The local-to-global principle for triangulated categories via dimension functions

Stevenson, Greg

doi:10.1016/j.jalgebra.2016.12.002

Cited by 4 publications

(4 citation statements)

References 11 publications

(13 reference statements)

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“…Remark. This notion of "weakly visible" point coincides with the notion of "visible" point in [Ste14,Ste17]. In contrast, we follow the terminology of [BF11] and say that a point P is visible if its closure {P} ⊆ X is a In particular, we note that a T 1 spectral space (a.k.a.…”

Section: Balmer-favi Supportmentioning

confidence: 99%

“…For a rigidly-compactly generated tt-category T with Spc(T c ) noetherian, the local-to-global principle always holds without needing any additional hypotheses.3.24.Example. Stevenson[Ste14,Ste17] proves that the derived category D(R) of an absolutely flat ring satisfies the local-to-global principle if and only if R is semi-artinian or noetherian. Note that the spectrum Spec(R) of an absolutely flat ring is always weakly noetherian.…”

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confidence: 99%

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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: Balmer-favi Supportmentioning

confidence: 99%

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confidence: 99%

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Barthel,

Heard,

Sanders

2021

Preprint

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“…In this section, we relate our support to that of Balmer, Favi, and Stevenson in [2,38]. Following [38], a point p in a spectral space X is visible if there exists Thomason subsets V, U ⊆ X such that…”

Section: Visible Pointsmentioning

confidence: 99%

“…Their support takes values in the Balmer spectrum of the compact objects, and their definition is valid whenever this space is topologically Noetherian. Greg Stevenson studied this support in [38] and applied these results to the derived category of an absolutely flat ring in [39] and [42]. These results suggest that even though Neeman's classification fails, support detects some semblance of order in the localising subcategories.…”

Section: Introductionmentioning

confidence: 99%

Support and vanishing for non-Noetherian rings and tensor triangulated categories

Sanders

2017

Preprint

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We define and characterise small support for complexes over non-Noetherian rings and in this context prove a vanishing theorem for modules. Our definition of support makes sense for any rigidly compactly generated tensor triangulated category. Working in this generality, we establish basic properties of support and investigate when it detects vanishing. We use pointless topology to relate support, the topology of the Balmer spectrum, and the structure of the idempotent Bousfield lattice.

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Adelic models of tensor-triangulated categories

Balchin

Greenlees

2020

Advances in Mathematics

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Given a suitable stable monoidal model category C and a specialization closed subset V of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over V and the part supported over V c spliced with the Tate object. Using this one can show that C is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra from [18] to a Quillen equivalence. In addition, a close analysis of the one step case highlights important features needed for general torsion models which we will return to in future work. Contents 10. The torsion model for rational T-spectra 28 References 34

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The local-to-global principle for triangulated categories via dimension functions

Cited by 4 publications

References 11 publications

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Support and vanishing for non-Noetherian rings and tensor triangulated categories

Adelic models of tensor-triangulated categories

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