Mod $p$ sheaves on Witt flags

Abstract

We characterize Cohen--Macaulay and $φ$-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 $φ$-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 $φ$-split special fiber.