On K\"ahler differentials of divided powers algebras

TL;DR

This paper establishes that K"ahler differentials of augmented divided powers algebras in prime characteristic represent Beck derivations, linking algebraic and geometric perspectives and applying results to modular Lie theory and Witt algebras.

Contribution

It proves that K"ahler differentials represent Beck derivations in prime characteristic and provides a geometric interpretation, advancing understanding in divided powers algebra cohomology.

Findings

K"ahler differentials represent Beck derivations in prime characteristic.

Geometric interpretation for sheaves of relative differentials.

Application to modular Lie theory and Witt algebras.

Abstract

The Quillen-Barr-Beck cohomology of augmented algebras with divided powers is defined as the derived functor of Beck derivations. The main theorem of this paper states that the K\"ahler differentials of an augmented algebra with divided powers in prime characteristic represents Beck derivations. We give a geometrical interpretation of this statement for the sheaf of relative differentials. As an application in modular Lie theory we prove that any special derivation of a divided powers algebra is a Beck derivation and we apply the theorem to Witt algebras.