Frobenius-Witt differentials and regularity

TL;DR

This paper extends the concept of Frobenius-Witt differentials to rings over Z_{(p)} and establishes a regularity criterion, connecting to prior work on p-differentials and applications in algebraic geometry.

Contribution

It introduces a new construction of Frobenius-Witt differentials over Z_{(p)} and proves a regularity criterion, expanding the framework for studying ring regularity.

Findings

Established a regularity criterion using Frobenius-Witt differentials.

Extended the module of total p-differentials to rings over Z_{(p)}.

Connected the construction to sheaf theory and algebraic geometry applications.

Abstract

T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $Z/p^2Z$. We study the same construction for a ring over $Z_{(p)}$ and prove a regularity criterion. For a local ring, the tensor product with the residue field is constructed in a different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we use the sheaf of FW-differentials to define the cotangent bundle and the micro-support of an etale sheaf.