Rigidity properties of the cotangent complex
File(s)
Accepted version
Author(s)
Briggs, Benjamin
Iyengar, Srikanth
Type
Journal Article
Abstract
This work concerns maps ϕ: R → S of commutative noetherian
rings, locally of finite flat dimension. It is proved that the Andr´e-Quillen
homology functors are rigid, namely, if Dn(S/R; −) = 0 for some n ≥ 2,
then Dn(S/R; −) = 0 for all n ≥ 2 and ϕ is locally complete intersection.
This extends Avramov’s theorem that draws the same conclusion assuming
Dn(S/R; −) vanishes for all n ≫ 0, confirming a conjecture of Quillen. The
rigidity of Andr´e-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 Andr´e-Quillen cohomology to Hochschild cohomology
defined using the universal Atiyah class of ϕ.
rings, locally of finite flat dimension. It is proved that the Andr´e-Quillen
homology functors are rigid, namely, if Dn(S/R; −) = 0 for some n ≥ 2,
then Dn(S/R; −) = 0 for all n ≥ 2 and ϕ is locally complete intersection.
This extends Avramov’s theorem that draws the same conclusion assuming
Dn(S/R; −) vanishes for all n ≫ 0, confirming a conjecture of Quillen. The
rigidity of Andr´e-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 Andr´e-Quillen cohomology to Hochschild cohomology
defined using the universal Atiyah class of ϕ.
Date Issued
2023
Date Acceptance
2021-10-23
Citation
Journal of the American Mathematical Society, 2023, 36 (1), pp.291-310
ISSN
0894-0347
Publisher
American Mathematical Society (AMS)
Start Page
291
End Page
310
Journal / Book Title
Journal of the American Mathematical Society
Volume
36
Issue
1
Copyright Statement
© Copyright 2022 American Mathematical Society
Briggs, Benjamin, and Srikanth Iyengar. "Rigidity properties of the cotangent complex." Journal of the American Mathematical Society 36.1 (2023): 291-310.
Briggs, Benjamin, and Srikanth Iyengar. "Rigidity properties of the cotangent complex." Journal of the American Mathematical Society 36.1 (2023): 291-310.
License URL
Identifier
http://dx.doi.org/10.1090/jams/1000
Publication Status
Published
Date Publish Online
2022-02-22