# On the decomposition of the De Rham complex on formal schemes

## TL;DR

This paper proves that the truncated De Rham complex on certain formal schemes over a positive characteristic field decomposes, and fully decomposes when the scheme's dimension equals the characteristic, also establishing the Cartier isomorphism.

## Contribution

It demonstrates the decomposability of the De Rham complex on formal schemes in positive characteristic and establishes the Cartier isomorphism in this context.

## Key findings

- De Rham complex decomposes up to characteristic p
- Full De Rham complex decomposes when dimension equals p
- Cartier isomorphism established for smooth morphisms

## Abstract

We show that, for a pseudo-proper smooth noetherian formal scheme $\mathfrak{X}$ over a positive characteristic $p$ field, its truncated De Rham complex up to the characteristic $p$ is decomposable. Moreover, if the dimension of $\mathfrak{X}$ is exactly $p$, then the full De Rham complex is decomposable. Along the way we establish the Cartier isomorphism associated to a smooth morphism of positive characteristic noetherian formal schemes.