Prisms and prismatic cohomology

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 Zₚ(n) introduced in our previous joint work with Morrow.