We characterize Cohen--Macaulay and -rational perfect schemes in terms of their perverse \'etale 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 -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 -split special fiber.
Cass et al. (Mon,) studied this question.
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