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A Pride–Guba–Sapir exact sequence for the relation bimodule of an associative algebra

Part of: Special aspects of infinite or finite groups Semigroups Homological methods

Published online by Cambridge University Press:  10 March 2025

Benjamin Steinberg Open the ORCID record for Benjamin Steinberg [Opens in a new window]
Benjamin Steinberg*
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue, New York, USA
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Email: [email protected]
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Abstract

Given a presentation of a monoid $M$, combined work of Pride and of Guba and Sapir provides an exact sequence connecting the relation bimodule of the presentation (in the sense of Ivanov) with the first homology of the Squier complex of the presentation, which is naturally a $\mathbb ZM$-bimodule. This exact sequence was used by Kobayashi and Otto to prove the equivalence of Pride’s finite homological type property with the homological finiteness condition bi-$\mathrm {FP}_3$. Guba and Sapir used this exact sequence to describe the abelianization of a diagram group. We prove here a generalization of this exact sequence of bimodules for presentations of associative algebras. Our proof is more elementary than the original proof for the special case of monoids.

Keywords

relation bimoduleSquier complexhomological finiteness

MSC classification

Primary: 20M50: Connections of semigroups with homological algebra and category theory 20M05: Free semigroups, generators and relations, word problems
Secondary: 20F65: Geometric group theory 16E05: Syzygies, resolutions, complexes
Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 8
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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