On Kähler differentials of divided power algebras

Abstract

The Quillen–Barr–Beck cohomology of augmented algebras with a system of divided powers is defined as the derived functor of Beck derivations. The main theorem of this paper states that the Kähler differentials of an augmented algebra with a system of 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 the theory of modular Lie algebras we prove that any special derivation of a divided power algebra is a Beck derivation and we apply the theorem to Witt algebras.