Gorenstein homological invariant properties under Frobenius extensions

Zhao, Zhibing

doi:10.1007/s11425-018-9432-2

Cited by 14 publications

(9 citation statements)

References 25 publications

(41 reference statements)

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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: 75%

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: 75%

On Gorenstein homological dimension of groups

Luo¹,

Ren²

2022

Preprint

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“…Applying Theorem 3.2 to the forgetful functor U , we infer that a left S-module S M is Gorenstein projective if and only if the underlying R-module R M is Gorenstein projective. This result is due to [28, Theorem 2.2] and[35, Theorem 3.2].…”

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confidence: 92%

“…The following fact seems to be well known: for a finite group G, a module over RG is Gorenstein projective if and only if its underlying R-module is Gorenstein projective; compare [6, Subsection 8.2] and [10]. This motivates the work [27,28,35], where it is proved that Frobenius extensions between rings preserve and reflect Gorenstein projective modules; compare [17]. Similarly, the forgetful functor from the category of cochain complexes to the category of graded modules preserves and reflects Gorenstein projective objects [33,34,27].…”

Section: Introductionmentioning

confidence: 99%

Frobenius functors and Gorenstein homological properties

Chen

Ren

2020

Preprint

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We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. In the appendix, we give a direct proof of the following known result: for an abelian category with enough projectives and injectives, its global Gorenstein projective dimension coincides with its global Gorenstein injective dimension.

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“…There are other examples of Frobenius extensions including full matrix ring over a base ring, Azumaya algebra over the base ring, finite extensions of enveloping algebras of Lie super-algebra and finite extensions of enveloping algebras of Lie coloralgebras etc [11,23]. More examples of Frobenius extension can be found in [24,Example 2.4] . We refer to a lecture due to [19] for more details.…”

Section: Introductionmentioning

confidence: 99%