Mod $p$ sheaves on Witt flags

Abstract

We characterize Cohen--Macaulay and $\ddot{I}†$-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 $\ddot{I}†$-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 $\ddot{I}†$-split special fiber.

34 pages, various improvements and new section on abstract BSDH varieties