Amine Koubaa (Leibniz Universität Hannover): Beauville Laszlo theorems for discretely ringed adic spaces

Abstract: Let $(A,A^+)$ be a discretely ringed Huber pair, and let $f\in A$ be a non-zero divisor. Under mild conditions, Bhatt proves the existence of an equivalence of $\infty$-categories

\[
D(A)\cong D(\hat{A}_f)\times_{D(\hat{A}_f[1/f])} D(A[1/f]),
\]
which restricts to an equivalence of Categories, originally proven by Beauville and Laszlo,
\[
\textrm{Mod}(A)\cong \textrm{Mod}(\hat{A}_f)\times_{\textrm{Mod}(\hat{A}_f[1/f])} \textrm{Mod}(A[1/f]).
\]
To remedy the absence of a category of quasi-coherent sheaves on discretely ringed adic space, we prove a similar equivalence on the derived category of solid modules over analytic rings, introduced by Clausen and Scholze
\[
D((A,A^+)_\solid)\cong D((\hat{A}_f,\hat{A}_f^+)_\solid)\times_{D((\hat{A}_f[1/f],\hat{A}_f^+[1/f])_\solid)} D((A[1/f],A^+[1/f])_\solid).
\]
We conjecture a similar result holds for a category of “analytically coherent $\mathcal{O}^+$ modules” denoted by $\mathcal{C}^+(X)$ on a discretely ringed adic space $X$: We prove the result when $A=A^+$ is a dvr and $f$ is a uniformizer.