ holds for $\mathcal{G}'$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$. \end{lemma} \begin{proof} Follows from Lemma \ref{lemma-property-irreducible-higher-rank-cohomological} in exactly the same way that Lemma \ref{lemma-property} follows from Lemma \ref{lemma-property-irreducible}. \end{proof} \section{Finite morphisms and affines} \label{section-finite-affine} \noindent In this section we use the results of the preceding sections to show that the image of a Noetherian affine scheme under a finite morphism is affine. We will see later that this result holds more generally (see Limits, Lemma \ref{limits-lemma-affine}). \begin{lemma} \label{lemma-finite-morphism-Noetherian} Let $f : Y \to X$ be a morphism of schemes. Assume $f$ is finite, surjective and $X$ locally Noetherian. Let $Z \subset X$ be an integral closed subscheme with generic point $\xi$. Then there exists a coherent sheaf $\mathcal{F}$ on $Y$ such that the support of $f_*\mathcal{F}$ is equal to $Z$ and $(f_*\mathcal{F})_\xi$ is annihilated by $\mathfrak m_\xi$. \end{lemma} \begin{proof} Note that $Y$ is locally Noetherian by Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}. Because $f$ is surjective the fibre $Y_\xi$ is not empty. Pick $\xi' \in Y$ mapping to $\xi$. Let $Z' = \overline{\{\xi'\}}$. We may think of $Z' \subset Y$ as a reduced closed subscheme, see Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme}. Hence the sheaf $\mathcal{F} = (Z' \to Y)_*\mathcal{O}_{Z'}$ is a coherent sheaf on $Y$ (see Lemma \ref{lemma-finite-pushforward-coherent}). Look at the commutative diagram $$\xymatrix{ Z' \ar[r]_{i'} \ar[d]_{f'} & Y \ar[d]^f \\ Z \ar[r]^i & X }$$ We see that $f_*\mathcal{F} = i_*f'_*\mathcal{O}_{Z'}$. Hence the stalk of $f_*\mathcal{F}$ at $\xi$ is the stalk of $f'_*\mathcal{O}_{Z'}$ at $\xi$. Note that since $Z'$ is integral with generic point $\xi'$ we have that $\xi'$ is the only point of $Z'$ lying over $\xi$, see Algebra, Lemmas \ref{algebra-lemma-finite-is-integral} and \ref{algebra-lemma-integral-no-inclusion}. Hence the stalk of $f'_*\mathcal{O}_{Z'}$ at $\xi$ equal $\mathcal{O}_{Z', \xi'} = \kappa(\xi')$. In particular the stalk of $f_*\mathcal{F}$ at $\xi$ is not zero. This combined with the fact that $f_*\mathcal{F}$ is of the form $i_*f'_*(\text{something})$ implies the lemma. \end{proof} \begin{lemma} \label{lemma-affine-morphism-projection-ideal} Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Let $\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$. If the morphism $f$ is affine then $\mathcal{I}f_*\mathcal{F} = f_*(f^{-1}\mathcal{I}\mathcal{F})$. \end{lemma} \begin{proof} The notation means the following. Since $f^{-1}$ is an exact functor we see that $f^{-1}\mathcal{I}$ is a sheaf of ideals of $f^{-1}\mathcal{O}_X$. Via the map $f^\sharp : f^{-1}\mathcal{O}_X \to \mathcal{O}_Y$ this acts on $\mathcal{F}$. Then $f^{-1}\mathcal{I}\mathcal{F}$ is the subsheaf generated by sums of local sections of the form $as$ where $a$ is a local section of $f^{-1}\mathcal{I}$ and $s$ is a local section of $\mathcal{F}$. It is a quasi-coherent $\mathcal{O}_Y$-submodule of $\mathcal{F}$ because it is also the image of a natural map $f^*\mathcal{I} \otimes_{\mathcal{O}_Y} \mathcal{F} \to \mathcal{F}$. \medskip\noindent Having said this the proof is straightforward. Namely, the question is local and hence we may assume $X$ is affine. Since $f$ is affine we see that $Y$ is affine too. Thus we may write $Y = \Spec(B)$, $X = \Spec(A)$, $\mathcal{F} = \widetilde{M}$, and $\mathcal{I} = \widetilde{I}$. The assertion of the lemma in this case boils down to the statement that $$I(M_A) = ((IB)M)_A$$ where $M_A$ indicates the $A$-module associated to the $B$-module $M$. \end{proof} \begin{lemma} \label{lemma-image-affine-finite-morphism-affine-Noetherian} Let $f : Y \to X$ be a morphism of schemes. 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 will in particular imply that $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals of $\mathcal{O}_X$. Then it follows that $X$ is affine from either Lemma \ref{lemma-quasi-compact-h1-zero-covering} or Lemma \ref{lemma-quasi-separated-h1-zero-covering}. \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 $Z \subset X$ be an integral closed subscheme with generic point $\xi$. Let $\mathcal{F}$ be a coherent sheaf on $Y$ such that the support of $f_*\mathcal{F}$ is equal to $Z$ and $(f_*\mathcal{F})_\xi$ is annihilated by $\mathfrak m_\xi$, see Lemma \ref{lemma-finite-morphism-Noetherian}. We claim that taking $\mathcal{G} = f_*\mathcal{F}$ works. We only have to verify part (3)(c) of Lemma \ref{lemma-property-higher-rank-cohomological}. Hence assume that $\mathcal{J} \subset \mathcal{O}_X$ is a quasi-coherent sheaf of ideals such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$. A finite morphism is affine hence by Lemma \ref{lemma-affine-morphism-projection-ideal} we see that $\mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F})$. Also, as pointed out in the proof of Lemma \ref{lemma-affine-morphism-projection-ideal} the sheaf $f^{-1}\mathcal{J}\mathcal{F}$ is a quasi-coherent $\mathcal{O}_Y$-module. Since $Y$ is affine we see that $H^1(Y, f^{-1}\mathcal{J}\mathcal{F}) = 0$, see Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero}. Since $f$ is finite, hence affine, we see that $$H^1(X, \mathcal{J}\mathcal{G}) = H^1(X, f_*(f^{-1}\mathcal{J}\mathcal{F})) = H^1(Y, f^{-1}\mathcal{J}\mathcal{F}) = 0$$ by Lemma \ref{lemma-relative-affine-cohomology}. Hence the quasi-coherent subsheaf $\mathcal{G}' = \mathcal{J}\mathcal{G}$ satisfies $\mathcal{P}$. This verifies property (3)(c) of Lemma \ref{lemma-property-higher-rank-cohomological} as desired. \end{proof}