Frobenius--Witt cotangent complexes

TL;DR

The paper introduces the Frobenius--Witt cotangent complex, a derived arithmetic variant of the cotangent complex, and explores its properties and relation to regularity in noetherian local rings.

Contribution

It defines the Frobenius--Witt cotangent complex, analyzes its properties, and links it to regularity criteria, extending Saito's work with computations in perfectoid rings.

Findings

Established the Frobenius--Witt cotangent complex as an arithmetic analog of the cotangent complex.

Connected the complex to regularity of noetherian local rings.

Performed computations in perfectoid rings to support theoretical results.

Abstract

We introduce the notion of the Frobenius--Witt cotangent complex, which can be considered as a derived variant of the module of Frobenius--Witt differentials defined by T. Saito. This new object also can be seen as an arithmetic variant of the notion of cotangent complex. We explain the suitability of these two viewpoints through a series of propositions. Furthermore, we establish a relationship between Frobenius--Witt cotangent complexes and the regularity of noetherian local rings, which can be considered as a derived variant of Saito's regularity criterion. This proof relies heavily on computations of Frobenius--Witt cotangent complexes in the case of perfectoid rings.