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Frobenius-Witt differentials and regularity

Takeshi Saito

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.

Frobenius-Witt differentials and regularity

Abstract

T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total -differentials for a ring over . We study the same construction for a ring over 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.

Paper Structure

This paper contains 4 sections, 30 theorems, 46 equations.

Table of Contents

  1. Frobenius-Witt derivation
  2. Frobenius-Witt differentials
  3. Regularity criterion
  4. Relation with cotangent complex

Key Result

Lemma 1.2

Let $A$ be a ring and $w\colon A\to M$ be an FW-derivation. 1. We have $w(1)=0$. Let $a\in A$ and $n\in {\mathbf Z}$. Then, we have If $n\geqq 0$, we have 2. For $n\in {\mathbf Z}$, we have In particular, we have $w(0)=0$. 3. Assume that $A$ is a ring over ${\mathbf Z}_{(p)}$. Then, for any $a\in A$, we have $p\cdot w(a)=0$.

Theorems & Definitions (34)

  • Definition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Proposition 1.6
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • ...and 24 more