In this paper using the Quillen–Barr–Beck (co-)homology theory of universal algebras we define (co-)homology groups for commutative algebras with divided powers in prime characteristic. In particular, we determine for A a commutative 𝔽p-algebra with divided powers, the category of Beck A-modules and the group of Beck derivations. We construct the abelianization functor and we define (co-)homology. Moreover, we determine the cohomology in low dimensions and we interpret the first cohomology in terms of extensions.
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