Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Hochschild and cotangent complexes of operadic algebras
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by Truong Hoang;
Trans. Amer. Math. Soc.
DOI: https://doi.org/10.1090/tran/9447
Published electronically: April 25, 2025

Abstract:

We make use of the cotangent complex formalism developed by Lurie to formulate Quillen cohomology of algebras over an enriched operad. Additionally, we introduce a spectral Hochschild cohomology theory for enriched operads and algebras over them. We prove that both the Quillen and Hochschild cohomologies of algebras over an operad can be controlled by the corresponding cohomologies of the operad itself. When passing to the category of simplicial sets, we assert that both these cohomology theories for operads, as well as their associated algebras, can be calculated in the same framework of spectrum valued functors on the twisted arrow $\infty$-category of the operad of interest. Moreover, we provide a convenient cofiber sequence relating the Hochschild and cotangent complexes of an $E_n$-space, establishing an unstable analogue of a significant result obtained by Francis and Lurie.
References
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Bibliographic Information
  • Truong Hoang
  • Affiliation: Department of Mathematics, Hanoi FPT University, Vietnam
  • ORCID: 0009-0001-7976-5385
  • Email: truonghm@fe.edu.vn
  • Received by editor(s): January 11, 2024
  • Received by editor(s) in revised form: December 22, 2024, and February 17, 2025
  • Published electronically: April 25, 2025
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc.
  • MSC (2020): Primary 55P42, 18M60, 18N60, 18M75
  • DOI: https://doi.org/10.1090/tran/9447