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Let $(X, \mathcal{O}_X)$ be a regular noetherian scheme of finite Krull dimension (over a field $k$ if needed).

Is it true that any $\mathcal{O}_X$-module (not necessarily quasi-coherent) has a finite resolution by injective $\mathcal{O}_X$-modules?

This is suggested by the remark on page 136 in Hartshorne's "Residues and Duality" but I could not find a reference.

Similarly, has any $\mathcal{O}_X$-module a finite resolution by flat $\mathcal{O}_X$-modules?

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  • $\begingroup$ According to one of Serre's theorems, a noetherian local ring is regular if and only if it has finite global dimension (in which case it is equal to the Krull dimension). The latter implies that both the injective dimension of all modules and the projective dimension of all finite modules are finite. $\endgroup$
    – Sándor Kovács
    May 5, 2019 at 4:49

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