Frobenius functors and Gorenstein homological properties

Chen, Xiaowu; Ren, Wei

doi:10.48550/arxiv.2008.11467

Cited by 4 publications

(6 citation statements)

References 34 publications

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“…If H is a subgroup of G with finite index, then RH → RG is a Frobenius extension of rings. The Gorenstein (co)homological properties under Frobenius extensions have been extensively studied recently, see [9,16,[20][21][22]28]. From this point of view, the above result follows from [16,Corollary 4.3], which extends [22,Theorem 3.3].…”

Section: Lemma 22 ([23 Corollary 312]) Let R Be a Ring Then The Class...mentioning

confidence: 76%

On Gorenstein homological dimension of groups

Luo¹,

Ren²

2022

Preprint

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Let G be a group and R a commutative ring. We define the Gorenstein homological dimension of G over R, denoted by Ghd R G, as the Gorenstein flat dimension of trivial RG-module R. It is proved that for any flat extensionWe show a Gorenstein homological version of Serre's theorem, i.e. Ghd R G = Ghd R H for any subgroup H of G with finite index. As applications, we give a homological characterization for groups by showing that G is a finite group if and only if Ghd R G = 0, and moreover, we use Ghd R G to give an upper bound of Gorenstein flat dimension of modules over the group ring RG.

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Section: Lemma 22 ([23 Corollary 312]) Let R Be a Ring Then The Class...mentioning

confidence: 76%

On Gorenstein homological dimension of groups

Luo¹,

Ren²

2022

Preprint

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“…This is equivalent to that A is of finite global Gorenstein projective dimension (i.e. A = GP <∞ (A)); see for example [8,Theorem A.6]. If the cateogry of R-modules is a Gorenstein category, then R is called a Gorenstein regular ring (also named left-Gorenstein ring in [3]).…”

Section: Preliminariesmentioning

confidence: 99%

“…The exactness of the functor H forces that F preserves the projective objects. It follows from [8,Lemma 3.6] that F (GP(A)) ⊆ GP(B). Hence, the condition (2) of the above lemma satisfies.…”

Section: Proof It Follows From [8 Corollary 33] That a Is A Gorenstei...mentioning

confidence: 99%

Frobenius functors, stable equivalences and $K$-theory of Gorenstein projective modules

Ren¹

2022

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Owing to the difference in K-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition such that the Frobenius pair of faithful functors between two abelian categories is a Quillen equivalence, which is equivalent to that the Frobenius functors induce mutually inverse equivalences between stable categories of Gorenstein projective objects.We show that the category of Gorenstein projective objects is a Waldenhausen category, then Gorenstein K-groups are introduced and characterized. As applications, we show that stable equivalences of Morita type preserve Gorenstein K-groups, CM-finite and CM-free. Two specific examples are presented to illustrate our results, where Gorenstein K 0 and K 1 -groups are calculated.

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“…We refer to [35] for more details. Here, we avoid using categorical arguments on Frobenius functors, unlike [13,41], but give a quite direct proof for clarity and completeness.…”

Section: Gorenstein Cohomological Dimension Under Changes Of Groupsmentioning

confidence: 99%

Gorenstein cohomological dimension and stable categories for groups

Ren¹

2022

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First we study the Gorenstein cohomological dimension Gcd R G of groups G over coefficient rings R, under changes of groups and rings; a characterization for finiteness of Gcd R G is given; a Gorenstein version of Serre's theorem is proved, i.e. Gcd R H = Gcd R G for any subgroup H of G with finite index; characterizations for finite groups are given. These generalize the results in literatures, which were obtained over Z or rings of finite global dimension, to more general rings. Moreover, we establish a model structure on the weakly idempotent complete exact category F ib consisting of fibrant RGmodules, and show that the homotopy category Ho(F ib) is triangle equivalent to the stable category Cof (RG) of Benson's cofibrant modules, as well as the stable module category StMod(RG). For any commutative ring R of finite global dimension, if either G is of type Φ R or is in Kropholler's large class such that Gcd R G is finite, then Ho(F ib) is equivalent to the stable category of Gorenstein projective RG-modules, the singular category, and the homotopy category of totally acyclic complexes of projective RG-modules.

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Frobenius functors and Gorenstein homological properties

Cited by 4 publications

References 34 publications

On Gorenstein homological dimension of groups

On Gorenstein homological dimension of groups

Frobenius functors, stable equivalences and $K$-theory of Gorenstein projective modules

Gorenstein cohomological dimension and stable categories for groups

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