On local categories of finite groups

Xu, Fei

doi:10.1007/s00209-011-0971-y

Cited by 10 publications

(22 citation statements)

References 9 publications

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“…The definition of the left and right Kan extensions depend on the so-called over-categories and under-categories, respectively. Despite their seemingly abstract definitions, they are quite computable and thus play an important role in category representations and cohomology, see [29,31,32], as well as Sections 2.5, 3.4 and 4.1.…”

Section: Category Algebras and Their Representationsmentioning

confidence: 99%

“…In summary, finite category cohomology behaves very much like the special case of finite group cohomology, except the finite generation property. The ordinary cohomology ring of a category algebra is usually far from finitely generated, but it is so when C = G ∝ P is finite as it is isomorphic to the equivariant cohomology ring H * G (BP, k), see Section 2.5 and [23,31]. The functor Res C D introduced earlier leads to a restriction on cohomology…”

Section: Category Cohomology and Spectrum For Any Two Kc-modules It mentioning

confidence: 99%

“…As an example, for a group G the only overcategory Id G /• is the Cayley graph and thus B G * is the bar resolution of k in group cohomology. For the convenience of the reader we recall from [31] how we obtain equivariant cohomology from category cohomology. Since we can explicitly calculate the unique overcategory as π/• ∼ = Id G / • ×P, it follows that…”

Section: Because the Two Restrictions Res G∝pmentioning

confidence: 99%

“…Although many algebras do not satisfy (Fg.1) and (Fg.2), we can show that Snashall-Solberg theory works for block algebras of a transporter category algebra k (G ∝ P). From [31] we know that Ext * k (G∝P) (k, k) is a Noetherian graded commutative ring such that Ext * k (G∝P) (M, N) is finitely generated over it, for any pair of M, N ∈ k (G ∝ P)-mod. It was also showed there that Ext * k (G∝P) e (k (G ∝ P), k (G ∝ P)) is Noetherian.…”

Section: Varieties In Blocksmentioning

confidence: 99%

“…There are various ways to construct modules for a transporter category algebra. In [31] we examined a context of three categories…”

Section: 4mentioning

confidence: 99%

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Support varieties for transporter category algebras

2014

Journal of Pure and Applied Algebra

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Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP, while the k-category algebra of the transporter category is the (Gorenstein) skew group algebra on the G-algebra kP.We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson's support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually non selfinjective) algebras to which Snashall-Solberg's (Hochschild) support variety theory applies. Various properties will be developed. Particularly we establish a Quillen stratification for modules. G ∼ = Ext * kG (k, k), Carlson [11] extended Quillen's work by attaching to every finitely generated kG-module M a subvariety of V G = V G,• , denoted by V G (M ) = MaxSpecH G /I G (M ), called the (cohomological) support variety of M , where I G (M ) is the kernel of the following map φ M = − ⊗ k M : H * G ∼ = Ext * kG (k, k) → Ext * kG (M, M ). Especially since φ k is the identity, V G = V G (k). Following Carlson's construction, Avrunin and Scott [5] quickly generalized the Quillen stratification from V G to V G (M ). By showing that support

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Section: Category Algebras and Their Representationsmentioning

confidence: 99%

Section: Category Cohomology and Spectrum For Any Two Kc-modules It mentioning

confidence: 99%

Section: Because the Two Restrictions Res G∝pmentioning

confidence: 99%

Section: Varieties In Blocksmentioning

confidence: 99%

“…There are various ways to construct modules for a transporter category algebra. In [31] we examined a context of three categories…”

Section: 4mentioning

confidence: 99%

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Support varieties for transporter category algebras

2014

Journal of Pure and Applied Algebra

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Spectra of tensor triangulated categories over category algebras

2014

Arch. Math.

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Abstract. Let C be a finite EI category and k be a field. We consider the category algebra kC. Suppose K(C) = D b (kC-mod) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category and we compute its spectrum in the sense of Balmer. When C = G ∝ P is a finite transporter category, the category algebra becomes Gorenstein so we can define the stable module category CMk(G ∝ P), of maximal Cohen-Macaulay modules, as a quotient category of K(G ∝ P). Since CMk(G ∝ P) is also tensor triangulated, we compute its spectrum as well.These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories, despite the fact that the previously mentioned tensor categories are not rigid.

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LHS-spectral sequences for regular extensions of categories

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2024

J. Homotopy Relat. Struct.

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On local categories of finite groups

Cited by 10 publications

References 9 publications

Support varieties for transporter category algebras

Support varieties for transporter category algebras

Spectra of tensor triangulated categories over category algebras

LHS-spectral sequences for regular extensions of categories

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