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Transactions of the American Mathematical Society

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

 
 

 

Depth and detection for Noetherian unstable algebras


Author: Drew Heard
Journal: Trans. Amer. Math. Soc. 373 (2020), 7429-7454
MSC (2010): Primary 55S10, 13C15, 57T05
DOI: https://doi.org/10.1090/tran/8157
Published electronically: August 5, 2020
MathSciNet review: 4155212
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Abstract: For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when $R$ is the mod $p$ cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable $R$-modules. Moreover, we explain the results in the case of the $p$-local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.


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Additional Information

Drew Heard
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany
Address at time of publication: Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway
MR Author ID: 1112422
ORCID: 0000-0002-0895-3354
Email: drew.k.heard@ntnu.no

Received by editor(s): July 23, 2019
Received by editor(s) in revised form: March 18, 2020
Published electronically: August 5, 2020
Additional Notes: This work was supported by SFB 1085 “Higher Invariants” (Universität Regensburg), funded by the Deutsche Forschungsgemeinschaft.
Article copyright: © Copyright 2020 American Mathematical Society