## Chapter 7

Last update: 4 June 2013

## Operations on sheaves, quasi-coherent and coherent sheaves

Let $\left(X,𝒪\right)$ 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 $𝒪\left(\mathrm{𝔘\right)}\text{-module}$ 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 $ℱ\left(U\right)/ℱ\prime \left(U\right)\text{.}$ $U↦ℱ\left(U\right)/ℱ\prime \left(U\right),$ 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 $\underset{\to }{\text{lim}}$ 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 $U↦ℱ\prime \left(U\right)$ 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 $𝒢\left(U\right)\text{.}$ $U↦\text{im}\left(\phi \left(U\right)\right)$ is a presheaf (not necessarily a sheaf). Let $ℋ$ be the sheaf associated with this presheaf. Then $ℋ$ is a subsheaf of $𝒢,$ called the image of $𝒪\text{.}$ Again by the exactness of $\underset{\to }{\text{lim}}$ we have ${ℋ}_{x}=\text{Im}\left({\phi }_{x}\right)\text{.}$ Also $ℋ$ is isomorphic to the quotient $ℱ/ℱ\prime ,$ where $ℱ\prime$ is the kernel of $\phi \text{.}$

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

$(Ker(φ))x≅ Ker(φx); (Im(φ))x≅ Im(φx); (Coker(φ))x≅ Coker(φx).$

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

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

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

Lemma (7.2). The "section functor" $\Gamma \left(ℱ\right)$ $\text{(}=\Gamma \left(X,ℱ\right)=ℱ\left(X\right)\text{)}$ is left exact: if $0\to ℱ\to 𝒢\to ℋ$ is exact, then $0\to \Gamma \left(ℱ\right)\to \Gamma \left(𝒢\right)\to \Gamma \left(ℋ\right)$ 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 $U↦\underset{i\in I}{⨁}{ℱ}_{i}\left(U\right)\text{.}$ 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 $U↦ℱ\left(U\right){\otimes }_{𝒪\left(U\right)}𝒢\left(U\right)\text{.}$ Since $\otimes$ commutes with $\underset{\to }{\text{lim}},$ 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. ${\text{Hom}}_{𝒪}\left(ℱ,𝒢\right)$ 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 $ℋℴ𝓂\text{.}$ The presheaf $U↦{\text{Hom}}_{𝒪|U}$ is easily checked to be a sheaf, denoted by ${ℋℴ𝓂}_{𝒪}\left(ℱ,𝒢\right)\text{.}$ Thus $\Gamma \left(X,{ℋℴ𝓂}_{𝒪}\left(ℱ,𝒢\right)\right)={\text{Hom}}_{𝒪}\left(ℱ,𝒢\right)\text{.}$ ${ℋℴ𝓂}_{𝒪}\left(ℱ,𝒢\right)$ has a natural $𝒪\text{-Module}$ structure. Both Hom and $ℋℴ𝓂$ are left exact in each variable (contravariant in the 1st variable, covariant in the 2nd). We have ${ℋℴ𝓂}_{𝒪}\left(𝒪,𝒢\right)\cong 𝒢\text{.}$

Let ${f}_{x}\in {\left({ℋℴ𝓂}_{𝒪}\left(ℱ,𝒢\right)\right)}_{x}\text{.}$ 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 ${\text{Hom}}_{{𝒪}_{X}}\left({ℱ}_{x},{𝒢}_{x}\right)\text{.}$ Hence we have ${𝒪}_{x}\text{-module}$ homomorphism

$(ℋℴ𝓂𝒪(ℱ,𝒢))x →Hom𝒪x(ℱx,𝒢x)$

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

(xi) Direct image. Let $\Psi =\left(\psi ,\theta \right):\left(X,{𝒪}_{X}\right)\to \left(Y,{𝒪}_{Y}\right)$ 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 $𝒪\to \Gamma \left({\psi }^{-1}\left(V\right),ℱ\prime \right)\to \Gamma \left({\psi }^{-1}\left(V\right),ℱ\right)\to \Gamma \left({\psi }^{-1}\left(V\right),{ℱ}^{\prime \prime }\right)$ 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 ${\Psi }_{*}\left(ℱ\right)=ℱ\left(X\right)=\Gamma \left(X,ℱ\right)\text{.}$ Thus ${\Psi }_{*}\left(ℱ\right)=ℱ\left(X\right)=\Gamma \left(X,ℱ\right)\text{.}$ 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 $u\left(t\right)=t·\left(s|U\right)\text{.}$ 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 $\text{ker} f=\text{ker} i\circ f$ is of finite type. $\square$

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

$𝒢 ⟶h ℋ ⟶ 0 ↗f 𝒪p (over a neighbourhood of x∈X)$

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 ${s}_{i}$ $\left(1\le i\le p\right)$ belonging to $ℋ\left(U\right)$ $\text{(}U$ some open neighbourhood of $x\text{).}$ Explicitly, $f\left(U\right)$ maps $𝒪{\left(U\right)}^{p}$ into $ℋ\left(U\right),$ 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 ${h}_{x}\left({g}_{i,x}\right)={\left({s}_{i}\right)}_{x}$ $\left(1\le i\le p\right)\text{.}$ Each ${g}_{i,x}$ is represented by say ${g}_{i}^{\prime }\in ℋ\left({U}_{i}\right)\text{;}$ $h\left({g}_{i}^{\prime }\right)$ 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 $0\to ℱ\stackrel{f}{\to }𝒢\stackrel{g}{\to }ℋ\to 0$ is an exact sequence of $𝒪\text{-Modules}$ on $X,$ and if any two of $ℱ,𝒢,ℋ$ are coherent, then so is the third.

 Proof. (1) $𝒢,ℋ$ 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 $ℋ$ is coherent, the kernel of $g\circ u$ is of finite type, hence we have an exact sequence $𝒪q⟶v𝒪p ⟶guℋ⟶0 (over some neighbourhood U of x).$ Hence a commutative diagram with exact rows: $𝒪q ⟶v 𝒪p ⟶gu ℋ ⟶ 0 w \stackrel{\phantom{AAAA}}{⤏} ww ↓u ↓id 0 ⟶ ℱ ⟶f 𝒢 ⟶g ℋ ⟶ 0 (over U).$ 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 $ℋ\text{.}$ Let $x\in X$ and let $u:{𝒪}^{p}\to ℋ$ 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: $0 ⟶ ℱ ⟶f ℱ ⟶g ℋ ⟶ 0 e↑ t↑ ↓u 0 ⟶ 𝒪q ⇄rh 𝒪p+q ⇄sk 𝒪p ⟶ 0 (over a neighourhood of x).$ 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: $0 ⟶ ℱ ⟶f ℱ ⟶g ℋ ⟶ 0 e↑ t↑ ↓u 0 ⟶ 𝒪q ⟶h 𝒪p+q ⟶k 𝒪p ⟶ 0 k↑ ↑kw 𝒪n ⟶id 𝒪n (over a neighourhood of x).$ 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 $ℋ$ is coherent. (3) $ℱ,ℋ$ coherent. Since $ℋ$ and $ℋ$ are of finite type we have $0 ⟶ ℱ ⟶f ℱ ⟶g ℋ ⟶ 0 u↓ v \stackrel{\phantom{AAAA}}{⤏} vv ↓w 0 ⟶ 𝒪q ⟶ 𝒪p+q ⟶ 𝒪p ⟶ 0 (over a neighourhood of x).$ 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 $ℋ$ is coherent we have an exact sequence of the form ${𝒪}^{s}\stackrel{v}{⟶}{𝒪}^{r}\stackrel{gu}{⟶}ℋ$ (over some open neighbourhood $u$ of $x\text{),}$ hence a diagram $0 ⟶ ℱ ⟶f ℱ ⟶g ℋ ⟶ 0 w \stackrel{\phantom{AAAA}}{⤏} ww ↑u ↑id 0 ⟶ 𝒪s ⟶v 𝒪r ⟶gu ℋ ⟶ 0 (over U).$ 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: $0 ⟶ ℱ ⟶f ℱ ⟶g ℋ ⟶ 0 w↑ ↑u ↑id 0 ⟶ 𝒪s ⟶v 𝒪r ⟶gu ℋ p↑ ↑vp 𝒪t ⟶id 𝒪t (over a neighourhood of x).$ 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 $0⟶Ker(φ)⟶ℱ ⟶Im(φ)⟶0 0⟶Im(φ)⟶𝒢⟶ Coker(φ)⟶0.$ $\square$

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 ${\text{ℋℴ𝓂}}_{𝒪}\left(ℱ,𝒢\right)$ are coherent.

 Proof. Consider $ℱ{\otimes }_{𝒪}𝒢\text{.}$ Let $x\in X\text{;}$ since $ℱ$ is coherent there is an exact sequence. $(✶) 𝒪q⟶𝒪p⟶ℱ⟶0 (over some open neighourbood U of x);$ hence, as tensoring with $𝒢$ is right exact and $𝒪\otimes 𝒢\cong 𝒢,$ an exact sequence $𝒢q⟶𝒢p⟶ℱ⊗𝒢 ⟶0(over U);$ 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 ${\text{ℋℴ𝓂}}_{𝒪}\left(ℱ,𝒢\right),$ operate on $\left(✶\right)$ ${\text{ℋℴ𝓂}}_{𝒪}\left( ,𝒢\right)\text{.}$ The argument is similar. $\square$

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

$(ℋℴ𝓂𝒪(ℱ,𝒢))x→ Hom𝒪x(ℱx,𝒢x)$

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 $0 ⟶ (ℋℴ𝓂𝒪(ℱ,𝒢))x→ ⟶ (ℋℴ𝓂𝒪(𝒪p,𝒢))x→ ⟶ (ℋℴ𝓂𝒪(𝒪q,𝒢))x→ ↓ ↓ ↓ 0 ⟶ Hom𝒪x(ℱx,𝒢x) ⟶ Hom𝒪x(𝒪xp,𝒢x) ⟶ Hom𝒪x(𝒪xq,𝒢x).$ Since ${\text{ℋℴ𝓂}}_{𝒪}\left({𝒪}^{p},𝒢\right)={𝒢}^{p},$ 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".