Abstract
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.
Citation
Bhargav Bhatt. Peter Scholze. "Prisms and prismatic cohomology." Ann. of Math. (2) 196 (3) 1135 - 1275, November 2022. https://doi.org/10.4007/annals.2022.196.3.5
Information