On K\"ahler differentials of divided powers algebras

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.