Saturated de Rham-Witt complexes with unit-root coefficients

TL;DR

This paper generalizes the saturated de Rham-Witt complex to include unit-root F-crystal coefficients, providing a new universal construction that simplifies and extends previous theories, with applications to crystalline cohomology.

Contribution

It introduces a universal construction of the saturated de Rham-Witt complex with unit-root coefficients, extending its applicability and establishing key properties and comparisons.

Findings

Existence and quasicoherence of the new complex

Comparison with classical de Rham-Witt and crystalline cohomology

Extension to non-smooth varieties

Abstract

The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew, is a variant of the classical de Rham-Witt complex which provides a conceptual simplification of the construction and which is expected to produce better results for non-smooth varieties. In this paper, we introduce a generalization of the saturated de Rham-Witt complex which allows coefficients in a unit-root $F$-crystal. We define our complex by a universal property in a category of so-called de Rham-Witt modules. We prove a number of results about it, including existence, quasicoherence, and comparisons to the de Rham-Witt complex of Bhatt-Lurie-Mathew and (in the smooth case) to crystalline cohomology and the classical de Rham-Witt complex with coefficients.