17.10 Quasi-coherent modules
In this section we introduce an abstract notion of quasi-coherent $\mathcal{O}_ X$-module. This notion is very useful in algebraic geometry, since quasi-coherent modules on a scheme have a good description on any affine open. However, we warn the reader that in the general setting of (locally) ringed spaces this notion is not well behaved at all. The category of quasi-coherent sheaves is not abelian in general, infinite direct sums of quasi-coherent sheaves aren't quasi-coherent, etc, etc.
Definition 17.10.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We say that $\mathcal{F}$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules if for every point $x \in X$ there exists an open neighbourhood $x\in U \subset X$ such that $\mathcal{F}|_ U$ is isomorphic to the cokernel of a map
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ U \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ U \]
The category of quasi-coherent $\mathcal{O}_ X$-modules is denoted $\mathit{QCoh}(\mathcal{O}_ X)$.
The definition means that $X$ is covered by open sets $U$ such that $\mathcal{F}|_ U$ has a presentation of the form
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ U \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ U \longrightarrow \mathcal{F}|_ U \longrightarrow 0. \]
Here presentation signifies that the displayed sequence is exact. In other words
for every point $x$ of $X$ there exists an open neighbourhood such that $\mathcal{F}|_ U$ is generated by global sections, and
for a suitable choice of these sections the kernel of the associated surjection is also generated by global sections.
Lemma 17.10.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. The direct sum of two quasi-coherent $\mathcal{O}_ X$-modules is a quasi-coherent $\mathcal{O}_ X$-module.
Proof.
Omitted.
$\square$
Lemma 17.10.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a quasi-coherent $\mathcal{O}_ Y$-module is quasi-coherent.
Proof.
Arguing as in the proof of Lemma 17.8.2 we may assume $\mathcal{G}$ has a global presentation by direct sums of copies of $\mathcal{O}_ Y$. We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma 17.3.3. Thus if we have an exact sequence
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ Y \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ Y \longrightarrow \mathcal{G} \longrightarrow 0 \]
then upon applying $f^*$ we obtain the exact sequence
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \longrightarrow f^*\mathcal{G} \longrightarrow 0. \]
This implies the lemma.
$\square$
This gives plenty of examples of quasi-coherent sheaves.
Lemma 17.10.5. Let $(X, \mathcal{O}_ X)$ be ringed space. Let $\alpha : R \to \Gamma (X, \mathcal{O}_ X)$ be a ring homomorphism from a ring $R$ into the ring of global sections on $X$. Let $M$ be an $R$-module. The following three constructions give canonically isomorphic sheaves of $\mathcal{O}_ X$-modules:
Let $\pi : (X, \mathcal{O}_ X) \longrightarrow (\{ *\} , R)$ be the morphism of ringed spaces with $\pi : X \to \{ *\} $ the unique map and with $\pi $-map $\pi ^\sharp $ the given map $\alpha : R \to \Gamma (X, \mathcal{O}_ X)$. Set $\mathcal{F}_1 = \pi ^*M$.
Choose a presentation $\bigoplus _{j \in J} R \to \bigoplus _{i \in I} R \to M \to 0$. Set
\[ \mathcal{F}_2 = \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j \in J} \mathcal{O}_ X \to \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \right). \]
Here the map on the component $\mathcal{O}_ X$ corresponding to $j \in J$ given by the section $\sum _ i \alpha (r_{ij})$ where the $r_{ij}$ are the matrix coefficients of the map in the presentation of $M$.
Set $\mathcal{F}_3$ equal to the sheaf associated to the presheaf $U \mapsto \mathcal{O}_ X(U) \otimes _ R M$, where the map $R \to \mathcal{O}_ X(U)$ is the composition of $\alpha $ and the restriction map $\mathcal{O}_ X(X) \to \mathcal{O}_ X(U)$.
This construction has the following properties:
The resulting sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}_ M = \mathcal{F}_1 = \mathcal{F}_2 = \mathcal{F}_3$ is quasi-coherent.
The construction gives a functor from the category of $R$-modules to the category of quasi-coherent sheaves on $X$ which commutes with arbitrary colimits.
For any $x \in X$ we have $\mathcal{F}_{M, x} = \mathcal{O}_{X, x} \otimes _ R M$ functorial in $M$.
Given any $\mathcal{O}_ X$-module $\mathcal{G}$ we have
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}_ M, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _ R(M, \Gamma (X, \mathcal{G})) \]
where the $R$-module structure on $\Gamma (X, \mathcal{G})$ comes from the $\Gamma (X, \mathcal{O}_ X)$-module structure via $\alpha $.
Proof.
The isomorphism between $\mathcal{F}_1$ and $\mathcal{F}_3$ comes from the fact that $\pi ^*$ is defined as the sheafification of the presheaf in (3), see Sheaves, Section 6.26. The isomorphism between the constructions in (2) and (1) comes from the fact that the functor $\pi ^*$ is right exact, so $\pi ^*(\bigoplus _{j \in J} R) \to \pi ^*(\bigoplus _{i \in I} R) \to \pi ^*M \to 0$ is exact, $\pi ^*$ commutes with arbitrary direct sums, see Lemma 17.3.3, and finally the fact that $\pi ^*(R) = \mathcal{O}_ X$.
Assertion (1) is clear from construction (2). Assertion (2) is clear since $\pi ^*$ has these properties. Assertion (3) follows from the description of stalks of pullback sheaves, see Sheaves, Lemma 6.26.4. Assertion (4) follows from adjointness of $\pi _*$ and $\pi ^*$.
$\square$
Definition 17.10.6. In the situation of Lemma 17.10.5 we say $\mathcal{F}_ M$ is the sheaf associated to the module $M$ and the ring map $\alpha $. If $R = \Gamma (X, \mathcal{O}_ X)$ and $\alpha = \text{id}_ R$ we simply say $\mathcal{F}_ M$ is the sheaf associated to the module $M$.
Lemma 17.10.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Set $R = \Gamma (X, \mathcal{O}_ X)$. Let $M$ be an $R$-module. Let $\mathcal{F}_ M$ be the quasi-coherent sheaf of $\mathcal{O}_ X$-modules associated to $M$. If $g : (Y, \mathcal{O}_ Y) \to (X, \mathcal{O}_ X)$ is a morphism of ringed spaces, then $g^*\mathcal{F}_ M$ is the sheaf associated to the $\Gamma (Y, \mathcal{O}_ Y)$-module $\Gamma (Y, \mathcal{O}_ Y) \otimes _ R M$.
Proof.
The assertion follows from the first description of $\mathcal{F}_ M$ in Lemma 17.10.5 as $\pi ^*M$, and the following commutative diagram of ringed spaces
\[ \xymatrix{ (Y, \mathcal{O}_ Y) \ar[r]_-\pi \ar[d]_ g & (\{ *\} , \Gamma (Y, \mathcal{O}_ Y)) \ar[d]^{\text{induced by }g^\sharp } \\ (X, \mathcal{O}_ X) \ar[r]^-\pi & (\{ *\} , \Gamma (X, \mathcal{O}_ X)) } \]
(Also use Sheaves, Lemma 6.26.3.)
$\square$
Lemma 17.10.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $x \in X$ be a point. Assume that $x$ has a fundamental system of quasi-compact neighbourhoods. Consider any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$. Then there exists an open neighbourhood $U$ of $x$ such that $\mathcal{F}|_ U$ is isomorphic to the sheaf of modules $\mathcal{F}_ M$ on $(U, \mathcal{O}_ U)$ associated to some $\Gamma (U, \mathcal{O}_ U)$-module $M$.
Proof.
First we may replace $X$ by an open neighbourhood of $x$ and assume that $\mathcal{F}$ is isomorphic to the cokernel of a map
\[ \Psi : \bigoplus \nolimits _{j \in J} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ X. \]
The problem is that this map may not be given by a “matrix”, because the module of global sections of a direct sum is in general different from the direct sum of the modules of global sections.
Let $x \in E \subset X$ be a quasi-compact neighbourhood of $x$ (note: $E$ may not be open). Let $x \in U \subset E$ be an open neighbourhood of $x$ contained in $E$. Next, we proceed as in the proof of Lemma 17.3.5. For each $j \in J$ denote $s_ j \in \Gamma (X, \bigoplus \nolimits _{i \in I} \mathcal{O}_ X)$ the image of the section $1$ in the summand $\mathcal{O}_ X$ corresponding to $j$. There exists a finite collection of opens $U_{jk}$, $k \in K_ j$ such that $E \subset \bigcup _{k \in K_ j} U_{jk}$ and such that each restriction $s_ j|_{U_{jk}}$ is a finite sum $\sum _{i \in I_{jk}} f_{jki}$ with $I_{jk} \subset I$, and $f_{jki}$ in the summand $\mathcal{O}_ X$ corresponding to $i \in I$. Set $I_ j = \bigcup _{k \in K_ j} I_{jk}$. This is a finite set. Since $U \subset E \subset \bigcup _{k \in K_ j} U_{jk}$ the section $s_ j|_ U$ is a section of the finite direct sum $\bigoplus _{i \in I_ j} \mathcal{O}_ X$. By Lemma 17.3.2 we see that actually $s_ j|_ U$ is a sum $\sum _{i \in I_ j} f_{ij}$ and $f_{ij} \in \mathcal{O}_ X(U) = \Gamma (U, \mathcal{O}_ U)$.
At this point we can define a module $M$ as the cokernel of the map
\[ \bigoplus \nolimits _{j \in J} \Gamma (U, \mathcal{O}_ U) \longrightarrow \bigoplus \nolimits _{i \in I} \Gamma (U, \mathcal{O}_ U) \]
with matrix given by the $(f_{ij})$. By construction (2) of Lemma 17.10.5 we see that $\mathcal{F}_ M$ has the same presentation as $\mathcal{F}|_ U$ and therefore $\mathcal{F}_ M \cong \mathcal{F}|_ U$.
$\square$
Example 17.10.9. Let $X$ be countably many copies $L_1, L_2, L_3, \ldots $ of the real line all glued together at $0$; a fundamental system of neighbourhoods of $0$ being the collection $\{ U_ n\} _{n \in \mathbf{N}}$, with $U_ n \cap L_ i = (-1/n, 1/n)$. Let $\mathcal{O}_ X$ be the sheaf of continuous real valued functions. Let $f : \mathbf{R} \to \mathbf{R}$ be a continuous function which is identically zero on $(-1, 1)$ and identically $1$ on $(-\infty , -2) \cup (2, \infty )$. Denote $f_ n$ the continuous function on $X$ which is equal to $x \mapsto f(nx)$ on each $L_ j = \mathbf{R}$. Let $1_{L_ j}$ be the characteristic function of $L_ j$. We consider the map
\[ \bigoplus \nolimits _{j \in \mathbf{N}} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{j, i \in \mathbf{N}} \mathcal{O}_ X, \quad e_ j \longmapsto \sum \nolimits _{i \in \mathbf{N}} f_ j 1_{L_ i} e_{ij} \]
with obvious notation. This makes sense because this sum is locally finite as $f_ j$ is zero in a neighbourhood of $0$. Over $U_ n$ the image of $e_ j$, for $j > 2n$ is not a finite linear combination $\sum g_{ij} e_{ij}$ with $g_{ij}$ continuous. Thus there is no neighbourhood of $0 \in X$ such that the displayed map is given by a “matrix” as in the proof of Lemma 17.10.8 above.
Note that $\bigoplus \nolimits _{j \in \mathbf{N}} \mathcal{O}_ X$ is the sheaf associated to the free module with basis $e_ j$ and similarly for the other direct sum. Thus we see that a morphism of sheaves associated to modules in general even locally on $X$ does not come from a morphism of modules. Similarly there should be an example of a ringed space $X$ and a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ such that $\mathcal{F}$ is not locally of the form $\mathcal{F}_ M$. (Please email if you find one.) Moreover, there should be examples of locally compact spaces $X$ and maps $\mathcal{F}_ M \to \mathcal{F}_ N$ which also do not locally come from maps of modules (the proof of Lemma 17.10.8 shows this cannot happen if $N$ is free).
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