Homological support of big objects in tensor-triangulated categories

Balmer, Paul

doi:10.5802/jep.135

Cited by 11 publications

(9 citation statements)

References 23 publications

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“…In the companion paper [BHS21], we compare the Balmer-Favi notion of support with the homological support introduced by Balmer [Bal20]. In particular, we will prove that they coincide in a strong sense whenever T is stratified.…”

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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Section: Introductionmentioning

confidence: 99%

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Barthel,

Heard,

Sanders

2021

Preprint

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“…Remark. For our purposes, the next most significant example is the homological support function introduced by Balmer [Bal20b]. We briefly recall the construction; more details can be found in [Bal20b] and [BKS19].…”

Section: 2mentioning

confidence: 99%

“…For example, the Balmer-Favi notion of support Supp T does not provide an extension of the universal support on T c to the whole of T without some such hypothesis. Recently, Balmer [Bal20b] has extended homological support to big objects. The resulting notion of support (Spc h (T c ), Supp h T ) does not require any noetherian hypotheses and extends the pull-back supp h T c = φ −1 (supp T c ) of the universal support to the whole of T.…”

Section: Introductionmentioning

confidence: 99%

Stratification and the comparison between homological and tensor triangular support

Barthel¹,

Heard²,

Sanders³

2021

Preprint

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We compare the homological support and tensor triangular support for 'big' objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support.

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“…The recent work [BKS19,Bal20b,Bal20a] introduced and explored homological residue fields as an alternative that exists in broad generality and is always tensorfriendly. They consist of symmetric monoidal homological functors (1.1) hB : T → ĀB from T to 'simple' abelian categories ĀB .…”

Section: Introductionmentioning

confidence: 99%

“…One such functor exists for each so-called homological prime B, as recalled in Section 2. These homological residue fields collectively detect the nilpotence of maps [Bal20b] and they give rise to a support theory for not necessarily compact objects [Bal20a]. Homological residue fields are undeniably useful but they are defined in a rather abstract manner and it is not clear how they relate to the tensor-triangulated residue fields F : T → F that partially exist in examples.…”

Section: Introductionmentioning

confidence: 99%

Computing homological residue fields in algebra and topology

Balmer¹,

Cameron²

2020

Preprint

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Homological support of big objects in tensor-triangulated categories

Abstract: Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.

Cited by 11 publications

References 23 publications

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Stratification and the comparison between homological and tensor triangular support

Computing homological residue fields in algebra and topology

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