Finite generation of Tate cohomology

Carlson, Jon; Chebolu, Sunil K.; Mináč, Ján

doi:10.1090/s1088-4165-2011-00385-x

Cited by 5 publications

(11 citation statements)

References 11 publications

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“…They would like to thank the University of Alberta, the organisers of the summer school (A. Adem, J. Kuttler and A. Pianzola), and V. Chernousov for their hospitality. We are grateful to J. Carlson for his work in [8], which inspired us to consider almost split sequences in the last section. Calculations using Peter Webb's reps package [22] for GAP [15] were useful in our search for non-trivial ghosts.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Chebolu¹,

Christensen²,

Mináč³

2012

Can. math. bull.

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Abstract. Let G be a finite group, and let k be a field whose characteristic p divides the order of G. Freyd's generating hypothesis for the stable module category of G is the statement that a map between finite-dimensional kG-modules in the thick subcategory generated by k factors through a projective if the induced map on Tate cohomology is trivial. We show that if G has periodic cohomology, then the generating hypothesis holds if and only if the Sylow p-subgroup of G is C 2 or C 3 . We also give some other conditions that are equivalent to the GH for groups with periodic cohomology.

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Section: Acknowledgmentsmentioning

confidence: 99%

Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Chebolu¹,

Christensen²,

Mináč³

2012

Can. math. bull.

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“…The Tate side: the conditions (ii') and (iii') entering in Theorems B(c), C(b) and D(b) are strong conditions on T. The question of what finite tensor categories satisfy them is currently under intense study [6,18,39] and even the case of module categories of finite groups is still open. In these situations Theorems B(c), C(b) and D(b) give important information about cohomological support of elements of T computed with respect to Tate cohomology.…”

Section: Theorem Cmentioning

confidence: 99%

On the spectrum and support theory of a finite tensor category

Nakano¹,

Vashaw²,

Yakimov³

2021

Preprint

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Finite tensor categories (FTCs) T are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras, which play a key role in many areas of mathematics and mathematical physics. There are two fundamentally different support theories for them: a cohomological one and a universal one based on the noncommutative Balmer spectra of their stable (triangulated) categories T.In this paper we introduce the new notion of the categorical center C • T of the cohomology ring R • T of an FTC, T. This enables us to put forward a complete and detailed program for determining the exact relationship between the two support theories, based on C • T of the cohomology ring R • T of an FTC, T. More specifically, we construct a continuous map from the noncommutative Balmer spectrum of an FTC, T, to the Proj of the categorical center C • T , and prove that this map is surjective under a weaker finite generation assumption for T than the one conjectured by Etingof-Ostrik. Under stronger assumptions, we prove that (i) the map is homeomorphism and (ii) the two-sided thick ideals of T are classified by the specialization closed subsets of Proj C • T . We conjecture that both results hold for all FTCs. Many examples are presented that demonstrate how in important cases C • T arises as a fixed point subring of R • T and how the two-sided thick ideals of T are determined in a uniform fashion (while previous methods dealt on a case-by-case basis with case specific methods). The majority of our results are proved in the greater generality of monoidal triangulated categories.

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“…If every eventual map is a ghost map, then the above theorem tells us that every finitely generated kG-module has finitely generated Tate cohomology. By [8,Theorem 4.1], G has periodic cohomology.…”

Section: Eventual Ghosts and Groups With Periodic Cohomologymentioning

confidence: 99%

“…The answer is that this happens if and only if G has periodic cohomology. This question is related to the finite generation of Tate cohomology studied in [8].…”

Section: Introductionmentioning

confidence: 99%

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Ghosts and Strong Ghosts in the Stable Category

Carlson¹,

Chebolu²,

Mináč³

2016

Can. math. bull.

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Abstract. Suppose that G is a nite group and k is a eld of characteristic p > . A ghost map is a map in the stable category of nitely generated kG-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow p-subgroup of G is cyclic of order or . In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We nd all groups satisfying a strong form of Freyd's generating hypothesis and show that ghosts can be detected on a nite range of degrees of Tate cohomology. We also consider maps which mimic ghosts in high degrees.

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Finite generation of Tate cohomology

Cited by 5 publications

References 11 publications

Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Freyd's Generating Hypothesis for Groups with Periodic Cohomology

On the spectrum and support theory of a finite tensor category

Ghosts and Strong Ghosts in the Stable Category

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