The Convergent Topos in Characteristic p

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The purpose of this note is to investigate some of the foundational questions concerning convergent cohomology as introduced in [?] and [?], using the language and techniques of Grothendieck topologies. In particular, if X is a scheme of finite type over a perfect field k of characteristic p and with Witt ring W, we define the “convergent topos (X/W) conv , and we study the cohomology of its structure sheaf O X/W and of K X := QO X/W . Since the topos (X/W) conv is not noetherian, formation of cohomology does not commute with tensor products, and these are potentially quite different.