Affine weakly regular tensor triangulated categories

Dell’Ambrogio, Ivo; Stanley, Donald

doi:10.2140/pjm.2016.285.93

Cited by 10 publications

(6 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There is an alternative, perhaps more conceptual path which consists in defining the Kasparov category as a certain localization of the Spanier-Whitehead category associated to the standard tensor category of G-C * -algebras and * -homomorphisms [20]. The triangulated structure of the Spanier-Whithead category is proved in [20,Theorem A.5.3]. The argument given there can be directly used to show that KK G is triangulated, because it makes use of only two facts, which we prove below.…”

Section: Triangulated Structure and Comparison With E -Theorymentioning

confidence: 99%

A Categorical Approach to the Baum–connes Conjecture for Étale Groupoids

Bönicke,

Proietti

2024

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the ‘weakly contractible’ objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of ‘compactly induced’ algebras with respect to certain proper subgroupoids related to isotropy. The resulting ‘strong’ Baum–Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized ‘going-down’ principle, injectivity results for groupoids that are amenable at infinity, the Baum–Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid $C^*$ -algebras under homotopy of twists.

show abstract

Section: Triangulated Structure and Comparison With E -Theorymentioning

confidence: 99%

A Categorical Approach to the Baum–connes Conjecture for Étale Groupoids

Bönicke,

Proietti

2024

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

show abstract

“…18.10. Definition (Dell'Ambrogio-Stanley [DS16]). A tensor-triangulated category T is said to be affine weakly regular if it satisfies the following two conditions:…”

Section: Definition (Beligiannis [Bel00] Krause [Kra00]mentioning

confidence: 99%

“…where κ(p) := R p /pR p denotes the algebraic residue field; see [DS16,§3]. This object plays a role analogous to g p ; in particular, Loc g p = Loc K(p) .…”

Section: Definition (Beligiannis [Bel00] Krause [Kra00]mentioning

confidence: 99%

See 1 more Smart Citation

Cosupport in tensor triangular geometry

Barthel¹,

Castellana²,

Heard³

et al. 2023

Preprint

View full text Add to dashboard Cite

We develop a theory of cosupport and costratification in tensor triangular geometry. We study the geometric relationship between support and cosupport, provide a conceptual foundation for cosupport as categorically dual to support, and discover surprising relations between the theory of costratification and the theory of stratification. We prove that many categories in algebra, topology and geometry are costratified by developing and applying descent techniques. An overarching theme is that cosupport is relevant for diverse questions in tensor triangular geometry and that a full understanding of a category requires knowledge of both its support and its cosupport.

show abstract

“…Remark 2.9. One can ask if the conclusion of Corollary 2.8 holds for general tensor triangular categories; see [DS16,Lemma 2.1] and [DS] for a discussion.…”

Section: E Laumentioning

confidence: 99%

The Balmer spectrum of certain Deligne–Mumford stacks

Lau¹

2023

Compositio Math.

View full text Add to dashboard Cite

We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$ .

show abstract

Affine weakly regular tensor triangulated categories

Cited by 10 publications

References 18 publications

A Categorical Approach to the Baum–connes Conjecture for Étale Groupoids

A Categorical Approach to the Baum–connes Conjecture for Étale Groupoids

Cosupport in tensor triangular geometry

The Balmer spectrum of certain Deligne–Mumford stacks

Contact Info

Product

Resources

About