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TRIPLE COHOMOLOGY AND DIVIDED POWERS ALGEBRAS IN PRIME CHARACTERISTIC

  • IOANNIS DOKAS (a1)
Abstract

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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References
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[1]Barr, M. and Beck, J., ‘Homology and standard constructions’, in: Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics, 80 (Springer, Berlin, 1969), pp. 245335.
[2]Cartan, H., ‘Algèbres de Eilenberg–MacLane et homotopie’, in: Seminaire Henri Cartan, 7ème année 1954–1955, 2ème édn (École Normale Supérieure, Paris, 1956).
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[5]Loday, J.-L., ‘Dialgebras’, in: Dialgebras and Related Operads, Lectures Notes in Math., 1763 (Springer, Berlin, 2001), pp. 766.
[6]Quillen, D., ‘On the (co-)homology of commutative rings’, in: Applications of Categorical Algebra, New York, 1968, Proc. Sympos. Pure Math., Vol. XVII (American Mathematical Society, Providence, RI, 1970), pp. 6587.
[7]Roby, N., ‘Les algèbres à puissances divisées’, Bull. Soc. Math. France 89 (1965), 7591.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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