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I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again.

This makes me wonder if there are any theorems in algebraic geometry of the form:

"If $X$ is an algebraic variety that is (insert properties here); $Y$ is an algebraic variety that is (insert properties here); $\mathcal S$ is a reflexive sheaf on $X$; and $f:X\rightarrow Y$ is a map that is (insert properties here); then $f_*\mathcal S$ is a locally-free sheaf on $Y$."

If indeed there are such theorems, which properties need to be inserted, regarding $X$, $Y$, and $f$?

I tend to think mostly about complex geometry, so I guess I would be most interested in the case when $X$ and $Y$ are complex varieties, if that makes a difference. It seems to me that the most natural situation would be when $Y$ is a nonsingular complex variety and $X$ is a singular, finite cover of $Y$, and $\mathcal S$ is locally free away from the singular locus of $X$. Can something be said in that situation?

References to specific theorems in the literature (or names of the people responsible for them) are always appreciated, too.

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1 Answer 1

A standard example of this kind of theorem is: If $X$ and $Y$ are surfaces, $Y$ is nonsingular, $X$ is normal, $f$ is finite, and $S$ is reflexive, then $f_*S$ is locally free. The reason is that reflexive over normal implies depth $\geq 2$ at closed points. Then $f_*S$ has depth $\geq 2$ at closed points too (by finiteness of $f$). Finally, if you have a maximal Cohen-Macaulay module over a regular local ring, then it is free (see Lemma Tag 00NT).

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