The Stacks project

Lemma 26.7.4. Let $(X, \mathcal{O}_ X) = (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ be an affine scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is isomorphic to the sheaf associated to the $R$-module $\Gamma (X, \mathcal{F})$.

Proof. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Since every standard open $D(f)$ is quasi-compact we see that $X$ is a locally quasi-compact, i.e., every point has a fundamental system of quasi-compact neighbourhoods, see Topology, Definition 5.13.1. Hence by Modules, Lemma 17.10.8 for every prime $\mathfrak p \subset R$ corresponding to $x \in X$ there exists an open neighbourhood $x \in U \subset X$ such that $\mathcal{F}|_ U$ is isomorphic to the quasi-coherent sheaf associated to some $\mathcal{O}_ X(U)$-module $M$. In other words, we get an open covering by $U$'s with this property. By Lemma 26.5.1 for example we can refine this covering to a standard open covering. Thus we get a covering $\mathop{\mathrm{Spec}}(R) = \bigcup D(f_ i)$ and $R_{f_ i}$-modules $M_ i$ and isomorphisms $\varphi _ i : \mathcal{F}|_{D(f_ i)} \to \mathcal{F}_{M_ i}$ for some $R_{f_ i}$-module $M_ i$. On the overlaps we get isomorphisms

\[ \xymatrix{ \mathcal{F}_{M_ i}|_{D(f_ if_ j)} \ar[rr]^{\varphi _ i^{-1}|_{D(f_ if_ j)}} & & \mathcal{F}|_{D(f_ if_ j)} \ar[rr]^{\varphi _ j|_{D(f_ if_ j)}} & & \mathcal{F}_{M_ j}|_{D(f_ if_ j)}. } \]

Let us denote these $\psi _{ij}$. It is clear that we have the cocycle condition

\[ \psi _{jk}|_{D(f_ if_ jf_ k)} \circ \psi _{ij}|_{D(f_ if_ jf_ k)} = \psi _{ik}|_{D(f_ if_ jf_ k)} \]

on triple overlaps.

Recall that each of the open subspaces $D(f_ i)$, $D(f_ if_ j)$, $D(f_ if_ jf_ k)$ is an affine scheme. Hence the sheaves $\mathcal{F}_{M_ i}$ are isomorphic to the sheaves $\widetilde M_ i$ by Lemma 26.7.1 above. In particular we see that $\mathcal{F}_{M_ i}(D(f_ if_ j)) = (M_ i)_{f_ j}$, etc. Also by Lemma 26.7.1 above we see that $\psi _{ij}$ corresponds to a unique $R_{f_ if_ j}$-module isomorphism

\[ \psi _{ij} : (M_ i)_{f_ j} \longrightarrow (M_ j)_{f_ i} \]

namely, the effect of $\psi _{ij}$ on sections over $D(f_ if_ j)$. Moreover these then satisfy the cocycle condition that

\[ \xymatrix{ (M_ i)_{f_ jf_ k} \ar[rd]_{\psi _{ij}} \ar[rr]^{\psi _{ik}} & & (M_ k)_{f_ if_ j} \\ & (M_ j)_{f_ if_ k} \ar[ru]_{\psi _{jk}} } \]

commutes (for any triple $i, j, k$).

Now Algebra, Lemma 10.23.5 shows that there exist an $R$-module $M$ such that $M_ i = M_{f_ i}$ compatible with the morphisms $\psi _{ij}$. Consider $\mathcal{F}_ M = \widetilde M$. At this point it is a formality to show that $\widetilde M$ is isomorphic to the quasi-coherent sheaf $\mathcal{F}$ we started out with. Namely, the sheaves $\mathcal{F}$ and $\widetilde M$ give rise to isomorphic sets of glueing data of sheaves of $\mathcal{O}_ X$-modules with respect to the covering $X = \bigcup D(f_ i)$, see Sheaves, Section 6.33 and in particular Lemma 6.33.4. Explicitly, in the current situation, this boils down to the following argument: Let us construct an $R$-module map

\[ M \longrightarrow \Gamma (X, \mathcal{F}). \]

Namely, given $m \in M$ we get $m_ i = m/1 \in M_{f_ i} = M_ i$ by construction of $M$. By construction of $M_ i$ this corresponds to a section $s_ i \in \mathcal{F}(U_ i)$. (Namely, $\varphi ^{-1}_ i(m_ i)$.) We claim that $s_ i|_{D(f_ if_ j)} = s_ j|_{D(f_ if_ j)}$. This is true because, by construction of $M$, we have $\psi _{ij}(m_ i) = m_ j$, and by the construction of the $\psi _{ij}$. By the sheaf condition of $\mathcal{F}$ this collection of sections gives rise to a unique section $s$ of $\mathcal{F}$ over $X$. We leave it to the reader to show that $m \mapsto s$ is a $R$-module map. By Lemma 26.7.1 we obtain an associated $\mathcal{O}_ X$-module map

\[ \widetilde M \longrightarrow \mathcal{F}. \]

By construction this map reduces to the isomorphisms $\varphi _ i^{-1}$ on each $D(f_ i)$ and hence is an isomorphism. $\square$


Comments (1)

Comment #4975 by Rubén Muñoz--Bertrand on

Typo at the beginning of the proof : '" is a locally quasi-compact'".


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