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 OX/W and of KX := Q ⊗ OX/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.
Finite Type Canonical Isomorphism Coherent Sheaf Zariski Topology Zariski Open Subset
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