Lifting (co)stratifications between tensor triangulated categories

Shaul, Liran; Williamson, Jordan

doi:10.48550/arxiv.2012.05190

Cited by 2 publications

(4 citation statements)

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“…The former is a significant strengthening of [BHS21, Thm. 6.4] and relies on work of Balmer [Bal16], while the latter is inspired by a recent result of Shaul-Williamson [SW20] on BIK-stratification. The terminology used in stating the next theorem is introduced in Section 2.…”

Section: Subsets Of Spec H (H * (G; O)) \ Spec(o)mentioning

confidence: 99%

Stratifying integral representations of finite groups

Barthel¹

2021

Preprint

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We classify the localizing tensor ideals of the integral stable module category for any finite group G. This results in a generic classification of Z[G]-lattices of finite and infinite rank and globalizes the modular case established in celebrated work of Benson, Iyengar, and Krause. Further consequences include a verification of the generalized telescope conjecture in this context, a tensor product formula for integral cohomological support, as well as a generalization of Quillen's stratification theorem for group cohomology. Our proof makes use of novel descent techniques for stratification in tensortriangular geometry that are of independent interest.

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Section: Subsets Of Spec H (H * (G; O)) \ Spec(o)mentioning

confidence: 99%

Stratifying integral representations of finite groups

Barthel¹

2021

Preprint

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“…We further remark that it follows from [31,Theorem 4.5] that this coincides with the set of primes p, such that M ⊗ L A H 0 (Ap)/p • H 0 (Ap) 0. However, (5.16) has the advantage that κ(A, p) has finite flat dimension (while H 0 (Ap)/p • H 0 (Ap) always has infinite flat dimension over A), making computations of these derived tensor products much easier.…”

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confidence: 84%

“…Proof. By [31,Theorem 4.15], there is a bijection between localizing (resp. colocalizing) subcategories of D(K) and subsets of Spec(H 0 (K)) given by sending every such category to the union of its supports (resp.…”

Section: It Follows Thatmentioning

confidence: 99%

“…In any case, we would like to mention that even if one works with this definition, it is still the case that not every sequence-regular DG-ring satisfies this definition. Indeed, in [31,Theorem 4.27], there is an example of a commutative noetherian DG-ring K with bounded cohomology, such that H 0 (K) is a field (and hence, K is sequence-regular), but there exist a non-zero object in D b f (K) which is not virtually small.…”

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Sequence-regular commutative DG-rings

Shaul

2021

Preprint

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We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A, m) such that the maximal ideal m ⊆ H 0 (A) can be generated by an A-regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular.

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Lifting (co)stratifications between tensor triangulated categories

Cited by 2 publications

References 0 publications

Stratifying integral representations of finite groups

Stratifying integral representations of finite groups

Sequence-regular commutative DG-rings

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