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
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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