A characterization of categories of coherent sheaves of certain algebraic stacks

TL;DR

This paper extends Tannakian reconstruction to algebraic stacks, characterizing categories of coherent sheaves and applying the results to p-adic Galois representations and Adams Hopf algebroids.

Contribution

It proves a generalized Tannakian recognition theorem for algebraic stacks and explores its implications for filtered modules and Hopf algebroids.

Findings

Characterization of categories of coherent sheaves on algebraic stacks

Extension of Tannakian recognition theorem to stacks

New insights into p-adic Galois representations and Adams Hopf algebroids

Abstract

Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in this case the category of its finite dimensional representations. Lurie's result that the pseudofunctor which sends a geometric stack to its category of quasi-coherent sheaves is fully faithful provides a conceptual explanation for why this works. In this paper we prove a generalized Tannakian recognition theorem, in order to characterize a part of the image of the extension of the above pseudofunctor to algebraic stacks in the sense of Naumann. This allows us to further investigate a conjecture by Richard Pink about categories of filtered modules, which were defined by Fontaine and Laffaille to construct p-adic Galois representations. In order to do this we give a new characterization of Adams Hopf algebroids, which also allows us to answer a question posed by Mark Hovey.