Stratification and duality for homotopical groups

Barthel, Tobias; Castellana, Natàlia; Heard, Drew; Valenzuela, Gabriel

doi:10.1016/j.aim.2019.106733

Cited by 12 publications

(9 citation statements)

References 56 publications

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“…We once again thank the referee for the proof that the Krull dimension is additive. (2). As noted by the referee, we can also prove this directly.…”

Section: The P-central Defect Of a Noetherian Unstable Algebramentioning

confidence: 54%

“…Proof This will be a consequence of Theorems 5.1 and 6.3, but we first explain why we are able to prove this without assuming anything about the Duflot algebra, using an observation of Nick Kuhn. 2 The point is that for a group we can always assume that the Duflot algebra is polynomial (this has already been observed by Kuhn in the case of compact Lie groups, see [37,Page 160]). Indeed, since the action of G on F p is trivial H 1 G ∼ = Hom Z (G, Z/ p) (these homomorphisms need be continuous in the case G is a profinite group).…”

Section: Group Theorymentioning

confidence: 97%

“…(1) G is a compact Lie group. (2) G is a discrete group for which there exists a mod p acyclic G-CW complex with finitely many G-cells and finite isotropy groups. (3) G is a profinite group such that the continuous mod p cohomology H * G is finitely generated as an F p -algebra.…”

Section: The Topological Nilpotence Degree For the Mod P Cohomology Omentioning

confidence: 99%

“…The filtration of M given by the M i is separated, and so the fact that each gr i M is a free B-module implies that M is also a free B-module. Let R be a connected Noetherian unstable algebra with center (C, g), and fix a Duflot algebra B of R.(1) CEss(R) is a finitely-generated free B-module, i.e., a Cohen-Macaulay module (2). The composite P C CEss(R) → CEss(R)…”

mentioning

confidence: 99%

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The topological nilpotence degree of a Noetherian unstable algebra

Heard

2021

Sel. Math. New Ser.

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We investigate the topological nilpotence degree, in the sense of Henn–Lannes–Schwartz, of a connected Noetherian unstable algebra R. When R is the mod p cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian p-subgroups. By replacing centralizers of elementary abelian p-subgroups with components of Lannes’ T-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn’s result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn’s result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac–Moody groups. In fact, our results apply much more generally, for example, we establish results for p-local compact groups in the sense of Broto–Levi–Oliver, for connected H-spaces with Noetherian mod p cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson’s depth conjecture in the case of a Noetherian unstable algebra of minimal depth.

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“…We once again thank the referee for the proof that the Krull dimension is additive. (2). As noted by the referee, we can also prove this directly.…”

Section: The P-central Defect Of a Noetherian Unstable Algebramentioning

confidence: 54%

Section: Group Theorymentioning

confidence: 97%

Section: The Topological Nilpotence Degree For the Mod P Cohomology Omentioning

confidence: 99%

mentioning

confidence: 99%

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The topological nilpotence degree of a Noetherian unstable algebra

Heard

2021

Sel. Math. New Ser.

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“…In general, the BIK notion of support does not classify the localizing ⊗-ideals of T but, inspired by ideas of [HPS97], BIK develop a powerful condition called stratification which is sufficient to obtain such a classification. Indeed this work of BIK culminates in the celebrated classification of localizing ⊗-ideals for the big stable module categories of finite groups [BIK11a] and has since seen many applications [Sha12,DS16,BCHV19a,BIKP18].…”

Section: Introductionmentioning

confidence: 99%

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Barthel,

Heard,

Sanders

2021

Preprint

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We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensorideals of certain categories of spectral G-Mackey functors for all finite groups G. Our theory of stratification is based on the approach of Stevenson which uses the Balmer-Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-toglobal principle and establish that the Balmer-Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson-Iyengar-Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite étale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria-Sanders-Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height 0 and height ∞ chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors noting that it bijects onto the height ≤ n chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.

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Lifting (co)stratifications between tensor triangulated categories

Shaul,

Williamson

2023

Isr. J. Math.

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Stratification and duality for homotopical groups

Cited by 12 publications

References 56 publications

The topological nilpotence degree of a Noetherian unstable algebra

The topological nilpotence degree of a Noetherian unstable algebra

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Lifting (co)stratifications between tensor triangulated categories

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