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
Let us denote these \psi _{ij}. It is clear that we have the cocycle condition
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
namely, the effect of \psi _{ij} on sections over D(f_ if_ j). Moreover these then satisfy the cocycle condition that
commutes (for any triple i, j, k).
Now Algebra, Lemma 10.24.5 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
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
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
There are also: