Mod $p$ sheaves on Witt flags
Robert Cass, João Lourenço
TL;DR
This work develops a bridge between mod $p$ perverse étale sheaves and singularity theory in mixed and equal characteristic by establishing intrinsic criteria for Cohen–Macaulayness and $\varphi$-rationality on perfect schemes via ${\mathbb F}_p$-perverse sheaves. It introduces and exploits global $+$-regularity and its inversion-of-adjunction principles, including asymptotic variants, to prove uniform global regularity results for affine Schubert varieties and their deperfections. The authors apply these results to affine flag varieties, Demazure resolutions, and local models, showing that Schubert varieties admit globally $+$-regular deperfections and that the special fibers of local models are $\varphi$-split, thereby removing several previously restrictive hypotheses. Overall, the paper provides a uniform, characteristic-agnostic framework for CM/$\varphi$-rationality and global regularity that unifies classical and affine Schubert geometry with applications to Beilinson–Drinfeld Grassmannians and local models, advancing Bhatt’s program on singularities in equal and mixed characteristic.
Abstract
We characterize Cohen--Macaulay and $\varphi$-rational perfect schemes in terms of their perverse étale mod $p$ sheaves. Using inversion of adjunction, we prove that sufficiently small Schubert varieties in the Witt affine flag variety are perfections of globally $+$-regular varieties, and hence they are $\varphi$-rational. Our methods apply uniformly to all affine Schubert varieties in equicharacteristic, as well as classical Schubert varieties, thereby answering a question of Bhatt. As a corollary, we deduce that scheme-theoretic local models always have $\varphi$-split special fiber.