The Stacks project

58.17 Quasi-coherent sheaves

We can apply the descent of modules to study quasi-coherent sheaves.

Proposition 58.17.1. For any quasi-coherent sheaf $\mathcal{F}$ on $S$ the presheaf

\[ \begin{matrix} \mathcal{F}^ a : & \mathit{Sch}/S & \to & \textit{Ab} \\ & (f: T \to S) & \mapsto & \Gamma (T, f^*\mathcal{F}) \end{matrix} \]

is an $\mathcal{O}$-module which satisfies the sheaf condition for the fpqc topology.

Proof. This is proved in Descent, Lemma 35.8.1. We indicate the proof here. As established in Lemma 58.15.6, it is enough to check the sheaf property on Zariski coverings and faithfully flat morphisms of affine schemes. The sheaf property for Zariski coverings is standard scheme theory, since $\Gamma (U, i^\ast \mathcal{F}) = \mathcal{F}(U)$ when $i : U \hookrightarrow S$ is an open immersion.

For $\left\{ \mathop{\mathrm{Spec}}(B)\to \mathop{\mathrm{Spec}}(A)\right\} $ with $A\to B$ faithfully flat and $\mathcal{F}|_{\mathop{\mathrm{Spec}}(A)} = \widetilde{M}$ this corresponds to the fact that $M = H^0\left((B/A)_\bullet \otimes _ A M \right)$, i.e., that

\begin{align*} 0 \to M \to B \otimes _ A M \to B \otimes _ A B \otimes _ A M \end{align*}

is exact by Lemma 58.16.4. $\square$

There is an abstract notion of a quasi-coherent sheaf on a ringed site. We briefly introduce this here. For more information please consult Modules on Sites, Section 18.23. Let $\mathcal{C}$ be a category, and let $U$ be an object of $\mathcal{C}$. Then $\mathcal{C}/U$ indicates the category of objects over $U$, see Categories, Example 4.2.13. If $\mathcal{C}$ is a site, then $\mathcal{C}/U$ is a site as well, namely the coverings of $V/U$ are families $\{ V_ i/U \to V/U\} $ of morphisms of $\mathcal{C}/U$ with fixed target such that $\{ V_ i \to V\} $ is a covering of $\mathcal{C}$. Moreover, given any sheaf $\mathcal{F}$ on $\mathcal{C}$ the restriction $\mathcal{F}|_{\mathcal{C}/U}$ (defined in the obvious manner) is a sheaf as well. See Sites, Section 7.25 for details.

Definition 58.17.2. Let $\mathcal{C}$ be a ringed site, i.e., a site endowed with a sheaf of rings $\mathcal{O}$. A sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ is called quasi-coherent if for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a covering $\{ U_ i \to U\} _{i\in I}$ of $\mathcal{C}$ such that the restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is isomorphic to the cokernel of an $\mathcal{O}$-linear map of free $\mathcal{O}$-modules

\[ \bigoplus \nolimits _{k \in K} \mathcal{O}|_{\mathcal{C}/U_ i} \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}|_{\mathcal{C}/U_ i}. \]

The direct sum over $K$ is the sheaf associated to the presheaf $V \mapsto \bigoplus _{k \in K} \mathcal{O}(V)$ and similarly for the other.

Although it is useful to be able to give a general definition as above this notion is not well behaved in general.

Remark 58.17.3. In the case where $\mathcal{C}$ has a final object, e.g. $S$, it suffices to check the condition of the definition for $U = S$ in the above statement. See Modules on Sites, Lemma 18.23.3.

Theorem 58.17.4 (Meta theorem on quasi-coherent sheaves). Let $S$ be a scheme. Let $\mathcal{C}$ be a site. Assume that

  1. the underlying category $\mathcal{C}$ is a full subcategory of $\mathit{Sch}/S$,

  2. any Zariski covering of $T \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ can be refined by a covering of $\mathcal{C}$,

  3. $S/S$ is an object of $\mathcal{C}$,

  4. every covering of $\mathcal{C}$ is an fpqc covering of schemes.

Then the presheaf $\mathcal{O}$ is a sheaf on $\mathcal{C}$ and any quasi-coherent $\mathcal{O}$-module on $(\mathcal{C}, \mathcal{O})$ is of the form $\mathcal{F}^ a$ for some quasi-coherent sheaf $\mathcal{F}$ on $S$.

Proof. After some formal arguments this is exactly Theorem 58.16.2. Details omitted. In Descent, Proposition 35.8.11 we prove a more precise version of the theorem for the big Zariski, fppf, étale, smooth, and syntomic sites of $S$, as well as the small Zariski and étale sites of $S$. $\square$

In other words, there is no difference between quasi-coherent modules on the scheme $S$ and quasi-coherent $\mathcal{O}$-modules on sites $\mathcal{C}$ as in the theorem. More precise statements for the big and small sites $(\mathit{Sch}/S)_{fppf}$, $S_{\acute{e}tale}$, etc can be found in Descent, Section 35.8. In this chapter we will sometimes refer to a “site as in Theorem 58.17.4” in order to conveniently state results which hold in any of those situations.


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