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In his paper "Supersingular K3 surfaces", Artin states the following theorem (Theorem 3.1) without proof:

Let $\pi:X \to S = \mathrm{Spec}(k)$ be a smooth proper surface with $k$ an algebraically closed field. Then the functors $R^q_{fl} \pi_* \mu_n$ are represented by finite type group schemes over $k$.

Has the proof appeared in the literature somewhere? (It does not seem to have been published by Artin himself.)

I am also interested in explicit computations of these group schemes and extensions to morphisms of relative dimension $>2$, other coefficients, and more general bases.

Any references will be greatly appreciated.

(I am aware of Milne's paper on flat duality, but my main interest is in the infinitesimal structure of these group schemes and not the corresponding quasi-algebraic group.)

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  • $\begingroup$ This talk looks relevant (Illusie Conference, June 8): Martin Olsson, Representability results for flat cohomology Let be $f$ a proper morphism of schemes over a field of positive characteristic, and let $G$ be a finite flat abelian group scheme. In this talk I will discuss recent representability results for the derived pushforwards $R^if_*G$. Key ingredients in proving our results is the development of a theory of compactly supported flat cohomology and description in terms of the cotangent complex in some cases. $\endgroup$
    – user166831
    Commented Jun 1, 2021 at 1:09
  • $\begingroup$ @anon: Thanks, this does look promising! $\endgroup$
    – naf
    Commented Jun 1, 2021 at 13:25

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In the case of a K3 surface, the representability of $R^2 \pi_\ast \mu_p$ (for $p$ an arbitrary integer) is proven in the paper "Twistor Spaces for Supersingular K3s" by Daniel Bragg and Max Lieblich.

What you want is Theorem 2.1.6: If $\pi : X \to S$ is a relative K3, then the fppf sheaf $R^2 \pi_\ast \mu_p$ is representable by a group algebraic space, locally of finite presentation over $S$.

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  • $\begingroup$ Thanks. Artin does not assume that $X$ is a K3, so I was hoping more general results would be known, but this reference looks like a good starting point. $\endgroup$
    – naf
    Commented Apr 26, 2021 at 1:09
  • $\begingroup$ I started reading the proof, but got confused: at the bottom of p. 8 they define a map $\chi$ which they use to get a presentation of the sheaf $R^2 \pi_* \mu_p$. However, there seems to be no statement about the surjectivity of this map so I don't see how this works. One has to perhaps use that the fibres are surfaces and de Jong's theorem saying that the period is equal to the index, but nothing is said about this in the paper. $\endgroup$
    – naf
    Commented Apr 26, 2021 at 11:04
  • $\begingroup$ @naf Would you like to continue this correspondence over email? How can I reach you? $\endgroup$
    – David Benjamin Lim
    Commented Apr 26, 2021 at 12:17
  • $\begingroup$ I deleted my email after keeping it here for a week. In case you didn't see it you can ping me here. $\endgroup$
    – naf
    Commented May 4, 2021 at 4:14
  • $\begingroup$ @naf Sorry was a little busy recently. I have just emailed you. $\endgroup$
    – David Benjamin Lim
    Commented May 4, 2021 at 5:57
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This is Proposition 1.8 in Artin, Mazur "Formal groups arising from algebraic varieties", Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 10 (1977) no. 1, pp. 87-131, available from numdam: http://www.numdam.org/item/?id=ASENS_1977_4_10_1_87_0 . Note that you need the vanishing of $R^{q-1}$.

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  • $\begingroup$ Thanks. Which Proposition 1.8 did you mean? There are two (at least) and I don't see how either of them immediately gives an answer to my question. $\endgroup$
    – naf
    Commented Apr 27, 2021 at 10:00
  • $\begingroup$ Sorry, I meant Prop. 1.8 in Chap. II. But indeed it only treats smooth groups. In chapter IV they seem to make claims about p-torsion which may imply what you want. $\endgroup$
    – Piotr Achinger
    Commented Apr 27, 2021 at 13:28
  • $\begingroup$ Thanks for the clarification. My question was actually motivated by reading Chap IV, since I wanted to know if the results there could be formulated and proved over a more general base. $\endgroup$
    – naf
    Commented Apr 27, 2021 at 13:32

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