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Finite generation for Hochschild cohomology of Gorenstein monomial algebras

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Abstract

We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall–Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann–Holloway–Snashall–Solberg–Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.

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Notes

  1. This follows from binaturality of \({\text {Ext}}^*_{\Lambda ^e}(\Lambda , M)\) making it a right module over \({\text {Ext}}^*_{\Lambda ^e}(\Lambda , \Lambda )\); note that a priori this only makes the map \(- \smile \chi \) commute with the long exact sequence, but we use \(\chi \smile -\) for consistency with the theorem’s claim and use graded-commutativity.

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Acknowledgements

We are grateful to Ben Briggs for useful discussions and to Ira Gessel for sending a copy of [19]. A crucial thank you is due to Joe Chuang and Alastair King who kindly shared with us their unpublished manuscript from fifteen years ago that highlights the importance of previous work of Gruenberg [28] for computing higher structures on Ext algebras of monomial algebras; the beautiful example of Section 1.3 is also coming from their unpublished note (and is reproduced here with their permission). Their work on higher structures ultimately led to a paper on functorial non-minimal resolutions of general associative algebras [12], and their ideas related to the specific case of monomial algebras never made it to that paper. However, it was of utmost importance for the genesis of this paper, and we cannot thank them enough for sharing their work with us. We also are indebted to Bernhard Keller who documented the existence of unpublished work of Chuang and King in his survey [37]. The second named author was supported by Simons Foundation (through a postdoctoral fellowship at Hamilton Mathematics Institute). Preparation of the final version of this paper was supported by Institut Universitaire de France, by the University of Strasbourg Institute for Advanced Study (USIAS) through the Fellowship USIAS-2021-061 within the French national program “Investment for the future” (IdEx-Unistra), and by the French national research agency project ANR-20-CE40-0016.

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Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000, Strasbourg, France

    Vladimir Dotsenko

  2. Montreal, Canada

    Vincent Gélinas

  3. Institut für Mathematik, Johann von Neumann-Haus, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Pedro Tamaroff

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  1. Vladimir Dotsenko
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  2. Vincent Gélinas
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Correspondence to Vladimir Dotsenko.

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To Ed Green with deep admiration of his work on the homology theory of monomial algebras.

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Dotsenko, V., Gélinas, V. & Tamaroff, P. Finite generation for Hochschild cohomology of Gorenstein monomial algebras. Sel. Math. New Ser. 29, 14 (2023). https://doi.org/10.1007/s00029-022-00817-8

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