Complexity and qoutient categories for group algebras

Carlson, Jon; Donovan, Peter; Wheeler, Wayne W.

doi:10.1016/0022-4049(94)90109-0

Cited by 15 publications

(5 citation statements)

References 4 publications

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“…Our first main result is such a theorem for an arbitrary M∆C through the use of localization functors. It can be thought of as analogous to block decompositions that encompass specific cases proved earlier in [21], [14]. Our results also provide a decomposition theorem for related indempotent algebra objects of an M∆C, which are algebra objects whose product map is an isomorphism, see Definition 2.3.1; these are the natural monoidal triangulated categorical analogues of idempotents of an algebra.…”

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confidence: 60%

Noncommutative tensor triangular geometry

Nakano¹,

Vashaw²,

Yakimov³

2022

American Journal of Mathematics

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In this paper the authors prove fundamental decomposition theorems pertaining to the internal structure of monoidal triangulated categories (M∆Cs). The tensor structure of an M∆C enables one to view these categories like (noncommutative) rings and to attempt to extend the key results for the latter to the categorical setting. The main theorem is an analogue of the Chinese Remainder Theorem involving the Verdier quotients for coprime thick ideals. This result is used to obtain orthogonal decompositions of the extended endomorphism rings of idempotent algebra objects of M∆Cs. The authors also provide topological characterizations on when an M∆C contains a pair of coprime proper thick ideals, and additionally, when the latter are complementary in the sense that their intersection is contained in the prime radical of the category.As an application of the aforementioned results, the authors establish for arbitrary M∆Cs a general version of Carlson's theorem on the connnectedness of supports for indecomposable objects. Examples of our results are given at the end of the paper for the derived category of schemes and for the stable module categories for finite group schemes.

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confidence: 60%

Noncommutative tensor triangular geometry

Nakano¹,

Vashaw²,

Yakimov³

2022

American Journal of Mathematics

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“…This follows from the theorem and the fact that every object in % r /^l r _ 1 is periodic. I It is proved in Carlson, Donovan and Wheeler [7] that the homomorphisms Hom*(M,N) in JIJ Jl c _ x have the following description. Let S M be the multiplicative subset of H*(G, k) consisting of the homogeneous elements which do not belong to any prime ideal of dimension c containing the annihilator of Ext£ G (ilf,.M).…”

Section: Notation and Statement Of Main Resultsmentioning

confidence: 99%

Decomposing the complexity quotient category

Benson

1996

Math. Proc. Camb. Phil. Soc.

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In the modular representation theory of finite groups, much recent effort has gone into describing cohomological properties of the category of finitely generated modules. In recent joint work of the author with Jon Carlson and Jeremy Rickard[3], it has become clear that for some purposes the finiteness restriction is undesirable. In particular, in the quotient category of kG-modules by the subcategory of modules of less than maximal complexity, it turns out that finitely generated modules can have infinitely generated summands, and that including these summands in the category repairs the lack of Krull–Schmidt property.

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“…Here, the exterior generators v i have degree one, and the Bockstein of v i is u i of degree two. It follows from the work of Carlson, Donovan and Wheeler in [12] and that of Carlson and Wheeler in [13] that as a ring and as a module over H * (E, k), we have…”

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confidence: 99%

Phantom maps and purity in modular representation theory, I

Benson¹,

Gnacadja²

1999

Fund. Math.

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Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → N between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need to compose to get a map which factors through a projective module. However, this bound is not sharp. For example, for the group Z/4 × Z/2 in characteristic two, the composite of 6 phantom maps always factors through a projective module, whereas the pure global dimension of the group algebra can be arbitrarily large.

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Complexity and qoutient categories for group algebras

Cited by 15 publications

References 4 publications

Noncommutative tensor triangular geometry

Noncommutative tensor triangular geometry

Decomposing the complexity quotient category

Phantom maps and purity in modular representation theory, I

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