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?