Chapter 7

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 4 June 2013

Operations on sheaves, quasi-coherent and coherent sheaves

Let $(X,𝒪)$ be a ringed space. An $𝒪\text{-Module}$ (note the capital $M\text{)}$ is a sheaf $ℱ$ of abelian groups such that, for each open set $U$ in $X,$ the group $ℱ\left(U\right)$ carries a structure of an $𝒪\left(U\right)\text{-module,}$ these structures being compatible with the restriction homomorphisms: explicitly, if $U\subseteq V$ are open sets in $X,$ then the restriction $\phi :ℱ\left(U\right)\to ℱ\left(V\right)$ is compatible with the restriction $\rho :𝒪\left(U\right)\to 𝒪\left(V\right),$ that is to say, if $f\in ℱ\left(U\right)$ and $a\in 𝒪\left(U\right)$ then $\phi \left(af\right)=\rho \left(a\right)·\phi \left(f\right)\text{.}$ Then each stalk ${ℱ}_x$ has a natural ${𝒪}_X\text{-module}$ structure, defined as follows: if $a_x\in {𝒪}_X,$ $f_x\in {ℱ}_x,$ say $a_x$ is the image of $a\in 𝒪\left(U\right),$ $f_x$ the image of $f\in ℱ\left(U\right)$ for some sufficiently small open neighbourhood $U$ of $x\text{;}$ then $a_x·f_x$ is the image of $af$ in ${ℱ}_x\text{.}$

In particular, $𝒪$ itself is an $𝒪\text{-Module.}$

Most of the concepts of module theory have their counterparts for Modules: -

(i) An $𝒪\text{-Module}$ homomorphism $\phi :ℱ\to 𝒢$ is a sheaf homomorphism (i.e. a family of homomorphisms $\phi \left(U\right):ℱ\left(U\right)\to 𝒢\left(U\right),$ commuting with the restrictions) such that each $\phi \left(U\right)$ is an $𝒪(\mathrm{𝔘)}\text{-module}$ homomorphism. Then each ${\phi }_x:{ℱ}_x\to {𝒢}_x$ is an ${𝒪}_X\text{-module}$ homomorphism.

(ii) Sub-Modules. A subsheaf $ℱ\prime$ of an $𝒪\text{-Module}$ $ℱ$ is a sub-Module of $ℱ$ if, for each open set $U$ in $X,$ $ℱ\prime \left(U\right)$ is a $\text{sub-}𝒪\left(U\right)\text{-module}$ of $ℱ\left(U\right)\text{.}$ Then each ${ℱ}_x^{\prime }$ is a $\text{sub-}{𝒪}_X\text{-module}$ of ${ℱ}_x,$ and the embedding $ℱ\prime \to ℱ$ is an $𝒪\text{-Module}$ homomorphism. In particular, a sub-Module of $𝒪$ is called an Ideal (with a capital I).

(iii) Quotient Modules. Let $ℱ$ be an $𝒪\text{-Module,}$ $ℱ\prime$ a sub-Module of $ℱ\text{.}$ For each open set $U$ in $X,$ form $ℱ\left(U\right)/ℱ\prime \left(U\right)\text{.}$ $U\mapsto ℱ\left(U\right)/ℱ\prime \left(U\right),$ with the induced restriction homomorphisms, is a presheaf, but not necessarily a sheaf. So we form the sheaf associated with this presheaf: this is the quotient Module ${ℱ}^{\prime \prime }=ℱ/ℱ\prime \text{.}$ Since $\underrightarrow{\text{lim}}$ is exact, we have ${ℱ}_x^{\prime \prime }={ℱ}_x/{ℱ}_x^{\prime }\text{.}$

(iv) Kernel. Let $\phi :ℱ\to 𝒢$ be an $𝒪\text{-Module}$ homomorphism. For each open set $U$ in $X$ let $ℱ\prime \left(U\right)$ be the kernel of $\phi \left(U\right):ℱ\left(U\right)\to 𝒢\left(U\right)\text{.}$ Then $U\mapsto ℱ\prime \left(U\right)$ is a sheaf $ℱ\prime ,$ called the kernel of $\phi \text{.}$ Clearly $ℱ\prime$ is an $𝒪\text{-Module.}$ We have ${ℱ}_x^{\prime }=\text{Ker}\left({\phi }_x\right)$ for all $x\in X\text{.}$

(v) Image. For each open set $U$ in $X$ we can form $\text{Im}\left(\phi \left(U\right)\right),$ which is a submodule of $𝒢\left(U\right)\text{.}$ $U\mapsto \text{im}\left(\phi \left(U\right)\right)$ is a presheaf (not necessarily a sheaf). Let $\mathscr{H}$ be the sheaf associated with this presheaf. Then $\mathscr{H}$ is a subsheaf of $𝒢,$ called the image of $𝒪\text{.}$ Again by the exactness of $\underrightarrow{\text{lim}}$ we have ${\mathscr{H}}_x=\text{Im}\left({\phi }_x\right)\text{.}$ Also $\mathscr{H}$ is isomorphic to the quotient $ℱ/ℱ\prime ,$ where $ℱ\prime$ is the kernel of $\phi \text{.}$

(vi) Cokernel. The cokernel of $\phi$ is $𝒢/\mathscr{H}\text{.}$ We have the formulas

$${\left(\text{Ker}\left(\phi \right)\right)}_x\cong \text{Ker}\left({\phi }_x\right)\text{;}{\left(\text{Im}\left(\phi \right)\right)}_x\cong \text{Im}\left({\phi }_x\right)\text{;}{\left(\text{Coker}\left(\phi \right)\right)}_x\cong \text{Coker}\left({\phi }_x\right)\text{.}$$

The class of $𝒪\text{-Modules}$ is an abelian category. Exact sequences are defined in the usual way.

Lemma (7.1). A sequence $ℱ\stackrel{\phi }{\to }𝒢\stackrel{\psi }{\to }\mathscr{H}$ is exact if and only if ${ℱ}_x\stackrel{{\phi }_x}{\to }{𝒢}_x\stackrel{{\psi }_x}{\to }{\mathscr{H}}_x$ is exact for all $x\in X\text{.}$

		Proof.
	$ℱ\to 𝒢\to \mathscr{H}$ is exact $\iff \text{Im}\left(\phi \right)=\text{Ker}\left(\psi \right)\iff {\left(\text{Im}\left(\phi \right)\right)}_x={\left(\text{Ker}\left(\psi \right)\right)}_x$ for all $x\in X\iff \text{Im}\left({\phi }_x\right)=\text{Ker}\left({\psi }_x\right)$ for all $x\in X\iff {ℱ}_x\to {𝒢}_x\to {\mathscr{H}}_x$ is exact for all $x\in X\text{.}$ $\square$

Lemma (7.2). The "section functor" $\Gamma \left(ℱ\right)$ $\text{(}=\Gamma (X,ℱ)=ℱ\left(X\right)\text{)}$ is left exact: if $0\to ℱ\to 𝒢\to \mathscr{H}$ is exact, then $0\to \Gamma \left(ℱ\right)\to \Gamma \left(𝒢\right)\to \Gamma \left(\mathscr{H}\right)$ is exact. This follows from (iii) above.

(vii) Direct sum. Let ${\left({ℱ}_i\right)}_{i\in I}$ be any family of $𝒪\text{-Modules.}$ Their direct sum $ℱ=\underset{i\in I}{⨁}{ℱ}_i$ is the sheaf $U\mapsto \underset{i\in I}{⨁}{ℱ}_i\left(U\right)\text{.}$ If each ${ℱ}_i$ is equal to $𝒪,$ we write ${𝒪}^{\left(I\right)}$ for the direct sum. In particular, if $I$ is finite and has $n$ elements, we write ${𝒪}^n$ for the direct sum of $n$ copies of $𝒪\text{.}$

(viii) Tensor product. If $ℱ,𝒢$ are $𝒪\text{-Modules,}$ their tensor product $ℱ{\otimes }_{𝒪}𝒢$ is defined to be the sheaf associated with the presheaf $U\mapsto ℱ\left(U\right){\otimes }_{𝒪\left(U\right)}𝒢\left(U\right)\text{.}$ Since $\otimes$ commutes with $\underrightarrow{\text{lim}},$ we have ${\left(ℱ{\otimes }_{𝒪}𝒢\right)}_x\cong {ℱ}_x{\otimes }_{{𝒪}_x}{𝒢}_x\text{.}$ This tensor product has all the usual properties: it is commutative, associative, distributive over $⨁,$ and is right exact in each variable (look at the stalks and use (7.1)). Also $ℱ{\otimes }_{𝒪}𝒪\cong ℱ\text{.}$

(ix) Global Hom. ${\text{Hom}}_{𝒪}(ℱ,𝒢)$ is the group of all $𝒪\text{-Module}$ homomorphisms $\phi :ℱ\to 𝒢\text{.}$ It has a natural $𝒪\left(X\right)\text{-module}$ structure: if $\phi :ℱ\to 𝒢$ and $s\in 𝒪\left(X\right),$ define $s\phi :ℱ\to 𝒢$ by $\left(s\phi \right)\left(U\right)=s|U·\phi \left(U\right)\text{.}$

(x) Sheaf $\mathscr{H}ℴ𝓂\text{.}$ The presheaf $U\mapsto {\text{Hom}}_{𝒪|U}$ is easily checked to be a sheaf, denoted by ${\mathscr{H}ℴ𝓂}_{𝒪}(ℱ,𝒢)\text{.}$ Thus $\Gamma (X,{\mathscr{H}ℴ𝓂}_{𝒪}(ℱ,𝒢))={\text{Hom}}_{𝒪}(ℱ,𝒢)\text{.}$ ${\mathscr{H}ℴ𝓂}_{𝒪}(ℱ,𝒢)$ has a natural $𝒪\text{-Module}$ structure. Both Hom and $\mathscr{H}ℴ𝓂$ are left exact in each variable (contravariant in the 1st variable, covariant in the 2nd). We have ${\mathscr{H}ℴ𝓂}_{𝒪}(𝒪,𝒢)\cong 𝒢\text{.}$

Let $f_x\in {\left({\mathscr{H}ℴ𝓂}_{𝒪}(ℱ,𝒢)\right)}_x\text{.}$ Then $f_x$ is represented by say $f:ℱ|U\to 𝒢|U,$ which gives rise to a homomorphism ${ℱ}_x\to {𝒢}_x,$ i.e. an element of ${\text{Hom}}_{{𝒪}_X}({ℱ}_x,{𝒢}_x)\text{.}$ Hence we have ${𝒪}_x\text{-module}$ homomorphism

$${\left({\mathscr{H}ℴ𝓂}_{𝒪}(ℱ,𝒢)\right)}_x\to {\text{Hom}}_{{𝒪}_x}({ℱ}_x,{𝒢}_x)$$

which in general is neither injective nor surjective (but see (7.9)).

(xi) Direct image. Let $\Psi =(\psi ,\theta ):(X,{𝒪}_X)\to (Y,{𝒪}_Y)$ be a morphism of ringed spaces. If $ℱ$ is an ${𝒪}_X\text{-Module}$ (thus a sheaf on $X\text{),}$ we define its direct image ${\Psi }_{*}ℱ,$ which is an ${𝒪}_Y\text{-Module}$ (thus a sheaf on $Y\text{)}$ as follows: ${\Psi }_{*}ℱ\left(V\right)=ℱ\left({\psi }^{-1}\left(V\right)\right)$ for each open set $V$ in $Y\text{;}$ $ℱ\left({\psi }^{-1}\left(V\right)\right)$ is an ${𝒪}_X\left({\psi }^{-1}\left(V\right)\right)\text{-module,}$ hence an ${𝒪}_Y\left(V\right)\text{-module}$ via the homomorphism $\theta \left(V\right)\to {𝒪}_Y\left(V\right)\to {𝒪}_X\left({\psi }^{-1}\left(V\right)\right)\text{.}$

${\Psi }_{*}$ is a left-exact functor from ${𝒪}_X\text{-Modules}$ to ${𝒪}_Y\text{-Modules.}$ For the section functor $\Gamma$ is left exact by (7.2). Hence if $0\to ℱ\prime \to ℱ\to {ℱ}^{\prime \prime }$ is an exact sequence of ${𝒪}_X\text{-Modules,}$ then $𝒪\to \Gamma ({\psi }^{-1}\left(V\right),ℱ\prime )\to \Gamma ({\psi }^{-1}\left(V\right),ℱ)\to \Gamma ({\psi }^{-1}\left(V\right),{ℱ}^{\prime \prime })$ is exact for each open $V\subseteq Y\text{;}$ hence $0\to {\Psi }_{*}ℱ\prime \to {\Psi }_{*}ℱ\to {\Psi }_{*}{ℱ}^{\prime \prime }$ is exact.

In particular, if $Y$ is the ringed space consisting of a single point and the ring $𝒪\left(X\right),$ then ${\Psi }_{*}\left(ℱ\right)=ℱ\left(X\right)=\Gamma (X,ℱ)\text{.}$ Thus ${\Psi }_{*}\left(ℱ\right)=ℱ\left(X\right)=\Gamma (X,ℱ)\text{.}$ Thus ${\Psi }_{*}$ is a 'relativization' of the section functor $\Gamma \text{.}$

Quasi-coherent and coherent sheaves

If $ℱ$ is an $𝒪\text{-Module,}$ a homomorphism $u:𝒪\to ℱ$ gives rise to $s=u\left(X\right)\left(I\right)\in ℱ\left(X\right),$ i.e. to a global section of $ℱ\text{.}$ Conversely, given $s\in ℱ\left(X\right)$ we may reconstruct $u\text{:}$ if $U$ is open in $X$ and $t\in 𝒪\left(U\right),$ then $u\left(t\right)=t·\left(s\right|U)\text{.}$ Hence we have a one-one correspondence between $𝒪\text{-Module}$ homomorphisms $𝒪\to ℱ$ and global sections of $ℱ,$ hence between $𝒪\text{-Module}$ homomorphisms $u:{𝒪}^{\left(I\right)}\to ℱ$ and families ${\left(s_i\right)}_{i\in I}$ of global sections of $ℱ,$ where $I$ is any index set. $u$ is an epimorphism if and only if each ${ℱ}_x$ is generated (as an ${𝒪}_x\text{-module)}$ by the ${\left(s_i\right)}_x$ (for $u$ is an epimorphism if and only if each $u_x:{\left({𝒪}_x\right)}^{\left(I\right)}\to {ℱ}_x$ is an epimorphism, by (7.1)).

$ℱ$ is said to be quasi-coherent if each $x\in X$ has an open neighbourhood $U$ such that $ℱ|U$ is the cokernel of a homomorphism ${𝒪}^{\left(I\right)}|U\to {𝒪}^{\left(J\right)}|U,$ where the index sets $I,J$ are of arbitrary cardinal (and depend on $U\text{).}$ Clearly $𝒪$ itself is quasi-coherent as an $𝒪\text{-Module.}$

Thus $ℱ$ is quasi-coherent if and only if $ℱ$ is locally generated by its sections and if the 'sheaf of relations' is locally generated by its sections.

An $𝒪\text{-module}$ $ℱ$ is of finite type if each $x\in X$ has an open neighbourhood $U$ such that $ℱ|U$ is generated by a finite set of sections of $ℱ$ over $U,$ i.e. if there exists an epimorphism ${𝒪}^p|U\to ℱ|U$ for some integer $p>0\text{.}$ If $ℱ,G$ are of finite type, then so are $ℱ\oplus 𝒢$ and $ℱ{\otimes }_{𝒪}𝒢$ (the latter because $\otimes$ is right exact). If $𝒢$ is of finite type and $𝒢$ is a homomorphic image of $ℱ,$ then $𝒢$ is of finite type.

$ℱ$ is said to be coherent if

(i)	$ℱ$ is of finite type;
(ii)	for each open set $U$ in $X$ and each homomorphism $\phi :{𝒪}^n|U\to ℱ|U$ (n a positive integer), $\text{Ker}\left(\phi \right)$ is of finite type.

Clearly a coherent sheaf is quasi-coherent. All these properties (quasi-coherence, finite type, coherence) are local with respect to the base-space $X\text{.}$

We shall use the following notation. If $U$ is an open set in $X,$ the phrase $\text{'}f:ℱ\to 𝒢$ (over $U\text{)'}$ shall mean $f:ℱ|U\to 𝒢|U\text{.}$ Similarly for diagrams of sheaves and homomorphisms.

Lemma (7.3). If $ℱ$ is a subsheaf of $𝒢,$ and $𝒢$ is of finite type and $𝒢$ is coherent, then $ℱ$ is coherent.

		Proof.
	Let $i:ℱ\to 𝒢$ be the embedding. If we have $f:{𝒪}^n\to ℱ$ (over $U\text{),}$ then $i\circ f:{𝒪}^n\to 𝒢$ (over $U\text{);}$ but $𝒢$ is coherent, hence $\text{ker}\hspace{0.17em}f=\text{ker}\hspace{0.17em}i\circ f$ is of finite type. $\square$

Lemma (7.4). Let $𝒢,H$ be $𝒪\text{-Modules.}$ If we have a diagram

$$\begin{array}{ccccccc}𝒢& \multicolumn{3}{c}{\stackrel{h}{⟶}}& \mathscr{H}& ⟶& 0\\ & & & \nearrow f\\ & & {𝒪}^p\end{array}\text{(over a neighbourhood of}\hspace{0.17em}x\in X\text{)}$$

with the row exact, then there exists an $𝒪\text{-Module}$ homomorphism $g:{𝒪}^p\to 𝒢$ (over a (smaller) neighbourhood of $x\text{),}$ such that $h\circ g=f\text{.}$

		Proof.
	The map $f$ defines $p$ sections $s_i$ $(1\le i\le p)$ belonging to $\mathscr{H}\left(U\right)$ $\text{(}U$ some open neighbourhood of $x\text{).}$ Explicitly, $f\left(U\right)$ maps $𝒪{\left(U\right)}^p$ into $\mathscr{H}\left(U\right),$ and $s_i$ is the image of the $i\text{th}$ generator $e_i$ of $𝒪{\left(U\right)}^p\text{.}$ Since $h$ is an epimorphism, there exist $g_{i,x}\in {𝒢}_x$ such that $h_x\left(g_{i,x}\right)={\left(s_i\right)}_x$ $(1\le i\le p)\text{.}$ Each $g_{i,x}$ is represented by say $g_i^{\prime }\in \mathscr{H}\left(U_i\right)\text{<mo>;</mo>}$ $h\left(g_i^{\prime }\right)$ agrees with $s_i$ at $x,$ hence in some open neighbourhood of $x,$ say $V_i$ $\text{(}\subset U_i\cap U\text{).}$ Let $V=V_1\cap \dots \cap V_p,$ then the $g_i=g_i^{\prime }|V$ define $g:{𝒪}^p\to 𝒢$ (over $V\text{),}$ and we have (over $V\text{)}$ $h\circ g\left(e_i\right)=h\left(g_i\right)=s_i|V=f\left(e_i\right),$ hence $h\circ g=f\text{.}$ $\square$

Theorem (7.5). If $0\to ℱ\stackrel{f}{\to }𝒢\stackrel{g}{\to }\mathscr{H}\to 0$ is an exact sequence of $𝒪\text{-Modules}$ on $X,$ and if any two of $ℱ,𝒢,\mathscr{H}$ are coherent, then so is the third.

		Proof.
	(1) $𝒢,\mathscr{H}$ coherent. By (7.3) it is enough to show that $ℱ$ is of finite type. Let $x\in X\text{.}$ Since $𝒢$ is of finite type we have an epimorphism $u:{𝒪}^p\to 𝒢$ (over some neighbourhood of $x\text{).}$ Since $\mathscr{H}$ is coherent, the kernel of $g\circ u$ is of finite type, hence we have an exact sequence $${𝒪}^q\stackrel{v}{⟶}{𝒪}^p\stackrel{gu}{⟶}\mathscr{H}⟶0\text{(over some neighbourhood}\hspace{0.17em}U\hspace{0.17em}\text{of}\hspace{0.17em}x\text{).}$$ Hence a commutative diagram with exact rows: $$\begin{array}{ccccccccc}& & {𝒪}^q& \stackrel{v}{⟶}& {𝒪}^p& \stackrel{gu}{⟶}& \mathscr{H}& ⟶& 0\\ & & w\begin{array}{c} ⤏AAAA \end{array}\phantom{ww}& & \begin{array}{c}\downarrow \end{array}u& & \begin{array}{c}\downarrow \end{array}\text{id}\\ 0& ⟶& ℱ& \underset{f}{⟶}& 𝒢& \underset{g}{⟶}& \mathscr{H}& ⟶& 0\end{array}\text{(over}\hspace{0.17em}U\text{).}$$ We wish to define $w:{𝒪}^q\to ℱ$ such that $fw=uv,$ and show that $w$ is an epimorphism (over $U\text{).}$ Since $guv=0,$ $\text{Im}\left(uv\right)\subseteq \text{Ker}\left(g\right)=\text{Im}\left(f\right),$ so we can define $w$ to be $f^{-1}uv\text{.}$ To show that $w$ is an epimorphism, let $y\in U,$ consider the corresponding diagram of stalks over $y,$ and verify that $w_y$ is an epimorphism by diagram-chasing. Hence by (7.1) $w$ is an epimorphism and therefore $ℱ$ is of finite type. (2) $ℱ,𝒢$ coherent. $𝒢$ is of finite type, hence so is $\mathscr{H}\text{.}$ Let $x\in X$ and let $u:{𝒪}^p\to \mathscr{H}$ be a homomorphism (over an open neighbourhood of $x\text{).}$ By (7.4) we can lift $u$ to $v:{𝒪}^p\to 𝒢$ (over a smaller open neighbourhood of $x\text{),}$ so that $gv=u\text{.}$ $ℱ$ is of finite type, hence we have say $e:{𝒪}^q\to ℱ$ (over some open neighbourhood of $x\text{).}$ Hence we have the following diagram: $$\begin{array}{ccccccccc}0& ⟶& ℱ& \stackrel{f}{⟶}& ℱ& \stackrel{g}{⟶}& \mathscr{H}& ⟶& 0\\ & & e\begin{array}{c}\uparrow \end{array}& & t\begin{array}{c}\uparrow \end{array}& & \begin{array}{c}\downarrow \end{array}u\\ 0& ⟶& {𝒪}^q& \underset{r}{\overset{h}{\rightleftarrows }}& {𝒪}^{p+q}& \underset{s}{\overset{k}{\rightleftarrows }}& {𝒪}^p& ⟶& 0\end{array}\text{(over a neighourhood of}\hspace{0.17em}x\text{).}$$ in which the rows are exact and the bottom row is split: $rh=1,$ $ks=1,$ $hr+sk=1\text{.}$ Define $t=fer+vk:{𝒪}^{p+q}\to 𝒢,$ then the diagram is commutative. Since $𝒢$ is coherent, the kernel of $t$ is of finite type and we can therefore enlarge the diagram: $$\begin{array}{ccccccccc}0& ⟶& ℱ& \stackrel{f}{⟶}& ℱ& \stackrel{g}{⟶}& \mathscr{H}& ⟶& 0\\ & & e\begin{array}{c}\uparrow \end{array}& & t\begin{array}{c}\uparrow \end{array}& & \begin{array}{c}\downarrow \end{array}u\\ 0& ⟶& {𝒪}^q& \underset{h}{⟶}& {𝒪}^{p+q}& \underset{k}{⟶}& {𝒪}^p& ⟶& 0\\ & & & & k\begin{array}{c}\uparrow \end{array}& & \begin{array}{c}\uparrow \end{array}kw\\ & & & & {𝒪}^n& \underset{\text{id}}{⟶}& {𝒪}^n\end{array}\text{(over a neighourhood of}\hspace{0.17em}x\text{).}$$ Verify that the right-hand column is exact, e.g. by considering the corresponding diagram of stalks over a point $y\in U\text{.}$ Hence $\mathscr{H}$ is coherent. (3) $ℱ,\mathscr{H}$ coherent. Since $\mathscr{H}$ and $\mathscr{H}$ are of finite type we have $$\begin{array}{ccccccccc}0& ⟶& ℱ& \stackrel{f}{⟶}& ℱ& \stackrel{g}{⟶}& \mathscr{H}& ⟶& 0\\ & & u\begin{array}{c}\downarrow \end{array}& & v\begin{array}{c} ⤏AAAA \end{array}\phantom{vv}& & \begin{array}{c}\downarrow \end{array}w\\ 0& ⟶& {𝒪}^q& ⟶& {𝒪}^{p+q}& ⟶& {𝒪}^p& ⟶& 0\end{array}\text{(over a neighourhood of}\hspace{0.17em}x\text{).}$$ with $u,w$ epimorphisms; hence as in (2) we can define $v:{𝒪}^{p+q}\to 𝒢\text{.}$ Since $u,w$ are epi, so is $v$ (by the 5 lemma). Hence $𝒢$ is of finite type. Now let $u:{𝒪}^r\to 𝒢$ be a homomorphism (over some open neighbourhood of $x\text{);}$ we have to show that $\text{Ker}\left(u\right)$ is of finite type. Since $\mathscr{H}$ is coherent we have an exact sequence of the form ${𝒪}^s\stackrel{v}{⟶}{𝒪}^r\stackrel{gu}{⟶}\mathscr{H}$ (over some open neighbourhood $u$ of $x\text{),}$ hence a diagram $$\begin{array}{ccccccccc}0& ⟶& ℱ& \stackrel{f}{⟶}& ℱ& \stackrel{g}{⟶}& \mathscr{H}& ⟶& 0\\ & & w\begin{array}{c} ⤏AAAA \end{array}\phantom{ww}& & \begin{array}{c}\uparrow \end{array}u& & \begin{array}{c}\uparrow \end{array}\text{id}\\ 0& ⟶& {𝒪}^s& \underset{v}{⟶}& {𝒪}^r& \underset{gu}{⟶}& \mathscr{H}& ⟶& 0\end{array}\text{(over}\hspace{0.17em}U\text{).}$$ here we have $guv=0,$ hence $\text{Im}\left(uv\right)\subseteq \text{Ker}\left(g\right)=\text{Im}\left(f\right),$ so we can define $w:{𝒪}^s\to ℱ$ (over $U\text{)}$ so that $uv=fw\text{.}$ Now $ℱ$ is coherent, hence the kernel of $w$ is of finite type, hence we can enlarge the diagram: $$\begin{array}{ccccccccc}0& ⟶& ℱ& \stackrel{f}{⟶}& ℱ& \stackrel{g}{⟶}& \mathscr{H}& ⟶& 0\\ & & w\begin{array}{c}\uparrow \end{array}& & \begin{array}{c}\uparrow \end{array}u& & \begin{array}{c}\uparrow \end{array}\text{id}\\ 0& ⟶& {𝒪}^s& \stackrel{v}{⟶}& {𝒪}^r& \stackrel{gu}{⟶}& \mathscr{H}\\ & & p\begin{array}{c}\uparrow \end{array}& & \begin{array}{c}\uparrow \end{array}vp\\ & & {𝒪}^t& \underset{\text{id}}{⟶}& {𝒪}^t\end{array}\text{(over a neighourhood of}\hspace{0.17em}x\text{).}$$ Here the first column (as well as the top row) is exact, and we verify (e.g. by diagram-chasing in the stalks) that the second column is exact. Hence $\text{Ker}\left(u\right)$ is of finite type and therefore $Q$ is coherent. $\square$

Corollary (7.6). $ℱ$ and $𝒢$ are coherent if and only if $ℱ\oplus 𝒢$ is coherent.

		Proof.
	If $ℱ,𝒢$ are coherent, the exact sequence $0\to ℱ\to ℱ\oplus 𝒢\to 𝒢\to 0$ shows that $ℱ\oplus 𝒢$ is coherent. If $ℱ\oplus 𝒢$ is coherent then $ℱ$ is of finite type because it is a homomorphic image of $ℱ\oplus 𝒢\text{;}$ it is also a subsheaf of $ℱ\oplus 𝒢,$ hence coherent by (7.3). $\square$

Corollary (7.7). If $\phi :ℱ\to 𝒢$ is a homomorphism of coherent $𝒪\text{-Modules,}$ then the kernel, image and cokernel of $\phi$ are all coherent.

		Proof.
	$\text{Im}\left(\phi \right)$ is a homomorphic image of $ℱ,$ hence is of finite type; it is also a sub-Module of $𝒢,$ hence coherent by (7.3). Now apply (7.5) to the exact sequences $$\begin{array}{c}0⟶\text{Ker}\left(\phi \right)⟶ℱ⟶\text{Im}\left(\phi \right)⟶0\\ 0⟶\text{Im}\left(\phi \right)⟶𝒢⟶\text{Coker}\left(\phi \right)⟶0\text{.}\end{array}$$ $\square$

Corollary (7.8). If ${ℱ}_1\stackrel{\phi }{⟶}{ℱ}_2⟶{ℱ}_3⟶{ℱ}_4\stackrel{\psi }{⟶}{ℱ}_5$ is an exact sequence in which all but ${ℱ}_3$ are coherent, then ${ℱ}_3$ is coherent.

		Proof.
	From (7.7) and the exact sequence $0⟶\text{Coker}\left(\phi \right)⟶{ℱ}_3⟶\text{Ker}\left(\psi \right)⟶0\text{.}$ $\square$

Proposition (7.9). If $ℱ,𝒢$ are coherent $𝒪\text{-Modules,}$ then $ℱ{\otimes }_{𝒪}𝒢$ and ${\text{ℋℴ𝓂}}_{𝒪}(ℱ,𝒢)$ are coherent.

		Proof.
	Consider $ℱ{\otimes }_{𝒪}𝒢\text{.}$ Let $x\in X\text{;}$ since $ℱ$ is coherent there is an exact sequence. $$\begin{array}{cc}\left(✶\right)& {𝒪}^q⟶{𝒪}^p⟶ℱ⟶0\text{(over some open neighourbood}\hspace{0.17em}U\hspace{0.17em}\text{of}\hspace{0.17em}x\text{);}\end{array}$$ hence, as tensoring with $𝒢$ is right exact and $𝒪\otimes 𝒢\cong 𝒢,$ an exact sequence $${𝒢}^q⟶{𝒢}^p⟶ℱ\otimes 𝒢⟶0\text{(over}\hspace{0.17em}U\text{);}$$ since $𝒢$ is coherent, so are ${𝒢}^p,{𝒢}^q$ by (7.6), hence $ℱ\otimes 𝒢$ is coherent by (7.7) and the fact that coherence is a local property. For operate on $\left(✶\right)$ ${\text{ℋℴ𝓂}}_{𝒪}(\hspace{0.17em},𝒢)\text{.}$ The argument is similar. $\square$

Proposition (7.10). If $ℱ,𝒢$ are $𝒪\text{-Modules}$ and $ℱ$ is coherent, the mapping

$${\left({\text{ℋℴ𝓂}}_{𝒪}(ℱ,𝒢)\right)}_x\to {\text{Hom}}_{{𝒪}_x}({ℱ}_x,{𝒢}_x)$$

is an isomorphism.

		Proof.
	From $\left(✶\right)$ we have ${𝒪}_x^q\to {𝒪}_x^p\to {ℱ}_x\to 0$ exact, hence by the left exactness of ${\text{ℋℴ𝓂}}_{𝒪}$ and ${\text{Hom}}_{{𝒪}_x}$ we have exact sequences $$\begin{array}{ccccccc}0& ⟶& {\left({\text{ℋℴ𝓂}}_{𝒪}(ℱ,𝒢)\right)}_x\to & ⟶& {\left({\text{ℋℴ𝓂}}_{𝒪}({𝒪}^p,𝒢)\right)}_x\to & ⟶& {\left({\text{ℋℴ𝓂}}_{𝒪}({𝒪}^q,𝒢)\right)}_x\to \\ & & \downarrow & & \downarrow & & \downarrow \\ 0& ⟶& {\text{Hom}}_{{𝒪}_x}({ℱ}_x,{𝒢}_x)& ⟶& {\text{Hom}}_{{𝒪}_x}({𝒪}_x^p,{𝒢}_x)& ⟶& {\text{Hom}}_{{𝒪}_x}({𝒪}_x^q,{𝒢}_x)\text{.}\end{array}$$ Since ${\text{ℋℴ𝓂}}_{𝒪}({𝒪}^p,𝒢)={𝒢}^p,$ the second and third vertical arrows are isomorphisms, hence so is the first. $\square$

If $𝒪$ itself is coherent as an $𝒪\text{-Module,}$ we shall say that $𝒪$ is a coherent sheaf of rings.

Proposition (7.11). Let $𝒪$ be a coherent sheaf of rings and let $𝒢$ be an $𝒪\text{-Module.}$ Then $ℱ$ is coherent if and only if it is locally finitely presented, i.e. for each $x\in X$ there is an exact sequence ${𝒪}^q\to {𝒪}^p\to ℱ\to 0$ over some neighbourhood of $x\text{.}$

		Proof.
	If $ℱ$ is coherent it is locally finitely presented (whether $𝒪$ is coherent or not). Conversely, if $𝒪$ is coherent, so are ${𝒪}^p$ and ${𝒪}^q$ by (7.6), hence so is $ℱ$ by (7.7) (since coherence is a local property). $\square$

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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