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).

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 $1$.

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 $f$.

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

We define

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

Let $A$ be a ring and $M$ an $A$-module. Then setting 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).

Let $A$ be a ring, and let $M$, $N$ be $A$-modules. Then the maps given by and, in the other direction, 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.

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 for suitable (possibly infinite) index sets $I$, $J$.

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.

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.

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.

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 $f^*(\widetilde{M}) = \widetilde{M\otimes _{A}B}$.

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 (or resp.).

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 $.

Let $X$ be a ringed space, and let ${\mathscr F}$ be an ${\mathscr O}_X$-module of finite type. Then the support of ${\mathscr F}$ is a closed subset of $X$.

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.

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.

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 $f^\sharp _x\colon {\mathscr O}_{Y, f(x)}\to {\mathscr O}_{X, x}$).