Cohomology of inverse semigroups

Abstract

D’Alarcao [3] and Coudron [2] investigated the following problem: Given a semilattice G of groups and an inverse semigroup S, what are the inverse semigroups U such that there is an idempotent separating surjective homomorphism from U to S with G as its kernel normal system? Their answer came out in terms of a certain action of S on G and a “factor system” condition, similar to the classical case of group extensions, but naturally more involved. Whereas Eilenberg and MacLane [5] could phrase the theory of group extensions in terms of cohomology theory, the corresponding extension problem for inverse semigroups was somehow left in the “wilderness,” similar to Schreier’s original paper [S] on group extensions. Only for a very special situation, cohomological notions have been introduced [9]. The purpose of this paper is to provide a cohomological framework for inverse semigroups which will not only fit the extension problem, but also discuss some apparently new notions such as complementation and inner automorphism for inverse semigroups. In Section 2 we introduce the category of S-modules for inverse semigroups S: an inverse semigroup S is represented as a semigroup of certain endomorphisms of a semilattice A of abelian groups. Sections 3 and 4 are devoted to the free, projective, and injective objects in this category. In Section 5 we apply some general results of cohomology theory for abelian categories to the category of S-modules, and in Section 6 we set up various projective resolutions of a standard S-module 2, . Section 7 links d’Alarcao’s and Coudron’s results with cohomology theory for the case where G consists of abelian groups. It is interesting to note that one has to introduce a dummy identity element in S to tackle the extension problem. Theorem 7.5 is an improvement on d’illarcao’s and Coudron’s results insofar as “factor systems” need not be defined on the whole of S x S but just a certain subset, in order to determine a unique extension. This fact can be neatly expressed in terms