In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds:

Statement: If $A$ is a commutative ring and $I$ is an injective $A$-module, then: $$ I \longrightarrow I_f \longrightarrow 0 \ $$ is surjective, for any function $f \in A$.

For example:

  • If $A$ is a domain, then $I$ is divisble, and the statement follows.
  • If $A$ is noetherian, then the statement also holds, because any injective module $I$ is the direct sum of injective envelopes $E(A/\mathfrak{p}_j)$, that are flabby sheaves.

My question is both whether the statement is true for arbitrary, commutative rings (I guess it is not), and whether there exists a class of rings (as in the case of domains), where the statement holds without much effort.


If the statement above is true, then it is easy to prove that any injective module $I$ gives rise to an acyclic quasi-coherent sheaf $\widetilde{I}$ over $ {\rm Spec}A $, where $\widetilde{I} (U) := I_U$.

Then, this would allow to easily prove the following basic theorem:

Theorem: Quasi-coherent modules $\widetilde{M}$ are acylcic sheaves over ${\rm Spec} A$.

Proof: Let $M$ be an $A$-module and take an injective resolution $$ 0 \to M \to I^\bullet \ . $$

Taking the associated sheaves, we have a resolution of the quasi-coherent sheaf $\widetilde{M}$, that is aciclyc because of the argument above, and hence it computes the cohomology of $\widetilde{M}$: $$ 0 \to \widetilde{M} \to \widetilde{I^\bullet} \ . $$

But taking global sections we obtain the original sequence (because $\Gamma ( {\rm Spec} A , \widetilde{M}) = M $, for any module) that is exact, so $\widetilde{M}$ is acyclic. $\square$


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