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TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

Published online by Cambridge University Press:  08 October 2021

BENJAMIN STEINBERG*
Affiliation:
Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, New York 10031, USA

Abstract

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.

We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Lisa Orloff Clark

The author was supported by a PSC CUNY grant.

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