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Hartshorne's Algebraic Geometry defines an $\mathcal O_X$-module $\mathscr F$ to be quasi-coherent if there is an open affine cover $(U_i=\operatorname{Spec} A_i)_{i\in\mathcal I}$ of $X$ such that each $\mathscr F\rvert_{U_i}$ is isomorphic to the sheaf of modules $\widetilde{M_i}$ associated to some $A_i$-module $M_i$. It is later proved that this is equivalent to the statement that for any open affine $U=\operatorname{Spec} A\subseteq X$, there exists an $A$-module $M$ with $\mathscr F\rvert_U\cong\widetilde M$.

On the other hand, Grothendieck's Éléments de Géométrie Algébrique defines $\mathscr F$ to be quasi-coherent if every point $x\in X$ admits an open neighborhood $U\subseteq X$ such that $\mathscr F\rvert_U$ is isomorphic to the cokernel of a morphism of free $\mathcal O_U$-modules.

Now I wonder how those definitions are equal. I tried proving it by myself, but both directions seem quite nontrivial to me... Any hints?

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  • $\begingroup$ Note that you can reduce to the affine case, and then the statement is Theorem 1.4.1 of EGA I. $\endgroup$ – user314 Jan 4 '16 at 17:38
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This is not so hard to prove, using the basic properties of sheafifying modules:

Hartshorne to Grothendieck:

If we choose a presentation $A_i^I \to A_i^J \to M_i$ and then sheafify, we obtain a presentation of $\mathcal F_{| U_i}$ as a cokernel of a morphism $\mathcal O_{U_i}^I \to \mathcal O_{U_i}^J,$ i.e. as the cokernel of a morphism of free $\mathcal O_{U_i}$-modules.

Grothendieck to Hartshorne:

Shrinking $U$ around $x$ if necessary, we may assume that $U = \mathrm{Spec} \, A$ is affine. By assumption, we have an exact sequence $$\mathcal O_U^I \to \mathcal O_U^J \to \mathcal F_{|U} \to 0.$$ Now passing to global sections with respect to the first map, we obtain a morphism of $A$-modules $$A^I \to A^J;$$ let $M$ be the cokernel of this map. Sheafifying the right-exact sequence $$A^I \to A^J \to M \to 0,$$ we find that $\widetilde{M}$ is identified with the cokernel of $\mathcal O_U^I \to \mathcal O_U^J,$ and hence is isomorphic to $\mathcal F_{| U}$.

Summary: both directions use the fact that the global sections of $\mathcal O_U$ on $U = $ Spec $A$ are naturally isomorphic to $A$, and that sheafification of $A$-modules is exact, but nothing more.

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  • $\begingroup$ As usual, I feel dumb now. Thank you very much. $\endgroup$ – HRpLMvKZW0geYs8h Jan 4 '16 at 21:11

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