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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2024 MCQ for Journal of the American Mathematical Society is 4.83.

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Rigidity properties of the cotangent complex
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by Benjamin Briggs and Srikanth B. Iyengar;
J. Amer. Math. Soc. 36 (2023), 291-310
DOI: https://doi.org/10.1090/jams/1000
Published electronically: February 22, 2022

Abstract:

This work concerns a map $\varphi \colon R\to S$ of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if $\mathrm {D}_n(S/R;-)=0$ for some $n\ge 1$, then $\mathrm {D}_i(S/R;-)=0$ for all $i\ge 2$ and ${\varphi }$ is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming $\mathrm {D}_n(S/R;-)$ vanishes for all $n\gg 0$, confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of $\varphi$.
References
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Bibliographic Information
  • Benjamin Briggs
  • Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
  • MR Author ID: 1281297
  • ORCID: 0000-0003-0003-9493
  • Email: briggs@math.utah.edu
  • Srikanth B. Iyengar
  • Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
  • MR Author ID: 616284
  • ORCID: 0000-0001-7597-7068
  • Email: iyengar@math.utah.edu
  • Received by editor(s): March 8, 2021
  • Received by editor(s) in revised form: October 23, 2021
  • Published electronically: February 22, 2022
  • Additional Notes: This work was partly supported by NSF grant DMS-2001368 (SBI)
  • © Copyright 2022 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 36 (2023), 291-310
  • MSC (2020): Primary 13D03; Secondary 13B10, 14A15, 14A30
  • DOI: https://doi.org/10.1090/jams/1000
  • MathSciNet review: 4495843