Chapter 0 — Preliminaries

§5. Quasi-coherent and Coherent Sheaves

5.1. Quasi-coherent sheaves

(5.1.1) Let $(X,{\mathcal{O}}_X)$ be a ringed space and $\mathcal{F}$ an ${\mathcal{O}}_X$-Module. To give a homomorphism $u:{\mathcal{O}}_X\to \mathcal{F}$ of ${\mathcal{O}}_X$-Modules is the same as to give the section $s=u(1)\in \Gamma (X,\mathcal{F})$. Indeed, once $s$ is given, for every section $t\in \Gamma (U,{\mathcal{O}}_X)$ one necessarily has $u(t)=t\cdot (s|U)$; we say that $u$ is defined by the section $s$. If $I$ is an arbitrary index set, consider the direct sum sheaf ${\mathcal{O}}_X^{(I)}$, and for $i\in I$ let $h_i$ be the canonical injection of the $i$-th factor. The map $u\mapsto (u\circ h_i)$ is an isomorphism of ${\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{O}}_X^{(I)},\mathcal{F})$ onto the product $({\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{O}}_X,\mathcal{F}){)}^I$. There is thus a canonical bijection between homomorphisms $u:{\mathcal{O}}_X^{(I)}\to \mathcal{F}$ and families of sections $(s_i{)}_{i\in I}$ of $\mathcal{F}$ over $X$. The homomorphism $u$ corresponding to $(s_i)$ sends $(a_i)\in (\Gamma (U,{\mathcal{O}}_X){)}^{(I)}$ to ${\sum }_{i\in I}{a}_i\cdot (s_i|U)$.

$\mathcal{F}$ is said to be generated by the family $(s_i)$ if the corresponding homomorphism ${\mathcal{O}}_X^{(I)}\to \mathcal{F}$ is surjective — equivalently, if for every $x\in X$, ${\mathcal{F}}_x$ is the ${\mathcal{O}}_x$-module generated by the $(s_i{)}_x$. $\mathcal{F}$ is generated by its sections over $X$ if it is generated by the family of all such sections (or a subfamily); equivalently, if there is a surjective homomorphism ${\mathcal{O}}_X^{(I)}\to \mathcal{F}$ for some $I$.

Note that an ${\mathcal{O}}_X$-Module $\mathcal{F}$ may be such that some point $x_0\in X$ has the property that $\mathcal{F}|U$ is not generated by its sections over $U$ for any open neighborhood $U$ of $x_0$: take $X=\mathbb{R}$, ${\mathcal{O}}_X$ the simple sheaf $\mathbb{Z}$, and $\mathcal{F}$ the sub-sheaf with ${\mathcal{F}}_0=0$ and ${\mathcal{F}}_x=\mathbb{Z}$ for $x\ne 0$, and $x_0=0$. The only section of $\mathcal{F}|U$ over $U$ is 0 for a neighborhood $U$ of 0.

(5.1.2) Let $f:X\to Y$ be a morphism of ringed spaces. If $\mathcal{F}$ is an ${\mathcal{O}}_X$-Module generated by its sections over $X$, then the canonical homomorphism $f\ast (f_{\ast }(\mathcal{F}))\to \mathcal{F}$ (4.4.3.3) is surjective: with the notation of (5.1.1), $s_i\otimes 1$ is a section of $f\ast (f_{\ast }(\mathcal{F}))$ over $X$, and its image in $\mathcal{F}$ is $s_i$. The example of (5.1.1) with $f$ the identity shows that the converse fails in general.

(5.1.3) An ${\mathcal{O}}_X$-Module $\mathcal{F}$ is said to be quasi-coherent if for every $x\in X$ there is an open neighborhood $U$ such that $\mathcal{F}|U$ is isomorphic to the cokernel of a homomorphism ${\mathcal{O}}_X^{(I)}|U\to {\mathcal{O}}_X^{(J)}|U$ for some index sets I, J. Clearly ${\mathcal{O}}_X$ itself is quasi-coherent, and every direct sum of quasi-coherent ${\mathcal{O}}_X$-Modules is quasi-coherent. An ${\mathcal{O}}_X$-Algebra $\mathcal{A}$ is quasi-coherent if it is quasi-coherent as an ${\mathcal{O}}_X$-Module.

(5.1.4) Let $f:X\to Y$ be a morphism of ringed spaces. If $\mathcal{G}$ is a quasi-coherent ${\mathcal{O}}_Y$-Module, then $f\ast (\mathcal{G})$ is a quasi-coherent ${\mathcal{O}}_X$-Module. Indeed, for every $x\in X$ there is an open neighborhood $V$ of $f(x)$ in $Y$ such that $\mathcal{G}|V$ is the cokernel of ${\mathcal{O}}_Y^{(I)}|V\to {\mathcal{O}}_Y^{(J)}|V$. Set $U=f^{-1}(V)$ and let $f_U=f|U$; then $f\ast (\mathcal{G})|U=f_U\ast (\mathcal{G}|V)$. Since $f_U\ast$ is right exact and commutes with direct sums, $f_U\ast (\mathcal{G}|V)$ is the cokernel of ${\mathcal{O}}_X^{(I)}|U\to {\mathcal{O}}_X^{(J)}|U$.

5.2. Sheaves of finite type

(5.2.1) An ${\mathcal{O}}_X$-Module $\mathcal{F}$ is said to be of finite type if every $x\in X$ admits an open neighborhood $U$ such that $\mathcal{F}|U$ is generated by finitely many sections over $U$; equivalently, $\mathcal{F}|U$ is a quotient sheaf of $({\mathcal{O}}_X|U{)}^p$ for some finite $p$. Every quotient of an ${\mathcal{O}}_X$-Module of finite type is of finite type; so is every finite direct sum and every finite tensor product of ${\mathcal{O}}_X$-Modules of finite type. An ${\mathcal{O}}_X$-Module of finite type need not be quasi-coherent, as one sees for ${\mathcal{O}}_X/\mathcal{F}$ where $\mathcal{F}$ is the sheaf of (5.1.1). If $\mathcal{F}$ is of finite type, then ${\mathcal{F}}_x$ is an ${\mathcal{O}}_x$-module of finite type for every $x\in X$; the example of (5.1.1) shows that this necessary condition is not in general sufficient.

(5.2.2) Let $\mathcal{F}$ be an ${\mathcal{O}}_X$-Module of finite type. If $s_i$ ($1\le i\le n$) are sections of $\mathcal{F}$ over an open neighborhood $U$ of a point $x\in X$ and the $(s_i{)}_x$ generate ${\mathcal{F}}_x$, then there is an open neighborhood $V\subset U$ of $x$ such that the $(s_i{)}_y$ generate ${\mathcal{F}}_y$ for every $y\in V$ (FAC, I, §2, no. 12, Prop. 1). In particular, the support of $\mathcal{F}$ is closed.

Similarly, if $u:\mathcal{F}\to \mathcal{G}$ is a homomorphism with $u_x=0$, there is a neighborhood $U$ of $x$ such that $u_y=0$ for every $y\in U$.

(5.2.3) Suppose $X$ is quasi-compact and let $\mathcal{F}$, $\mathcal{G}$ be ${\mathcal{O}}_X$-Modules with $\mathcal{G}$ of finite type, $u:\mathcal{F}\to \mathcal{G}$ a surjective homomorphism. Suppose moreover that $\mathcal{F}=\underrightarrow{\lim }{\mathcal{F}}_{\lambda }$ is the inductive limit of a system of ${\mathcal{O}}_X$-Modules. Then there exists an index $\mu$ such that the homomorphism ${\mathcal{F}}_{\mu }\to \mathcal{G}$ is surjective. Indeed, for every $x\in X$ there is a finite system of sections $s_i$ of $\mathcal{G}$ over an open neighborhood $U(x)$ of $x$ whose germs generate ${\mathcal{G}}_y$ for every $y\in U(x)$; there is also an open $V(x)\subset U(x)$ and sections $t_i$ of $\mathcal{F}$ over $V(x)$ with $s_i|V(x)=u(t_i)$. We may assume the $t_i$ are canonical images of sections of some ${\mathcal{F}}_{\lambda (x)}$ over $V(x)$. Cover $X$ by finitely many $V(x_k)$ and let $\mu$ exceed all the $\lambda (x_k)$; plainly $\mu$ answers the question.

Still assuming $X$ quasi-compact, if $\mathcal{F}$ is of finite type and generated by its sections over $X$ (5.1.1), then $\mathcal{F}$ is generated by a finite subfamily of those sections: cover $X$ by finitely many open $U_k$ such that for each $k$ there are finitely many sections $s_{ik}$ of $\mathcal{F}$ over $X$ whose restrictions to $U_k$ generate $\mathcal{F}|U_k$; plainly the $s_{ik}$ then generate $\mathcal{F}$.

(5.2.4) Let $f:X\to Y$ be a morphism of ringed spaces. If $\mathcal{G}$ is an ${\mathcal{O}}_Y$-Module of finite type, then $f\ast (\mathcal{G})$ is an ${\mathcal{O}}_X$-Module of finite type. Indeed, for every $x\in X$ there is an open $V$ containing $f(x)$ and a surjective homomorphism $v:{\mathcal{O}}_Y^p|V\to \mathcal{G}|V$. Set $U=f^{-1}(V)$ and $f_U=f|U$; then $f\ast (\mathcal{G})|U=f_U\ast (\mathcal{G}|V)$. Since $f_U\ast$ is right exact (4.3.1) and commutes with direct sums, $f_U\ast (v)$ is a surjective homomorphism ${\mathcal{O}}_X^p|U\to f\ast (\mathcal{G})|U$.

(5.2.5) An ${\mathcal{O}}_X$-Module $\mathcal{F}$ admits a finite presentation if for every $x\in X$ there is an open neighborhood $U$ such that $\mathcal{F}|U$ is isomorphic to the cokernel of a $({\mathcal{O}}_X|U)$-homomorphism ${\mathcal{O}}_X^p|U\to {\mathcal{O}}_X^q|U$ with $p,q>0$ finite. Such an ${\mathcal{O}}_X$-Module is of finite type and quasi-coherent. If $f:X\to Y$ is a morphism of ringed spaces and $\mathcal{G}$ admits a finite presentation, so does $f\ast (\mathcal{G})$, by the argument of (5.1.4).

(5.2.6) Let $\mathcal{F}$ be an ${\mathcal{O}}_X$-Module admitting a finite presentation (5.2.5). Then for every ${\mathcal{O}}_X$-Module $\mathcal{H}$, the canonical functorial homomorphism

(ℋom_{𝒪_X}(ℱ, ℋ))_x → Hom_{𝒪_x}(ℱ_x, ℋ_x)

is bijective (T, 4.1.1).

(5.2.7) Let $\mathcal{F}$, $\mathcal{G}$ be ${\mathcal{O}}_X$-Modules admitting finite presentations. If for some $x\in X$ the ${\mathcal{O}}_x$-modules ${\mathcal{F}}_x$ and ${\mathcal{G}}_x$ are isomorphic, then there is an open neighborhood $U$ of $x$ such that $\mathcal{F}|U$ and $\mathcal{G}|U$ are isomorphic. Indeed, if $\varphi :{\mathcal{F}}_x\to {\mathcal{G}}_x$ and $\psi :{\mathcal{G}}_x\to {\mathcal{F}}_x$ are inverse isomorphisms, then by (5.2.6) there is an open $V\ni x$ and sections $u$ (resp. $v$) of $\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})$ (resp. $\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{G},\mathcal{F})$) over $V$ with $u_x=\varphi$ (resp. $v_x=\psi$). Since $(u\circ v{)}_x$ and $(v\circ u{)}_x$ are the identity automorphisms, there is an open $U\subset V$ containing $x$ on which $(u\circ v)|U$ and $(v\circ u)|U$ are identities, whence the assertion.

5.3. Coherent sheaves

(5.3.1) An ${\mathcal{O}}_X$-Module $\mathcal{F}$ is coherent if:

a) $\mathcal{F}$ is of finite type; b) For every open $U\subset X$, every integer $n>0$, and every homomorphism $u:{\mathcal{O}}_X^n|U\to \mathcal{F}|U$, the kernel of $u$ is of finite type.

Both conditions are local.

For most of the proofs of the properties of coherent sheaves recalled below, see (FAC), I, §2.

(5.3.2) Every coherent ${\mathcal{O}}_X$-Module admits a finite presentation (5.2.5); the converse is not in general true, since ${\mathcal{O}}_X$ itself need not be coherent.

Every sub-${\mathcal{O}}_X$-Module of finite type of a coherent ${\mathcal{O}}_X$-Module is coherent; every finite direct sum of coherent ${\mathcal{O}}_X$-Modules is coherent.

(5.3.3) In an exact sequence $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ of ${\mathcal{O}}_X$-Modules, if two are coherent then so is the third.

(5.3.4) If $\mathcal{F}$ and $\mathcal{G}$ are coherent ${\mathcal{O}}_X$-Modules and $u:\mathcal{F}\to \mathcal{G}$ is a homomorphism, then $Im(u)$, $Ker(u)$, and $Coker(u)$ are coherent. In particular, sub-${\mathcal{O}}_X$-Modules $\mathcal{F}$, $\mathcal{G}$ of a coherent ${\mathcal{O}}_X$-Module have $\mathcal{F}+\mathcal{G}$ and $\mathcal{F}\cap \mathcal{G}$ coherent.

(5.3.5) If $\mathcal{F}$ and $\mathcal{G}$ are coherent ${\mathcal{O}}_X$-Modules, so are $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$ and $\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})$.

(5.3.6) Let $\mathcal{F}$ be coherent and $\mathcal{J}$ a coherent sheaf of ideals of ${\mathcal{O}}_X$. Then $\mathcal{J}\mathcal{F}$ is coherent, being the image of $\mathcal{J}{\otimes }_{{\mathcal{O}}_X}\mathcal{F}$ under the canonical map $\mathcal{J}{\otimes }_{{\mathcal{O}}_X}\mathcal{F}\to \mathcal{F}$ (5.3.4 and 5.3.5).

(5.3.7) An ${\mathcal{O}}_X$-Algebra $\mathcal{A}$ is coherent if it is a coherent ${\mathcal{O}}_X$-Module. In particular, ${\mathcal{O}}_X$ is a coherent sheaf of rings if and only if, for every open $U\subset X$ and every $u:{\mathcal{O}}_X^n|U\to {\mathcal{O}}_X|U$, the kernel of $u$ is a $({\mathcal{O}}_X|U)$-Module of finite type.

If ${\mathcal{O}}_X$ is a coherent sheaf of rings, every ${\mathcal{O}}_X$-Module $\mathcal{F}$ admitting a finite presentation (5.2.5) is coherent, by (5.3.4).

The annihilator of an ${\mathcal{O}}_X$-Module $\mathcal{F}$ is the kernel $\mathcal{J}$ of the canonical homomorphism ${\mathcal{O}}_X\to \mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{F})$ sending $s\in \Gamma (U,{\mathcal{O}}_X)$ to multiplication by $s$ in $\mathrm{Hom}(\mathcal{F}|U,\mathcal{F}|U)$. If ${\mathcal{O}}_X$ is coherent and $\mathcal{F}$ coherent, then $\mathcal{J}$ is coherent (5.3.4 and 5.3.5), and ${\mathcal{J}}_x$ is the annihilator of ${\mathcal{F}}_x$ for every $x\in X$ (5.2.6).

(5.3.8) Suppose ${\mathcal{O}}_X$ is coherent. Let $\mathcal{F}$ be a coherent ${\mathcal{O}}_X$-Module, $x\in X$, and $M$ a finite-type submodule of ${\mathcal{F}}_x$. There is an open neighborhood $U$ of $x$ and a coherent sub-$({\mathcal{O}}_X|U)$-Module $\mathcal{G}$ of $\mathcal{F}|U$ with ${\mathcal{G}}_x=M$ (T, 4.1, Lemma 1).

This, together with the properties of sub-${\mathcal{O}}_X$-Modules of a coherent ${\mathcal{O}}_X$-Module, imposes necessary conditions on the rings ${\mathcal{O}}_x$ for ${\mathcal{O}}_X$ to be coherent. For example (5.3.4), the intersection of two finite-type ideals of ${\mathcal{O}}_x$ must still be a finite-type ideal.

(5.3.9) Suppose ${\mathcal{O}}_X$ is coherent, and let $M$ be an ${\mathcal{O}}_x$-module admitting a finite presentation, so the cokernel of some $\varphi :{\mathcal{O}}_x^p\to {\mathcal{O}}_x^q$. There is an open neighborhood $U$ of $x$ and a coherent $({\mathcal{O}}_X|U)$-Module $\mathcal{F}$ with ${\mathcal{F}}_x\cong M$. Indeed, by (5.2.6) there is a section $u$ of $\mathcal{H}o{m}_{{\mathcal{O}}_X}({\mathcal{O}}_X^p,{\mathcal{O}}_X^q)$ over some $U$ with $u_x=\varphi$; the cokernel of $u:{\mathcal{O}}_X^p|U\to {\mathcal{O}}_X^q|U$ answers the question (5.3.4).

(5.3.10) Suppose ${\mathcal{O}}_X$ is coherent and $\mathcal{J}$ is a coherent sheaf of ideals of ${\mathcal{O}}_X$. For an $({\mathcal{O}}_X/\mathcal{J})$-Module $\mathcal{F}$ to be coherent, it is necessary and sufficient that it be coherent as an ${\mathcal{O}}_X$-Module. In particular ${\mathcal{O}}_X/\mathcal{J}$ is a coherent sheaf of rings.

(5.3.11) Let $f:X\to Y$ be a morphism of ringed spaces, and suppose ${\mathcal{O}}_X$ is coherent. Then for every coherent ${\mathcal{O}}_Y$-Module $\mathcal{G}$, $f\ast (\mathcal{G})$ is coherent. Indeed, with the notation of (5.2.4), one may assume $\mathcal{G}|V$ is the cokernel of $v:{\mathcal{O}}_Y^q|V\to {\mathcal{O}}_Y^p|V$. Since $f_U\ast$ is right exact, $f\ast (\mathcal{G})|U=f_U\ast (\mathcal{G}|V)$ is the cokernel of $f_U\ast (v):{\mathcal{O}}_X^q|U\to {\mathcal{O}}_X^p|U$.

(5.3.12) Let $Y\subset X$ be a closed subset, $j:Y\to X$ the canonical injection, ${\mathcal{O}}_Y$ a sheaf of rings on $Y$, and ${\mathcal{O}}_X=j_{\ast }({\mathcal{O}}_Y)$. An ${\mathcal{O}}_Y$-Module $\mathcal{G}$ is of finite type (resp. quasi-coherent, coherent) if and only if $j_{\ast }(\mathcal{G})$ is of finite type (resp. quasi-coherent, coherent) as an ${\mathcal{O}}_X$-Module.

5.4. Locally free sheaves

(5.4.1) Let $X$ be a ringed space. An ${\mathcal{O}}_X$-Module $\mathcal{F}$ is locally free if for every $x\in X$ there is an open neighborhood $U$ such that $\mathcal{F}|U$ is isomorphic to ${\mathcal{O}}_X^{(I)}|U$ for some $I$ (which may depend on $U$). If $I$ may always be taken finite, $\mathcal{F}$ is of finite rank; if $I$ may always be taken finite with the same number of elements $n$, $\mathcal{F}$ is of rank $n$. A locally free ${\mathcal{O}}_X$-Module of rank 1 is also called invertible (cf. (5.4.3)). If $\mathcal{F}$ is locally free of finite rank, then for every $x\in X$, ${\mathcal{F}}_x$ is a free ${\mathcal{O}}_x$-module of finite rank $n(x)$, and there is a neighborhood $U$ of $x$ on which $\mathcal{F}|U$ has rank $n(x)$; if $X$ is connected, $n(x)$ is constant.

Every locally free sheaf is quasi-coherent; if ${\mathcal{O}}_X$ is a coherent sheaf of rings, every locally free ${\mathcal{O}}_X$-Module of finite rank is coherent.

If $\mathcal{L}$ is locally free, then $\mathcal{L}{\otimes }_{{\mathcal{O}}_X}\mathcal{F}$ is exact in $\mathcal{F}$ on ${\mathcal{O}}_X$-Modules.

We shall mostly consider locally free ${\mathcal{O}}_X$-Modules of finite rank; when we speak of locally free sheaves without qualification, of finite rank is to be understood.

(5.4.2) If $\mathcal{L}$, $\mathcal{F}$ are two ${\mathcal{O}}_X$-Modules, there is a canonical functorial homomorphism

(5.4.2.1)    ℒ̌ ⊗_{𝒪_X} ℱ = ℋom_{𝒪_X}(ℒ, 𝒪_X) ⊗_{𝒪_X} ℱ → ℋom_{𝒪_X}(ℒ, ℱ)

defined as follows: for $U$ open and $(u,t)$ with $u\in \Gamma (U,\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{L},{\mathcal{O}}_X))=\mathrm{Hom}(\mathcal{L}|U,{\mathcal{O}}_X|U)$ and $t\in \Gamma (U,\mathcal{F})$, send $(u,t)$ to the element of $\mathrm{Hom}(\mathcal{L}|U,\mathcal{F}|U)$ carrying $s_x\in {\mathcal{L}}_x$ to $u_x(s_x)\cdot t_x\in {\mathcal{F}}_x$. If $\mathcal{L}$ is locally free of finite rank, this map is bijective: the question being local, reduce to $\mathcal{L}={\mathcal{O}}_X^n$; since $\mathcal{H}o{m}_{{\mathcal{O}}_X}({\mathcal{O}}_X^n,\mathcal{G})\cong {\mathcal{G}}^n$ canonically for every ${\mathcal{O}}_X$-Module $\mathcal{G}$, the assertion reduces to $\mathcal{L}={\mathcal{O}}_X$, which is immediate.

(5.4.3) If $\mathcal{L}$ is invertible, so is its dual $\check{\mathcal{L}}=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{L},{\mathcal{O}}_X)$, as one sees locally by reducing to $\mathcal{L}={\mathcal{O}}_X$. Moreover, there is a canonical isomorphism

(5.4.3.1)    ℋom_{𝒪_X}(ℒ, 𝒪_X) ⊗_{𝒪_X} ℒ ⥲ 𝒪_X.

Indeed, by (5.4.2) it suffices to define a canonical isomorphism $\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{L},\mathcal{L})\stackrel{\sim }{\to }{\mathcal{O}}_X$. For any ${\mathcal{O}}_X$-Module $\mathcal{F}$ there is a canonical homomorphism ${\mathcal{O}}_X\to \mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{F})$ (5.3.7); when $\mathcal{F}=\mathcal{L}$ is invertible, this homomorphism is bijective — locally the question reduces to $\mathcal{L}={\mathcal{O}}_X$, which is immediate.

Accordingly, write ${\mathcal{L}}^{-1}=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{L},{\mathcal{O}}_X)$, called the inverse of $\mathcal{L}$. The terminology "invertible sheaf" is motivated as follows for $X$ a single point and ${\mathcal{O}}_X$ a local ring $A$ with maximal ideal $\mathfrak{m}$. If $M$, $M^{\prime }$ are $A$-modules with $M$ of finite type and $M{\otimes }_A{M}^{\prime }\cong A$, then $(A/\mathfrak{m}){\otimes }_A(M{\otimes }_A{M}^{\prime })\cong (M/\mathfrak{m}M){\otimes }_{A/\mathfrak{m}}(M^{\prime }/\mathfrak{m}{M}^{\prime })$; this tensor product of $A/\mathfrak{m}$-vector spaces is $\cong A/\mathfrak{m}$, forcing $M/\mathfrak{m}M$ and $M^{\prime }/\mathfrak{m}{M}^{\prime }$ to be 1-dimensional. For any $z\in M\setminus \mathfrak{m}M$, $M=Az+\mathfrak{m}M$, so $M=Az$ by Nakayama (since $M$ is of finite type). The annihilator of $z$ annihilates $M{\otimes }_A{M}^{\prime }\cong A$, hence is (0); so $M\cong A$. In general, this shows: if $\mathcal{L}$ is an ${\mathcal{O}}_X$-Module of finite type with some ${\mathcal{O}}_X$-Module $\mathcal{F}$ satisfying $\mathcal{L}{\otimes }_{{\mathcal{O}}_X}\mathcal{F}\cong {\mathcal{O}}_X$, and the rings ${\mathcal{O}}_x$ are local, then ${\mathcal{L}}_x\cong {\mathcal{O}}_x$ for every $x\in X$. If ${\mathcal{O}}_X$ and $\mathcal{L}$ are coherent, one concludes that $\mathcal{L}$ is invertible by (5.2.7).

(5.4.4) If $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ are invertible ${\mathcal{O}}_X$-Modules, so is $\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{\mathcal{L}}^{\prime }$ (reducing locally to $\mathcal{L}={\mathcal{L}}^{\prime }={\mathcal{O}}_X$). For $n\ge 1$, write ${\mathcal{L}}^{\otimes n}$ for the tensor product of $n$ copies of $\mathcal{L}$; set ${\mathcal{L}}^{\otimes 0}={\mathcal{O}}_X$ and ${\mathcal{L}}^{\otimes (-n)}=({\mathcal{L}}^{-1}{)}^{\otimes n}$. There is a canonical functorial isomorphism

(5.4.4.1)    ℒ^{⊗m} ⊗_{𝒪_X} ℒ^{⊗n} ⥲ ℒ^{⊗(n+m)}

for any integers m, n: by the definitions one reduces to $m=-1,n=1$, the case treated in (5.4.3).

(5.4.5) Let $f:Y\to X$ be a morphism of ringed spaces. If $\mathcal{L}$ is a locally free (resp. invertible) ${\mathcal{O}}_X$-Module, then $f\ast (\mathcal{L})$ is a locally free (resp. invertible) ${\mathcal{O}}_Y$-Module — using that inverse images of locally isomorphic sheaves are locally isomorphic, that $f\ast$ commutes with finite direct sums, and that $f\ast ({\mathcal{O}}_X)={\mathcal{O}}_Y$ (4.3.4). There is a canonical functorial homomorphism $f\ast (\check{\mathcal{L}})\to \check{f\ast (\mathcal{L})}$ (4.4.6) which is bijective when $\mathcal{L}$ is locally free (reducing again to $\mathcal{L}={\mathcal{O}}_X$). Consequently, for $\mathcal{L}$ invertible, $f\ast ({\mathcal{L}}^{\otimes n})$ is canonically identified with $(f\ast (\mathcal{L}){)}^{\otimes n}$ for every integer $n$.

(5.4.6) Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-Module. Write ${\Gamma }_{\ast }(X,\mathcal{L})$ (or simply ${\Gamma }_{\ast }(\mathcal{L})$) for the abelian group ${\oplus }_{n\in \mathbb{Z}}\Gamma (X,{\mathcal{L}}^{\otimes n})$, made a graded ring by the multiplication sending $(s_n,s_m)\in \Gamma (X,{\mathcal{L}}^{\otimes n})\times \Gamma (X,{\mathcal{L}}^{\otimes m})$ to the section of ${\mathcal{L}}^{\otimes (n+m)}$ corresponding canonically (5.4.4.1) to $s_n\otimes s_m$. Associativity is immediate. Plainly ${\Gamma }_{\ast }(X,\mathcal{L})$ is a covariant functor in $\mathcal{L}$ with values in graded rings.

For any ${\mathcal{O}}_X$-Module $\mathcal{F}$, set

Γ_*(ℒ, ℱ) = ⊕_{n ∈ ℤ} Γ(X, ℱ ⊗_{𝒪_X} ℒ^{⊗n}),

made a graded module over ${\Gamma }_{\ast }(\mathcal{L})$ by sending $(s_n,u_m)\in \Gamma (X,{\mathcal{L}}^{\otimes n})\times \Gamma (X,\mathcal{F}\otimes {\mathcal{L}}^{\otimes m})$ to the section of $\mathcal{F}\otimes {\mathcal{L}}^{\otimes (n+m)}$ corresponding canonically (5.4.4.1) to $s_n\otimes u_m$. The module axioms hold by inspection. For $X,\mathcal{L}$ fixed, ${\Gamma }_{\ast }(\mathcal{L},\mathcal{F})$ is a covariant functor in $\mathcal{F}$ with values in graded ${\Gamma }_{\ast }(\mathcal{L})$-modules; for $X,\mathcal{F}$ fixed, a covariant functor in $\mathcal{L}$.

If $f:Y\to X$ is a morphism of ringed spaces, the canonical homomorphism (4.4.3.2) $\rho :{\mathcal{L}}^{\otimes n}\to f_{\ast }(f\ast ({\mathcal{L}}^{\otimes n}))$ defines $\Gamma (X,{\mathcal{L}}^{\otimes n})\to \Gamma (Y,f\ast ({\mathcal{L}}^{\otimes n}))$; since $f\ast ({\mathcal{L}}^{\otimes n})=(f\ast (\mathcal{L}){)}^{\otimes n}$, the canonical homomorphisms (4.4.3.2) and (5.4.4.1) yield a functorial homomorphism of graded rings ${\Gamma }_{\ast }(\mathcal{L})\to {\Gamma }_{\ast }(f\ast (\mathcal{L}))$. The same canonical homomorphism (4.4.3) similarly yields $\Gamma (X,\mathcal{F}\otimes {\mathcal{L}}^{\otimes n})\to \Gamma (Y,f\ast (\mathcal{F}\otimes {\mathcal{L}}^{\otimes n}))$; using

f*(ℱ ⊗ ℒ^{⊗n}) = f*(ℱ) ⊗ (f*(ℒ))^{⊗n}    (4.3.3.1),

these (for varying $n$) define a di-homomorphism of graded modules ${\Gamma }_{\ast }(\mathcal{L},\mathcal{F})\to {\Gamma }_{\ast }(f\ast (\mathcal{L}),f\ast (\mathcal{F}))$.

(5.4.7) One can show that there is a set $\mathfrak{M}$ (also written $\mathfrak{M}(X)$) of invertible ${\mathcal{O}}_X$-Modules such that every invertible ${\mathcal{O}}_X$-Module is isomorphic to a unique element of $\mathfrak{M}$.¹ Endow $\mathfrak{M}$ with the composition law sending $(\mathcal{L},{\mathcal{L}}^{\prime })$ to the unique element of $\mathfrak{M}$ isomorphic to $\mathcal{L}\otimes {\mathcal{L}}^{\prime }$. With this law, $\mathfrak{M}$ is a group isomorphic to the cohomology group $H^1(X,{\mathcal{O}}_X^{\ast })$, where ${\mathcal{O}}_X^{\ast }$ is the subsheaf of ${\mathcal{O}}_X$ whose sections over $U$ are the invertible elements of $\Gamma (U,{\mathcal{O}}_X)$ (so ${\mathcal{O}}_X^{\ast }$ is a sheaf of multiplicative abelian groups).

For this, note that for every open $U\subset X$, $\Gamma (U,{\mathcal{O}}_X^{\ast })$ is canonically identified with the automorphism group of the $({\mathcal{O}}_X|U)$-Module ${\mathcal{O}}_X|U$: a section $ϵ$ corresponds to the automorphism $u$ with $u_x(s_x)={ϵ}_x\cdot s_x$. Given an open cover $\mathfrak{U}=(U_{\lambda })$ of $X$, giving an automorphism ${\theta }_{\lambda \mu }$ of ${\mathcal{O}}_X|(U_{\lambda }\cap U_{\mu })$ for each pair amounts to giving a 1-cochain on $\mathfrak{U}$ with values in ${\mathcal{O}}_X^{\ast }$; the gluing condition (3.3.1) for the ${\theta }_{\lambda \mu }$ says exactly that the cochain is a cocycle. Likewise, giving an automorphism ${\omega }_{\lambda }$ of ${\mathcal{O}}_X|U_{\lambda }$ for each $\lambda$ is a 0-cochain, and its coboundary is the family $({\omega }_{\lambda }|U_{\lambda }\cap U_{\mu })\circ ({\omega }_{\mu }|U_{\lambda }\cap U_{\mu }{)}^{-1}$. To each 1-cocycle of $\mathfrak{U}$ we associate the element of $\mathfrak{M}$ isomorphic to the invertible ${\mathcal{O}}_X$-Module obtained by gluing via the $({\theta }_{\lambda \mu })$; cohomologous cocycles yield the same element of $\mathfrak{M}$ (3.3.2). This defines ${\varphi }_{\mathfrak{U}}:H^1(\mathfrak{U},{\mathcal{O}}_X^{\ast })\to \mathfrak{M}$. If $\mathfrak{V}$ refines $\mathfrak{U}$, the diagram

H¹(𝔘, 𝒪_X^*) ──φ_𝔘──→ 𝔐
     │              ↗
     ↓           φ_𝔙
H¹(𝔙, 𝒪_X^*)

(left arrow the canonical homomorphism (G, II, 5.7)) commutes, by (3.3.3). Passage to the inductive limit gives $H^1(X,{\mathcal{O}}_X^{\ast })\to \mathfrak{M}$, since Čech ${\check{H}}^1$ agrees with derived $H^1$ here (G, II, 5.9, Cor. of Th. 5.9.1). This map is surjective: every invertible $\mathcal{L}$ is obtained by gluing copies of ${\mathcal{O}}_X|U_{\lambda }$ for some cover. It is injective: this reduces to injectivity of each ${\varphi }_{\mathfrak{U}}$, which is (3.3.2). To see it is a group homomorphism: for invertible $\mathcal{L},{\mathcal{L}}^{\prime }$, choose a cover $(U_{\lambda })$ and $a_{\lambda }\in \Gamma (U_{\lambda },\mathcal{L})$, $a_{\lambda }^{\prime }\in \Gamma (U_{\lambda },{\mathcal{L}}^{\prime })$ with $\Gamma (U_{\lambda },\mathcal{L})=\Gamma (U_{\lambda },{\mathcal{O}}_X)\cdot a_{\lambda }$, $\Gamma (U_{\lambda },{\mathcal{L}}^{\prime })=\Gamma (U_{\lambda },{\mathcal{O}}_X)\cdot a_{\lambda }^{\prime }$. The corresponding cocycles $({ϵ}_{\lambda \mu })$, $({ϵ}_{\lambda \mu }^{\prime })$ satisfy $s_{\lambda }\cdot a_{\lambda }=s_{\mu }\cdot a_{\mu }$ iff $s_{\lambda }={ϵ}_{\lambda \mu }{s}_{\mu }$ (similarly for ${\mathcal{L}}^{\prime }$). Sections of $\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{\mathcal{L}}^{\prime }$ over $U_{\lambda }$ are finite sums of $s_{\lambda }{s}_{\lambda }^{\prime }\cdot (a_{\lambda }\otimes a_{\lambda }^{\prime })$; hence $({ϵ}_{\lambda \mu }{ϵ}_{\lambda \mu }^{\prime })$ is the cocycle of $\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{\mathcal{L}}^{\prime }$, completing the proof.²

(5.4.8) Let $f=(\psi ,\omega ):Y\to X$ be a morphism of ringed spaces. The functor $f\ast (\mathcal{L})$ on invertible ${\mathcal{O}}_X$-Modules defines (by abuse of language, still written $f\ast$) a map $\mathfrak{M}(X)\to \mathfrak{M}(Y)$. There is also a canonical homomorphism (T, 3.2.2)

(5.4.8.1)    H¹(X, 𝒪_X^*) → H¹(Y, 𝒪_Y^*).

Under the canonical identifications (5.4.7) $\mathfrak{M}(X)\cong H^1(X,{\mathcal{O}}_X^{\ast })$ (resp. $\mathfrak{M}(Y)\cong H^1(Y,{\mathcal{O}}_Y^{\ast })$), the homomorphism (5.4.8.1) is identified with $f\ast$. Indeed, if $\mathcal{L}$ comes from a cocycle $({ϵ}_{\lambda \mu })$ for a cover $(U_{\lambda })$ of $X$, it suffices to show $f\ast (\mathcal{L})$ comes from a cocycle whose cohomology class is the image under (5.4.8.1) of that of $({ϵ}_{\lambda \mu })$. If ${\theta }_{\lambda \mu }$ is the automorphism corresponding to ${ϵ}_{\lambda \mu }$, then $f\ast (\mathcal{L})$ is obtained by gluing the ${\mathcal{O}}_Y|{\psi }^{-1}(U_{\lambda })$ via the $f\ast ({\theta }_{\lambda \mu })$; one checks these correspond to the cocycle $({\omega }^{♯}({ϵ}_{\lambda \mu }))$ (after identifying ${ϵ}_{\lambda \mu }$ with its canonical image under $\rho$ (3.7.2), a section of $\psi \ast ({\mathcal{O}}_X^{\ast })$ over ${\psi }^{-1}(U_{\lambda }\cap U_{\mu })$).

(5.4.9) Let $\mathcal{E}$, $\mathcal{F}$ be ${\mathcal{O}}_X$-Modules with $\mathcal{F}$ locally free, and let $\mathcal{G}$ be an extension of $\mathcal{F}$ by $\mathcal{E}$ — an exact sequence $0\to \mathcal{E}\stackrel{j}{\to }\mathcal{G}\stackrel{p}{\to }\mathcal{F}\to 0$. Then for every $x\in X$ there is an open $U\ni x$ such that $\mathcal{G}|U\cong (\mathcal{E}|U)\oplus (\mathcal{F}|U)$. One reduces to $\mathcal{F}={\mathcal{O}}_X^n$; let $e_i$ ($1\le i\le n$) be the canonical sections of ${\mathcal{O}}_X^n$ (5.5.5). There is an open $U$ and sections $s_i\in \Gamma (U,\mathcal{G})$ with $p(s_i|U)=e_i|U$ ($1\le i\le n$). Let $f:\mathcal{F}|U\to \mathcal{G}|U$ be the homomorphism defined by the $s_i|U$ (5.1.1). For open $V\subset U$ and $s\in \Gamma (V,\mathcal{G})$, $s-f(p(s))\in \Gamma (V,\mathcal{E})$, whence the assertion.

(5.4.10) Let $f:X\to Y$ be a morphism of ringed spaces, $\mathcal{F}$ an ${\mathcal{O}}_X$-Module, and $\mathcal{L}$ a locally free ${\mathcal{O}}_Y$-Module of finite rank. There is a canonical isomorphism

(5.4.10.1)    f_*(ℱ) ⊗_{𝒪_Y} ℒ ⥲ f_*(ℱ ⊗_{𝒪_X} f*(ℒ)).

Indeed, for any ${\mathcal{O}}_Y$-Module $\mathcal{L}$ there is a canonical homomorphism

f_*(ℱ) ⊗_{𝒪_Y} ℒ ──1⊗ρ──→ f_*(ℱ) ⊗_{𝒪_Y} f_*(f*(ℒ)) ──α──→ f_*(ℱ ⊗_{𝒪_X} f*(ℒ))

($\rho$ from (4.4.3.2), $\alpha$ from (4.2.2.1)). To see this is an isomorphism when $\mathcal{L}$ is locally free, the question being local on $Y$, reduce to $\mathcal{L}={\mathcal{O}}_Y^n$; since $f_{\ast }$ and $f\ast$ commute with finite direct sums, reduce to $n=1$, where the assertion follows immediately from the definitions and $f\ast ({\mathcal{O}}_Y)={\mathcal{O}}_X$.

5.5. Sheaves on a space ringed in local rings

(5.5.1) A ringed space $(X,{\mathcal{O}}_X)$ is ringed in local rings if for every $x\in X$, ${\mathcal{O}}_x$ is a local ring; these are by far the most common ringed spaces we shall encounter. We then write ${\mathfrak{m}}_x$ for the maximal ideal of ${\mathcal{O}}_x$ and $k(x)$ for the residue field ${\mathcal{O}}_x/{\mathfrak{m}}_x$. For an ${\mathcal{O}}_X$-Module $\mathcal{F}$, an open $U\subset X$, a point $x\in U$, and a section $f\in \Gamma (U,\mathcal{F})$, we write $f(x)$ for the class of the germ $f_x\in {\mathcal{F}}_x$ mod ${\mathfrak{m}}_x{\mathcal{F}}_x$, and call it the value of $f$ at $x$. The relation $f(x)=0$ thus means $f_x\in {\mathfrak{m}}_x{\mathcal{F}}_x$; when it holds, we say (by abuse of language) that $f$ vanishes at $x$. Do not confuse this with $f_x=0$.

(5.5.2) Let $X$ be a space ringed in local rings, $\mathcal{L}$ an invertible ${\mathcal{O}}_X$-Module, and $f$ a section of $\mathcal{L}$ over $X$. At a point $x\in X$, the following are equivalent:

a) $f_x$ is a generator of ${\mathcal{L}}_x$; b) $f_x\notin {\mathfrak{m}}_x{\mathcal{L}}_x$ (i.e., $f(x)\ne >0$); c) There is a section $g$ of ${\mathcal{L}}^{-1}$ over some open $V\ni x$ such that the canonical image of $f>\otimes g$ in $\Gamma (V,{\mathcal{O}}_X)$ (5.4.3) is the unit section.

Indeed, the question being local, reduce to $\mathcal{L}={\mathcal{O}}_X$. Equivalence of a) and b) is then immediate, and c) ⟹ b). For b) ⟹ c): if $f_x\notin {\mathfrak{m}}_x$, $f_x$ is invertible in ${\mathcal{O}}_x$, say $f_x\cdot g_x=1_x$; by the definition of germs, there is a neighborhood $V\ni x$ and a section $g$ over $V$ with $fg=1$ on $V$.

By c), the set $X_f$ of $x$ satisfying the equivalent conditions a)–c) is open in $X$; in the terminology of (5.5.1), it is the set of $x$ where $f$ does not vanish.

(5.5.3) Under the hypotheses of (5.5.2), let ${\mathcal{L}}^{\prime }$ be a second invertible ${\mathcal{O}}_X$-Module. For $f\in \Gamma (X,\mathcal{L})$, $g\in \Gamma (X,{\mathcal{L}}^{\prime })$,

X_f ∩ X_g = X_{f ⊗ g}.

Reduce locally to $\mathcal{L}={\mathcal{L}}^{\prime }={\mathcal{O}}_X$; then $f\otimes g$ is canonically identified with the product fg, and the assertion is obvious.

(5.5.4) Let $\mathcal{F}$ be a locally free ${\mathcal{O}}_X$-Module of rank $n$. Then ${\bigwedge }^p\mathcal{F}$ is locally free of rank (n choose p) if $p\le n$, and zero if $p>n$ — the question being local, reduce to $\mathcal{F}={\mathcal{O}}_X^n$. For every $x\in X$, $({\bigwedge }^p\mathcal{F}{)}_x/{\mathfrak{m}}_x({\bigwedge }^p\mathcal{F}{)}_x$ is a $k(x)$-vector space of dimension (n choose p), canonically identified with ${\bigwedge }^p({\mathcal{F}}_x/{\mathfrak{m}}_x{\mathcal{F}}_x)$. For sections $s_1,\cdots ,s_p$ of $\mathcal{F}$ over an open $U$, set $s=s_1\wedge \cdots \wedge s_p\in \Gamma (U,{\bigwedge }^p\mathcal{F})$ (4.1.5); then $s(x)=s_1(x)\wedge \cdots \wedge s_p(x)$, so $s_1(x),\cdots ,s_p(x)$ are linearly dependent iff $s(x)=0$. Hence the set of $x\in X$ at which $s_1(x),\cdots ,s_p(x)$ are linearly independent is open in $X$: reducing to $\mathcal{F}={\mathcal{O}}_X^n$ and applying (5.5.2) to the image of $s$ under a projection of ${\bigwedge }^p\mathcal{F}={\mathcal{O}}_X^{(nchoosep)}$ onto one of its (n choose p) factors.

In particular, if $s_1,\cdots ,s_n$ are $n$ sections of $\mathcal{F}$ over $U$ with $s_1(x),\cdots ,s_n(x)$ linearly independent at every $x\in U$, then the homomorphism $u:{\mathcal{O}}_X^n|U\to \mathcal{F}|U$ defined by the $s_i$ (5.1.1) is an isomorphism: reduce to $\mathcal{F}={\mathcal{O}}_X^n$ and identify ${\bigwedge }^n\mathcal{F}$ with ${\mathcal{O}}_X$; then $s=s_1\wedge \cdots \wedge s_n$ is an invertible section of ${\mathcal{O}}_X$ over $U$, and an inverse to $u$ is given by Cramer's rule.

(5.5.5) Let $\mathcal{E}$, $\mathcal{F}$ be locally free ${\mathcal{O}}_X$-Modules (of finite rank), and let $u:\mathcal{E}\to \mathcal{F}$ be a homomorphism. For there to exist an open $U\ni x$ such that $u|U$ is injective and $\mathcal{F}|U$ is the direct sum of $u(\mathcal{E})|U$ and a locally free sub-$({\mathcal{O}}_X|U)$-Module, it is necessary and sufficient that $u_x:{\mathcal{E}}_x\to {\mathcal{F}}_x$ induce an injective $k(x)$-linear map ${\mathcal{E}}_x/{\mathfrak{m}}_x{\mathcal{E}}_x\to {\mathcal{F}}_x/{\mathfrak{m}}_x{\mathcal{F}}_x$. The condition is necessary: ${\mathcal{F}}_x$ is then the direct sum of the free ${\mathcal{O}}_x$-modules $u_x({\mathcal{E}}_x)$ and ${\mathcal{G}}_x$, so ${\mathcal{F}}_x/{\mathfrak{m}}_x{\mathcal{F}}_x$ is the direct sum of $u_x({\mathcal{E}}_x)/{\mathfrak{m}}_x{u}_x({\mathcal{E}}_x)$ and ${\mathcal{G}}_x/{\mathfrak{m}}_x{\mathcal{G}}_x$. The condition is sufficient: reduce to $\mathcal{E}={\mathcal{O}}_X^m$; let $s_1,\cdots ,s_m$ be the images by $u$ of the canonical sections $e_i$ of ${\mathcal{O}}_X^m$ (the $e_i$ are the sections with $(e_i{)}_y$ equal to the $i$-th element of the canonical basis of ${\mathcal{O}}_y^m$ for $y\in X$ — canonical sections of ${\mathcal{O}}_X^m$). By hypothesis $s_1(x),\cdots ,s_m(x)$ are linearly independent, so if $\mathcal{F}$ has rank $n$, there are $n-m$ sections $s_{m+1},\cdots ,s_n$ of $\mathcal{F}$ over some neighborhood $V$ of $x$ with $s_i(x)$ ($1\le i\le n$) a basis of ${\mathcal{F}}_x/{\mathfrak{m}}_x{\mathcal{F}}_x$. By (5.5.4) there is an open $U\subset V$ with the $s_i(y)$ ($1\le i\le n$) a basis of ${\mathcal{F}}_y/{\mathfrak{m}}_y{\mathcal{F}}_y$ for every $y\in U$. By (5.5.4), there is an isomorphism $\mathcal{F}|U\stackrel{\sim }{\to }{\mathcal{O}}_X^n|U$ sending $s_i|U$ ($1\le i\le m$) to $e_i|U$, finishing the proof.