Phantom maps and purity in modular representation theory, I

Benson, David J.; Gnacadja, Gilles

doi:10.4064/fm-161-1-2-37-91

Cited by 33 publications

(19 citation statements)

References 8 publications

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“…This material is taken mostly from Rickard's paper [12]. Variations on the theme and other accounts, can be found in [8], [6] and the last section of [5])…”

Section: Idempotent Modulesmentioning

confidence: 99%

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Negative cohomology and the endomorphism ring of the trivial module

Carlson¹

2020

Preprint

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Let k be a field of characteristic 2 and let H be a finite group or group scheme. We show that the negative Tate cohomology ring H ≤0 (H, k) can be realized as the endomorphism ring of the trivial module in a Verdier localization of the stable category of kG-modules for G an extension of H. This means in some cases that the endomorphism of the trivial module is a local ring with infinitely generated radical with square zero. This stands in stark contrast to some known calculations in which the endomorphism ring of the trivial module is the degree zero component of a localization of the cohomology ring of the group.

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“…This material is taken mostly from Rickard's paper [12]. Variations on the theme and other accounts, can be found in [8], [6] and the last section of [5])…”

Section: Idempotent Modulesmentioning

confidence: 99%

“…In the few cases where End C (k) has been computed, it turned out to be some sort of homogeneous localization of the cohomology ring of the group (see [12], [8], [5]). However, in all of those examples, the localization is with respect to a thick tensor ideal determined by a subvariety that is a hypersurface or union of hypersurfaces.…”

Section: Introductionmentioning

confidence: 99%

Negative cohomology and the endomorphism ring of the trivial module

Carlson¹

2020

Preprint

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“…Here they serve a dual purpose: the one just mentioned and that of replacing minimal A-approximations in case such approximations fail to exist. Our phantoms under (3) below are not to be confused with the phantom maps of algebraic topology, or the topology-inspired phantom maps in the modular representation theory of groups, as introduced by Benson and Gnacadja in [4].…”

Section: Background On Contravariant Finiteness and Phantomsmentioning

confidence: 99%

The homology of string algebras I

Huisgen-Zimmermann

Smalø

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

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Dedikert til vår venn og kollega Idun Reiten i andledning hennes sekstiårsdag Abstract. We show that string algebras are 'homologically tame' in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra Λ are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of Λ can be computed from a finite set of cyclic left ideals contained in the Jacobson radical. Second, our main result shows that the functorial finiteness status of the full subcategory P <∞ (Λ -mod) consisting of the finitely generated left Λ-modules of finite projective dimension is completely determined by a finite number of, possibly infinite dimensional, string modules -one for each simple Λ-module -which are algorithmically constructible from quiver and relations of Λ. Namely, P <∞ (Λ -mod) is contravariantly finite in Λ -mod precisely when all of these string modules are finite dimensional, in which case they coincide with the minimal P <∞ (Λ -mod)-approximations of the corresponding simple modules. Yet, even when P <∞ (Λ -mod) fails to be contravariantly finite, these 'characteristic' string modules encode, in an accessible format, all desirable homological information about Λ -mod.

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“…Neeman [N] introduced this concept into the context of triangulated categories. The theory also was developed in the stable category of a finite group ring in a series of works of Benson and Gnacadja [BG1,BG2,G].…”

Section: Introductionmentioning

confidence: 99%

Higher Ideal Approximation Theory

Asadollahi¹,

Sadeghi²

2020

Preprint

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Let C be an n-cluster tilting subcategory of an exact category (A , E ), where n ≥ 1 is an integer. It is proved by Jasso that if n > 1, then C although is no longer exact, but has a nice structure known as n-exact structure. In this new structure conflations are called admissible n-exact sequences and are E -acyclic complexes with n + 2 terms in C . Since their introduction by Iyama, cluster tilting subcategories has gained a lot of traction, due largely to their links and applications to many research areas, many of them unexpected. On the other hand, ideal approximation theory, that is a gentle generalization of the classical approximation theory and deals with morphisms and ideals instead of objects and subcategories, is an active area that has been the subject of several researches. Our aim in this paper is to introduce the so-called 'ideal approximation theory' into 'higher homological algebra'. To this end, we introduce some important notions in approximation theory into the theory of n-exact categories and prove some results. In particular, the higher version of the notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals will be introduced and studied. Our results motivate the definitions and show that n-exact categories are the appropriate context for the study of 'higher ideal approximation theory'.

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Phantom maps and purity in modular representation theory, I

Cited by 33 publications

References 8 publications

Negative cohomology and the endomorphism ring of the trivial module

Negative cohomology and the endomorphism ring of the trivial module

The homology of string algebras I

Higher Ideal Approximation Theory

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