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Question: let $f : X \to S$ be a smooth proper morphism of schemes. Under what circumstances is it true that $R^i f_* \mathcal{O}_X$ is a locally-free sheaf whose formation commutes with all base change?

Cases I understand:

(1) if $S$ is reduced and noetherian then $\dim_{\kappa(s)} H^0(X_s, \mathcal{O}_{X_s})$ is constant because the fibers are reduced (see Tag 0E1E) so by Grauert's theorem $f_* \mathcal{O}_X$ is locally free and commutes with all base change. Using the commutative with base change we then reduce to the noetherian setting by noetherian approximation.

(2) if furthermore $S$ is a smooth variety over a field of characteristic zero, then by Hodge theory and Ehresmann's lemma, all $R^i f_* \Omega^j_{X/S}$ are locally free and commute with base change.

(3) If $f$ is a relative smooth connected curve then this follows from Tag 0E6B for $i = 0$ and then for $i = 1$ by Cohomology and base change.

(4) Mumford's book shows this is true for abelian schemes.

Remy pointed out this fails over characteristic p dvrs for $i > 0$.

If $S$ is not reduced, even the case $i = 0$ is not clear to me. If $S$ is reduced, this question is equivalent to asking if top cohomology $H^r(X_s, \mathcal{O}_{X_s})$ is locally of constant dimension on $S$.

Remaining cases:

(1) is it true in general if $S$ has characteristic zero?

(2) does it always hold for $i = 0$?

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  • $\begingroup$ Do you mean that $S=Y$ here? $\endgroup$
    – Aphelli
    Commented Mar 6 at 21:29
  • $\begingroup$ @aphelli Yes, edited $\endgroup$
    – Ben C
    Commented Mar 6 at 21:44
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    $\begingroup$ In characteristic $p > 0$ (or mixed characteristic), the dimensions $h^i(X_s,\mathcal O_{X_s})$ can jump in smooth projective families (over a reduced base). Classical examples (going back to Serre's paper in the 1958 proceedings for Symposium internacional de topología algebraica in Mexico) can be constructed by certain Godeaux-like quotients of hypersurfaces by finite group actions, but surely there must be other examples. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 6 at 21:49
  • $\begingroup$ @R.vanDobbendeBruyn thanks for the reference, I am still interested in the other cases so I edited the question appropriately. $\endgroup$
    – Ben C
    Commented Mar 6 at 21:56

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I think that (1) was proven by Deligne in "Théorème de Lefschetz et critères de dégénérescence de suites spectrales" (Theorem 5.5 in http://www.numdam.org/item/PMIHES_1968__35__107_0/).

(2) for locally Noetherian base is contained in the book "FGA Explained" Proposition 8.5.16. There you actually only need to assume that the morphism is proper and flat with geometrically reduced fibers.

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  • $\begingroup$ I am not certain what you are claiming about (2). Clearly (2) fails in mixed characteristic and positive characteristic because of supersingular Enriques surfaces (among many other such smooth projective schemes). $\endgroup$
    – Jason Starr
    Commented Mar 7 at 17:20
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    $\begingroup$ @Jason Starr I am referring to (2) in the questions (at the end of the post). The claim is that $R^0f_*(\mathcal{O}_X)$ is locally free and its formation commutes with base-change. $\endgroup$
    – afh
    Commented Mar 8 at 1:01
  • $\begingroup$ I see. The OP listed different statements in the post as (2). Thank you for clarifying. $\endgroup$
    – Jason Starr
    Commented Mar 8 at 17:29

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