Prismatic cohomology and de Rham-Witt forms

TL;DR

This paper develops a new framework connecting prismatic cohomology with de Rham-Witt forms, providing explicit descriptions and isomorphisms that advance understanding in p-adic Hodge theory.

Contribution

It introduces a novel map analogous to Fontaine's and establishes an isomorphism between de Rham-Witt forms and prismatic cohomology in the perfect case.

Findings

Constructed an analogue of Fontaine's map for prisms.

Proved the de Rham-Witt forms are isomorphic to prismatic cohomology in the perfect case.

Provided explicit descriptions of prismatic cohomology for p-completed polynomial algebras.

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 for a $p$-completed polynomial algebra over $A/d$.