Ioannis Dokas - Academia.edu (original) (raw)

Papers by Ioannis Dokas

Homology, Homotopy and Applications, 2015

arXiv (Cornell University), Oct 13, 2011

$Research paper thumbnail of On K\"ahler differentials of divided powers algebras$

The Quillen-Barr-Beck cohomology of augmented algebras with divided powers is defined as the deri... more 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 Kähler 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.

Journal of Lie Theory, Nov 18, 2010

Communications in Algebra, 2006

Journal of the Australian Mathematical Society, 2009

In this paper using the Quillen–Barr–Beck (co-)homology theory of universal algebras we define (c... more 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.

Journal of Pure and Applied Algebra, 2004

In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to deÿne (... more In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to deÿne (co-) homology for the category RLie of restricted Lie algebras over a ÿeld k of characteristic p = 0. In contrast with the cases of groups, associative algebras and Lie algebras we do not obtain Hochschild (co-) homology shifted by 1. Precisely, we determine for L ∈ RLie the category of Beck L-modules and the group of Beck derivations of g ∈ RLie=L to a Beck L-module M. Moreover, we prove a classiÿcation theorem which gives a one-to-one correspondence between the one cohomology and the set of equivalent classes of p-extensions. Finally, a universal coe cient theorem is proved, relating the homology to the Hochschild homology via a short exact sequence. This shows that the new homology determines the Hochschild homology.

Glasgow Mathematical Journal, 2009

In this paper, we prove that any Zinbiel algebra can be endowed with the structure of commutative... more In this paper, we prove that any Zinbiel algebra can be endowed with the structure of commutative algebra with divided powers. We introduce the notion of universal enveloping Zinbiel algebra of a commutative algebra with divided powers algebras. We prove that the free divided powers algebra on a free module M, is the divided powers sub-algebra generated by M, of the divided powers algebra induced by the free Zinbiel algebra on M. Finally, we construct a basis for the enveloping Zinbiel algebra.

Given an exact sequence of restricted Lie algebras using Duskin's torsors theory, we establish an... more Given an exact sequence of restricted Lie algebras using Duskin's torsors theory, we establish an eight term exact sequence for Quillen-Barr-Beck cohomology of restricted Lie algebras. As an application we obtain for any extension of algebraic groups over an algebraic closed field of prime characteristic an eight term exact sequence for the corresponding restricted Lie algebras extension.