Auslander–Gorenstein algebras and precluster tilting

Iyama, Osamu; Solberg, Øyvind

doi:10.1016/j.aim.2017.11.025

Cited by 43 publications

(52 citation statements)

References 37 publications

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“…Recently Iyama and Solberg defined m-Auslander-Gorenstein algebra in [18]. They also showed that the notion of m-Auslander-Gorenstein algebra is left and right symmetric.…”

Section: 3mentioning

confidence: 99%

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Dominant dimension and tilting modules

et al. 2018

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We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension 2, Auslander algebras are classified by the existence of such tilting modules.In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting-cotilting module generated-cogenerated by projective-injective modules are precisely 1-Auslander-Gorenstein algebras.When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

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“…Recently Iyama and Solberg defined m-Auslander-Gorenstein algebra in [18]. They also showed that the notion of m-Auslander-Gorenstein algebra is left and right symmetric.…”

Section: 3mentioning

confidence: 99%

“…[18] An artin algebra Λ is called m-Auslander-Gorenstein 1 if injdim Λ Λ ≤ m + 1 ≤ domdim Λ.Proposition 2.4.9. [18, Proposition 4.1] Let Λ be an artin algebra.…”

mentioning

confidence: 99%

Dominant dimension and tilting modules

et al. 2018

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“…Recently, Iyama and Solberg defined m-Auslander-Gorenstein algebras in [7]. They also showed that the notion of m-Auslander-Gorenstein algebra is left and right symmetric.…”

Section: Properties Of the Subcategory C λmentioning

confidence: 99%

1-Auslander-Gorenstein algebras which are tilted

Zito

2020

Journal of Algebra

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“…Auslander-Solberg and Auslander correspondence. The Auslander-Solberg correspondence, which is defined by Iyama and Solberg [IS18], characterizes algebras Λ with domdim Λ ≥ k + 1 ≥ id Λ. In the case k = 1, this result is due to Auslander-Solberg [AS93a].…”

Section: Classical Casementioning

confidence: 99%

On faithfully balanced modules, F-cotilting and F-Auslander algebras

Sauter

2020

Journal of Algebra

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We revisit faithfully balanced modules. These are faithful modules having the double centralizer property. For finite-dimensional algebras our main tool is the category cogen 1 (M ) of modules with a copresentation by summands of finite sums of M on which Hom(−, M ) is exact. For a faithfully balanced module M the functor Hom(−, M ) is a duality on these categories -for cotilting modules this is the Brenner-Butler theorem. We also study new classes of faithfully balanced modules combining cogenerators and cotilting modules. Then we turn to relative homological algebra in the sense of Auslander-Solberg and define a relative version of faithfully balancedness which we call 1-Ffaithful. We find relative versions of the best known classes of faithfully balanced modules (including (co)generators ,(co)tilting and cluster tilting modules). Here we characterize the corresponding modules over the endomorphism ring of the faithfully balanced module -this is what we call a correspondence. Two highlights are the relative (higher) Auslander correspondence and the relative cotilting correspondence -the second is a generalization of a relative cotilting correspondence of Auslander-Solberg to an involution (as the usual cotilting correspondence is).

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Auslander–Gorenstein algebras and precluster tilting

Cited by 43 publications

References 37 publications

Dominant dimension and tilting modules

Dominant dimension and tilting modules

1-Auslander-Gorenstein algebras which are tilted

On faithfully balanced modules, F-cotilting and F-Auslander algebras

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