# 1 ${\mathscr O}_X$-modules

April 8,

2019

General references: [ GW ] Ch. 7, [ H ] II.5.

## Definition and basic properties

**(1.1) Definition of ${\mathscr O}_X$-modules**

**Definition 1.1**

Let $(X, {\mathscr O}_X)$ be a ringed space. An ${\mathscr O}_X$-module is a sheaf ${\mathscr F}$ of abelian groups on $X$ together with maps

giving each ${\mathscr F}(U)$ the structure of an ${\mathscr O}_X(U)$-module, and which are compatible with the restriction maps for open subsets $U’\subseteq U \subseteq X$.

An ${\mathscr O}_X$-module homomorphism ${\mathscr F}\to {\mathscr G}$ between ${\mathscr O}_X$-modules ${\mathscr F}$, ${\mathscr G}$ on $X$ is a sheaf morphism ${\mathscr F}\to {\mathscr G}$ such that for all open subsets $U\subseteq X$, the map ${\mathscr F}(U)\to {\mathscr G}(U)$ is a homomorphism of ${\mathscr O}_X(U)$-modules. We denote the set of ${\mathscr O}_X$-module homomorphisms from ${\mathscr F}$ to ${\mathscr G}$ by $\operatorname{Hom}_{{\mathscr O}_X}({\mathscr F}, {\mathscr G})$; this is an ${\mathscr O}_X(X)$-module (and in particular an abelian group).

We obtain the category (${\mathscr O}_X$-Mod) of ${\mathscr O}_X$-modules.

**Remark 1.2**

*fiber of ${\mathscr F}$ over $x$*.

**Constructions, examples 1.3**

${\mathscr O}_X$,

submodules and quotients,

$\oplus $, $\prod $, $\otimes $, (filtered) inductive limits,

kernels, cokernels, image, exactness; these are compatible with passing to the stalks,

restriction to open subsets: ${\mathscr F}_{X|U}$, $U\subseteq X$ open,

$\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits $, $-^\vee $,

The category of ${\mathscr O}_X$-modules is an abelian category.

**Definition 1.4**

Let ${\mathscr F}$ be an ${\mathscr O}_X$-module on the ringed space $X$. We call ${\mathscr F}$

*free*, if it is isomorphic to $\bigoplus _{i\in I}{\mathscr O}_X$ for some set $I$,*locally free*, if there exists an open covering $X=\bigcup _j U_j$ of $X$ such that ${\mathscr F}_{|U_j}$ is a free ${\mathscr O}_{U_j}$-module for each $j$.

The *rank* of a free ${\mathscr O}_X$-module is the cardinality of $I$ as above (we usually regard it in $\mathbb {Z}\cup \{ \infty \} $, without making a distinction between infinite cardinals). The *rank* of a locally free ${\mathscr O}_X$-module is a function $X\to \mathbb {Z}\cup \{ \infty \} $ which is locally constant on $X$ (i.e., on each connected component of $X$, there is an integer giving the rank).

An *invertible sheaf* or *line bundle* on $X$ is a locally free sheaf of rank $1$.

For ${\mathscr L}$ invertible, there is a natural isomorphism ${\mathscr L}\otimes _{{\mathscr O}_X}{\mathscr L}^\vee \cong {\mathscr O}_X$ (whence the name), cf. Problem 1. Hence $\otimes $ induces a group structure on the set of isomorphism classes of invertible sheaves in $X$. The resulting group is called the Picard group of $X$ and denoted by $\operatorname{Pic}(X)$.

**(1.2) Inverse image**

**Definition 1.5**

*direct image*or

*push-forward*of ${\mathscr F}$ under $f$.

**Definition 1.6**

Let $f\colon X\to Y$ be a morphism of ringed spaces, ${\mathscr F}$ an ${\mathscr O}_Y$-module.

We define

For $x\in X$, we have ${(f^*{\mathscr F})}_x \cong {\mathscr F}_{f(x)} \otimes _{{\mathscr O}_{Y, f(x)}}{\mathscr O}_{X, x}$.

We obtain functors $f_*$, $f^*$ between the categories of ${\mathscr O}_X$-modules and ${\mathscr O}_Y$-modules.

**Proposition 1.7**

April 10,

2019

## Quasi-coherent ${\mathscr O}_X$-modules

**(1.3) The ${\mathscr O}_{\operatorname{Spec}A}$-module attached to an $A$-module $M$**

**Definition 1.8**

**Remark 1.9**

**Proposition 1.10**

By applying the proposition to $M=A$, we also see that for an $A$-module $N$, $\widetilde{N}$ is zero if and only if $N$ is zero.

The construction $M\mapsto \widetilde{M}$ is compatible with exactness, kernels, cokernels, images, direct sums, filtered inductive limits. (Cf. [ GW ] Prop. 7.14 for a more precise statement.)

**(1.4) Quasi-coherent modules**

**Definition 1.11**

*quasi-coherent*, if every $x\in X$ has an open neighborhood $U$ such that there exists an exact sequence

For a morphism $f\colon X\to Y$ of ringed spaces and a quasi-coherent ${\mathscr O}_Y$-module ${\mathscr G}$, the pull-back $f^*{\mathscr G}$ is a quasi-coherent ${\mathscr O}_X$-module (since $f^{-1}$ is exact and tensor product is a right exact functor). The direct image $f_*$ preserves the property of quasi-coherence (only) under certain conditions.

Locally free ${\mathscr O}_X$-modules are quasi-coherent.

Clearly, for a ring $A$ and an $A$-module $M$, $\widetilde{M}$ is a quasi-coherent ${\mathscr O}_{\operatorname{Spec}A}$-module. We will see below that the converse is true as well:

For a ringed space $X$ and $f\in \Gamma (X, {\mathscr O}_X)$, we write $X_f := \{ x\in X;\ f_x \in {\mathscr O}_{X,x}^\times \} $, an open subset of $X$. We obtain a homomorphism

for every ${\mathscr O}_X$-module ${\mathscr F}$.

**Theorem 1.12**

For every affine open $\operatorname{Spec}A = U \subseteq X$, there exists an $A$-module $M$ such that ${\mathscr F}_{|U} \cong \widetilde{M}$.

There exists a covering $X = \bigcup _i U_i$ by affine open subschemes $U_i = \operatorname{Spec}A_i$ and $A_i$-modules $M_i$ such that ${\mathscr F}_{|U_i} \cong \widetilde{M_i}$ for all $i$.

The ${\mathscr O}_X$-module ${\mathscr F}$ is quasi-coherent.

For every affine open $\operatorname{Spec}A = U \subseteq X$ and every $f\in A$, the homomorphism ${\Gamma (U, {\mathscr F})}_f \to \Gamma (D(f), {\mathscr F})$ is an isomorphism.

April 15,

2019

**Corollary 1.13**

**Corollary 1.14**

Kernels, cokernels, images of ${\mathscr O}_X$-module homomorphisms between quasi-coherent ${\mathscr O}_X$-modules are quasi-coherent.

Direct sums of quasi-coherent ${\mathscr O}_X$-modules are quasi-coherent.

Let ${\mathscr F}$, ${\mathscr G}$ be quasi-coherent ${\mathscr O}_X$-module. Then ${\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr G}$ is quasi-coherent, and for every affine open $U\subseteq X$ we have

\[ \Gamma (U, {\mathscr F}\otimes {\mathscr G}) = \Gamma (U, {\mathscr F}) \otimes \Gamma (U, {\mathscr G}). \]

**(1.5) Direct and inverse image of quasi-coherent ${\mathscr O}_X$-module**

**Proposition 1.15**

Let $N$ be an $B$-module, then $f_*(\widetilde{N}) = \widetilde{N_{[A]}}$ where $N_{[A]}$ is $N$, considered as an $A$-module via $\Gamma (f)\colon A\to B$.

Let $M$ be an $A$-module, then $f^*(\widetilde{M}) = \widetilde{M\otimes _{A}B}$.

**(1.6) Finiteness conditions**

**Definition 1.16**

*of finite type*(or

*of finite presentation*, resp.), if every $x\in X$ has an open neighborhood $U\subseteq X$ such that there exists $n\ge 0$ (or $m, n\ge 0$, resp.) and a short exact sequence

On an affine scheme, this coincides with the corresponding definitions in terms of modules (via $M\mapsto \widetilde{M}$). Note that every ${\mathscr O}_X$-module of finite presentation is quasi-coherent. On a noetherian scheme, every quasi-coherent ${\mathscr O}_X$-module of finite type is of finite presentation.

**Proposition 1.17**

For all $x \in X$ and for each ${\mathscr O}_X$-module ${\mathscr G}$, the canonical homomorphism of ${\mathscr O}_{X,x}$-modules

\[ {\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr F},{\mathscr G})}_x \to \operatorname{Hom}_{{\mathscr O}_{X,x}}({\mathscr F}_x,{\mathscr G}_x) \]is bijective.

Let ${\mathscr F}$ and ${\mathscr G}$ be ${\mathscr O}_X$-modules of finite presentation. Let $x \in X$ be a point and let $\theta \colon {\mathscr F}_x \overset {\sim }{\to }{\mathscr G}_x$ be an isomorphism of ${\mathscr O}_{X,x}$-modules. Then there exists an open neighborhood $U$ of $x$ and an isomorphism $u\colon {\mathscr F}{}_{\vert }{}_{U} \overset {\sim }{\to }{\mathscr G}{}_{\vert }{}_{U}$ of ${\mathscr O}_U$-modules with $u_x = \theta $.

**Proposition 1.18**

**(1.7) Closed subschemes and quasi-coherent ideal sheaves**

**Proposition 1.19**

We hence obtain an inclusion-reversing bijection between the set of closed subschemes of a scheme $X$ and the set of quasi-coherent ideal sheaves in ${\mathscr O}_X$, mapping

a quasi-coherent ideal sheaf ${\mathscr I}$ to $Z:= (\operatorname{Supp}({\mathscr O}_X/{\mathscr I}), i^{-1}({\mathscr O}_X/{\mathscr I}))$, where $i\colon \operatorname{Supp}({\mathscr O}_X/{\mathscr I}) \to X$ denotes the inclusion,

a closed subscheme $Z\subseteq X$ to $\operatorname{Ker}({\mathscr O}_X \to i_*{\mathscr O}_Z)$, where $i\colon Z\to X$ denotes the inclusion morphism.

We denote the closed subscheme corresponding to a quasi-coherent ideal sheaf ${\mathscr I}$ by $V({\mathscr I})$.

April 17,

2019

**(1.8) Locally free sheaves on affine schemes**

There is an obvious “commutative algebra way” of writing down, for an $A$-module $M$, the condition that $\widetilde{M}$ is locally free.

**Theorem 1.20**

$\widetilde{M}$ is a locally free ${\mathscr O}_{\operatorname{Spec}A}$-module.

$M$ is locally free, i.e., there exist $f_1, \dots , f_n\in A$ generating the unit ideal such that for all $i$, the $A_{f_i}$-module $M_{f_i}$ is free.

For all ${\mathfrak p}\in \operatorname{Spec}A$, the $A_{{\mathfrak p}}$-module $M_{{\mathfrak p}}$ is free.

The $A$-module $M$ is flat.

We have the implications (i) $\Leftrightarrow $ (ii) $\Rightarrow $ (iii) $\Rightarrow $ (iv).

If $M$ is an $A$-module of finite presentation, then all the three properties are equivalent.

There is an obvious analogous theorem for ${\mathscr O}_X$-module on a scheme $X$, where we define

**Definition 1.21**

Let $X$ be a scheme. An ${\mathscr O}_X$-module ${\mathscr F}$ is called *flat*, if for all $x\in X$ the stalk ${\mathscr F}_x$ is a flat ${\mathscr O}_{X,x}$-module.

More generally, given an ${\mathscr O}_X$-module ${\mathscr F}$ and a morphism $f\colon X\to Y$ we say that ${\mathscr F}$ is $f$-flat or flat over $Y$, if for all $x\in X$ the stalk ${\mathscr F}_x$ is a flat ${\mathscr O}_{Y, f(x)}$-module (via $f^\sharp _x\colon {\mathscr O}_{Y, f(x)}\to {\mathscr O}_{X, x}$).

If $A$ is a domain, then every flat $A$-module $M$ is torsion-free (i.e., multiplication by $s$ is injective for all $s\in A\setminus \{ 0\} )$. The converse holds only rarely; it does hold if $A$ is a principal ideal domain and $M$ is finitely generated.

**Remark 1.22**

Let $A$ be a principal ideal domain. Then every finitely generated locally free (in the sense of condition (i$’$) in the theorem) $A$-module is free. (Use the structure theorem for finitely generated modules over principal ideal domains.)

It is a difficult theorem (conjectured by Serre, proved independently by Quillen and Suslin) that every locally free sheaf of finite type on $\mathbb {A}^n_k$, $k$ a field, is free. The same statement holds even for $k$ a discrete valuation ring.

It will not be relevant in the course, but to complete the picture (and since I did not remember the correct statement during the lecture …), let us remark that one can show that in the previous two items the hypothesis

*of finite type*can be omitted. In fact, whenever $R$ is a ring which is noetherian and such that $\operatorname{Spec}R$ is connected, then every locally free $R$-module which is*not finitely generated*is free. One way to show this to combine the paper [ Ba ] by H. Bass with the difficult theorem that the property of a module of being “projective” can be checked Zariski-locally on $\operatorname{Spec}A$ ( [ Stacks ] 058B), which shows that all locally free $R$-modules, finitely generated or not, are projective. Maybe there is also a more direct way, without talking about projective modules?Let $A$ be a noetherian unique factorization domain. Then every invertible sheaf on $\operatorname{Spec}A$ is free.

See the answers to this question (mathoverflow.net/q/54356) for examples of non-free locally free modules over $\operatorname{Spec}A$ for factorial (and even, in addition, regular) noetherian rings $A$.

Let $A$ be a domain, and let $M$ be a locally free $A$-module of rank $1$. Then $M$ is isomorphic to a

*fractional ideal*, i.e., to a finitely generated sub-$A$-module of $K:=\operatorname{Frac}(A)$. (Cf. Problem 8 for a converse statement in the case that $A$ is a Dedekind domain.)