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I am reading a paper of Lieblich on the unirationality of K3 surfaces and am trying to understand the result of Proposition 4.1:

Proposition 4.1: Let $k$ be an algebraically closed field of characteristic $p \geq 5$ and $X/k$ an (Artin) supersingular K3 surface. There exists $\alpha \in \operatorname{Br}(X \otimes_k k[[t]])$ such that:

  1. There is an Azumaya algebra of degree $p$ with class $\alpha$;

  2. $\alpha|_{t=0} = 0 \in \text{Br}(X);$

  3. The restriction of $\alpha$ to the formal scheme $X \times_k \operatorname{Spf} k[[t]]$ gives an isomorphism $$ \operatorname{Spf} k[[t]] \cong \widehat{\operatorname{Br}(X)}.$$

I'll state what I understand of the result and then ask my question. Let $F$ denote the functor $\widehat{\operatorname{Br}}(X)$. Since $F$ is pro-representable by $\operatorname{Spf} k[[t]]$ (since $X$ is Artin supersingular), the identity element in $\operatorname{Hom}( k[[t]], k[[t]])$ gives an element $\alpha \in \varprojlim F\big(k[[t]]/t^n\big)$. It is now a question of whether $\alpha$ lies in the image of the map $$F\big(k[[t]]\big) \to \varprojlim F\big(k[[t]]/t^n\big).$$

This is what Lieblich claims in the paper when he speaks of "algebraization" in the paragraph after Proposition 4.1. However, I can't seem to find this result in the paper "Rational Curves in the moduli of Supersingular K3s" which is where he claims they're proved. My guess is that the proof of Proposition 4.1 is buried in the proof of the representability of $R^2\pi_\ast \mu_p$ in the second paper (Here $\pi : X \to \operatorname{Spec} k$ is the structure map of supersingular K3). What is the relevant thing in the second paper that pertains to this proposition?

Edit: I'd also be interested in the relevant results in the second paper relating to Proposition 4.2 (A relative version of the Artin-Tate isomorphism) in the first paper.

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    $\begingroup$ I only looked at this quickly, but Lieblich says that the class is an formal deformation of a specified Azumaya algebra of specified degree on the closed fiber. In that case, the algebraization should follow from the Grothendieck existence theorem for coherent sheaves, cf. EGA III, Section 5. $\endgroup$
    – Jason Starr
    Commented Feb 6, 2017 at 11:18
  • $\begingroup$ @JasonStarr Thanks. I'll look at the other paper again today. Otherwise, I'll just email Max Lieblich. $\endgroup$
    – David Benjamin Lim
    Commented Feb 6, 2017 at 19:08
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    $\begingroup$ You must do what you feel is right, of course. I recommend that you have a look at EGA III, Section 5. $\endgroup$
    – Jason Starr
    Commented Feb 6, 2017 at 20:14
  • $\begingroup$ @JasonStarr Sure. There's also Proposition 4.2 that he says is proven in the latter paper that I'd like to understand. It's some kind of relative version of the Artin-Tate isomorphism. I emailed Max Lieblich today asking about both 4.1 and 4.2 $\endgroup$
    – David Benjamin Lim
    Commented Feb 7, 2017 at 5:36

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