Prismatic cohomology and de Rham–Witt forms

Abstract

For any prism $(A,d)$, we construct an analogue of Fontaine's map $W_r(A/d) \to A/d\phi(d)\cdots\phi^{r-1}(d)$. Subsequently, we define a canonical map from de Rham–Witt forms to prismatic cohomology in the perfect case and prove that it is an isomorphism. Using this result, we obtain an explicit description of the prismatic cohomology $H^i((S/A)_\Prism,\mathcal{O}_\Prism/d\phi(d)\cdots\phi^{n-1}(d))$, where $S$ is the $p$-completion of a polynomial algebra over $A/d$.