1 ${\mathscr O}_X$-modules

April 8,

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

\[ {\mathscr O}_X(U) \times {\mathscr F}(U) \to {\mathscr F}(U)\qquad \text{for each open} U\subseteq X \]

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
If ${\mathscr F}$ is an ${\mathscr O}_X$-module and $x\in X$, then the stalk ${\mathscr F}_x$ carries a natural ${\mathscr O}_{X,x}$-module structure. The $\kappa (x)$-vector space ${\mathscr F}(x):= {\mathscr F}_x\otimes _{{\mathscr O}_{X_x}}\kappa (x)$ is called the fiber of ${\mathscr F}$ over $x$.

Constructions, examples 1.3
Let $X$ be a ringed space, ${\mathscr F}$ an ${\mathscr O}_X$-module.
  1. ${\mathscr O}_X$,

  2. submodules and quotients,

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

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

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

  6. $\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 ${\mathscr F}$

  1. free, if it is isomorphic to $\bigoplus _{i\in I}{\mathscr O}_X$ for some set $I$,

  2. 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
Let $f\colon X\to Y$ be a morphism of ringed spaces, and let ${\mathscr F}$ be an ${\mathscr O}_X$-module. Then $f_* {\mathscr F}$ carries a natural ${\mathscr O}_Y$-module structure and is called the 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

\[ f^*{\mathscr F}:= f^{-1}{\mathscr F}\otimes _{f^{-1}{\mathscr O}_Y}{\mathscr O}_X. \]

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
Let $f\colon X\to Y$ be a morphism of ringed spaces. The functors $f_*$ is right adjoint to the functor $f^*$:
\[ \operatorname{Hom}_{{\mathscr O}_X}(f^*{\mathscr G}, {\mathscr F}) \cong \operatorname{Hom}_{{\mathscr O}_Y}({\mathscr G}, f_*{\mathscr F}) \]
for all ${\mathscr O}_X$-modules ${\mathscr F}$, all ${\mathscr O}_Y$-modules ${\mathscr G}$, functorially in ${\mathscr F}$ and ${\mathscr G}$.

April 10,

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


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

Definition 1.8
Let $A$ be a ring and $M$ an $A$-module. Then setting
\[ D(f) \mapsto M_f,\quad f\in A, \]
is well-defined and defines a sheaf on the basis of principal open sets in $\operatorname{Spec}A$. We denote the corresponding sheaf on $\operatorname{Spec}A$ by $\widetilde{M}$. It is an ${\mathscr O}_{\operatorname{Spec}A}$-module (by viewing each $M_f$ as an $A_f$-module in the natural way).

Remark 1.9
For an affine scheme $X$, in general not every ${\mathscr O}_X$-module has the above form. We will investigate this more closely soon.

Proposition 1.10
Let $A$ be a ring, and let $M$, $N$ be $A$-modules. Then the maps
\[ \operatorname{Hom}_A(M, N) \to \operatorname{Hom}_{{\mathscr O}_{\operatorname{Spec}A}}(\widetilde{M}, \widetilde{N}) \]
given by
\[ \varphi \mapsto \widetilde{f} := {(\varphi _f\colon M_f \to N_f)}_f \]
and, in the other direction,
\[ \Phi \mapsto \Gamma (\operatorname{Spec}A, \Phi ), \]
are inverse to each other. In other words, $\widetilde{\cdot }$ is a fully faithful functor from the category of $A$-modules to the category of ${\mathscr O}_{\operatorname{Spec}A}$-modules.

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
Let $X$ be a ringed space. An ${\mathscr O}_X$-module ${\mathscr F}$ is called quasi-coherent, if every $x\in X$ has an open neighborhood $U$ such that there exists an exact sequence
\[ {\mathscr O}_U^{(J)} \to {\mathscr O}_U^{(I)} \to {\mathscr F}_{|U} \to 0 \]
for suitable (possibly infinite) index sets $I$, $J$.

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

\[ {\Gamma (X, {\mathscr F})}_f \to \Gamma (X_f, {\mathscr F}) \]

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

Theorem 1.12
Let $X$ be a scheme and ${\mathscr F}$ an ${\mathscr O}_X$-module. The following are equivalent:
  1. For every affine open $\operatorname{Spec}A = U \subseteq X$, there exists an $A$-module $M$ such that ${\mathscr F}_{|U} \cong \widetilde{M}$.

  2. 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 $i$.

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

  4. 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,

Corollary 1.13
Let $A$ be a ring, $X=\operatorname{Spec}A$. The functor $\widetilde{\cdot }$ induces an exact equivalence between the categories of $A$-modules and of quasi-coherent ${\mathscr O}_X$-modules.

Corollary 1.14
Let $X$ be a scheme.
  1. Kernels, cokernels, images of ${\mathscr O}_X$-module homomorphisms between quasi-coherent ${\mathscr O}_X$-modules are quasi-coherent.

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

  3. 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

    \[ \Gamma (U, {\mathscr F}\otimes {\mathscr G}) = \Gamma (U, {\mathscr F}) \otimes \Gamma (U, {\mathscr G}). \]
In particular, by (1) and (2) the category of quasi-coherent ${\mathscr O}_X$-module is an abelian category, and the inclusion functor into the category of all ${\mathscr O}_X$-modules preserves kernels and cokernels and direct sums.

(1.5) Direct and inverse image of quasi-coherent ${\mathscr O}_X$-module
Proposition 1.15
Let $X=\operatorname{Spec}B$, $Y=\operatorname{Spec}A$ be affine schemes, and let $f\colon X\to Y$ be a scheme morphism.
  1. Let $N$ be an $B$-module, then $f_*(\widetilde{N}) = \widetilde{N_{[A]}}$ where $N_{[A]}$ is $N$, considered as an $A$-module via $\Gamma (f)\colon A\to B$.

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

(1.6) Finiteness conditions

Definition 1.16
We say that an ${\mathscr O}_X$-module ${\mathscr F}$ is 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
\[ {\mathscr O}_X^n\to {\mathscr F}\to 0 \]
\[ {\mathscr O}_X^m\to {\mathscr O}_X^n\to {\mathscr F}\to 0, \]

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
Let $X$ be a ringed space and let ${\mathscr F}$ be an ${\mathscr O}_X$-module of finite presentation.
  1. 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.

  2. 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 $.

Problem 2.

Proposition 1.18
Let $X$ be a ringed space, and let ${\mathscr F}$ be an ${\mathscr O}_X$-module of finite type. Then the support
\[ \operatorname{Supp}({\mathscr F}) =\{ x\in X;\ {\mathscr F}_x\ne 0\} \]
of ${\mathscr F}$ is a closed subset of $X$.

Problem 6.

(1.7) Closed subschemes and quasi-coherent ideal sheaves

Proposition 1.19
Let $X$ be a scheme. An ideal sheaf ${\mathscr I}\subseteq {\mathscr O}_X$ defines a closed subscheme if and only if ${\mathscr I}$ is a quasi-coherent ${\mathscr O}_X$-module.

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,

(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
Let $A$ be a ring and $M$ an $A$-module. Consider the following properties of $M$:
  1. $\widetilde{M}$ is a locally free ${\mathscr O}_{\operatorname{Spec}A}$-module.

  2. $M$ 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.

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

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

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

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

Part (1) is easy. Part (2) is more difficult. The implication (iv) $\Rightarrow $ (iii) follows from Prop. 1.17. See  [ GW ]  Prop. 7.40.

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
  1. 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.)

  2. 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.

  3. 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?

  4. Let $A$ be a noetherian unique factorization domain. Then every invertible sheaf on $\operatorname{Spec}A$ is free.

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

  6. 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.)