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.

我们引入了棱镜的概念，它可以被认为是对完美环概念的“缺陷”。使用棱镜，我们将一个环形站点——棱柱站点——附加到一个 $p$-adic 形式方案。由此产生的上同调理论专门用于（并且经常改进）大多数已知的积分 $p$-adic 上同调理论。

作为应用程序，我们证明了几乎纯定理的改进版本，允许沿任意闭合子集（不使用 adic 空间）产生分支，给出第二作者猜想的 $q$-de Rham 上同调的坐标自由描述，并解决在我们之前与 Morrow 的联合工作中引入的 $p$-adic Tate 扭曲 $\mathbf {Z}_p(n)$ 的消失猜想。