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.


Motivation:

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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