Let $\mathcal L$ be a sheaf over $X$ that is flat over a map $f: X \to Y$, that is separable and of finite type between Noetherian schemes $X,Y$.

Under what conditions is the pushforward $f_* \mathcal L$ locally free over $Y$? I believe that this is the case when $Y$ is regular and $\dim Y=1$, because local regular rings of $dim = 1$ are PIDs. I am wondering if there are more general conditions that also guarantee local freeness of $f_* \mathcal L$.

  • 1
    $\begingroup$ Without properness? $\endgroup$
    – Mohan
    Nov 17 '20 at 1:25
  • $\begingroup$ Sure, add properness if you have to. Do we need properness also for the 1-dimensional result? $\endgroup$
    – Hammerhead
    Nov 17 '20 at 4:42
  • 1
    $\begingroup$ Even in the one dimensional case without properness direct image will not be coherent . $\endgroup$
    – Mohan
    Nov 17 '20 at 14:47
  • $\begingroup$ Hm. In my situation $\mathcal L$ is actually coherent (finite vector bundle). Is that ok, without properness? Do you know a good reference for all this stuff? $\endgroup$
    – Hammerhead
    Nov 18 '20 at 3:20

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