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Prisms and prismatic cohomology. (English) Zbl 07611906

Summary: We introduce the notion of a prism, which may be regarded as a “deperfection” of the notion of a perfectoid ring. Using prisms, we attach a ringed site – the prismatic site – to a \(p\)-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral \(p\)-adic cohomology theories.
As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of \(q\)-de Rham cohomology as conjectured by the second author, and settle a vanishing conjecture for the \(p\)-adic Tate twists \(\mathbf{Z}_p(n)\) introduced in our previous joint work with Morrow.

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
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