Frobenius categories, Gorenstein algebras and rational surface singularities

AbstractWe give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

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On relative counterpart of Auslander’s conditions

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500159 ◽

2021 ◽

Author(s):

Driss Bennis ◽

Rachid El Maaouy ◽

J. R. García Rozas ◽

Luis Oyonarte

Keyword(s):

Noetherian Rings ◽

Projective Modules ◽

New Approach ◽

Gorenstein Rings ◽

Semidualizing Module ◽

Gorenstein Projective Modules ◽

Frobenius Category ◽

Relative Notion ◽

Frobenius Categories

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.

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RELATIONS BETWEEN MULTIPLICITY AND DIVISOR CLASS GROUP FOR RATIONAL SURFACE SINGULARITIES

International Journal of Mathematics ◽

10.1142/s0129167x10006318 ◽

2010 ◽

Vol 21 (07) ◽

pp. 915-938

Author(s):

R. V. GURJAR ◽

VINAY WAGH

Keyword(s):

Quadratic Forms ◽

Class Group ◽

Rational Surface ◽

Positive Definite ◽

Rational Singularity ◽

Divisor Class Group ◽

Divisor Class ◽

Rational Double Point ◽

Surface Singularities ◽

Rational Surface Singularity

In this paper we prove that a rational surface singularity with divisor class group ℤ/(2) is a rational double point. This generalizes a result by Brieskorn: if the divisor class group of a rational singularity is trivial then it is the E8 singularity [3]. We also prove several inequalities involving the integers e, δ, mi, [Formula: see text], where [Formula: see text] is the fundamental cycle. The proof of this result uses ideas from Minkowski's theory of reduction of positive-definite quadratic forms. We also give some interesting counterexamples to some of the related questions in this context.

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On symplectic fillings of links of rational surface singularities with reduced fundamental cycle

Nagoya Mathematical Journal ◽

10.1017/s0027763000008898 ◽

2004 ◽

Vol 175 ◽

pp. 51-57 ◽

Cited By ~ 1

Author(s):

Mohan Bhupal

Keyword(s):

Rational Surface ◽

Surface Singularity ◽

Fundamental Cycle ◽

Surface Singularities ◽

Rational Surface Singularity ◽

Symplectic Filling ◽

Rational Surface Singularities

AbstractWe prove that every symplectic filling of the link of a rational surface singularity with reduced fundamental cycle admits a rational compactification, possibly after a modification of the filling in a collar neighbourhood of the link.

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Morphism Categories of Gorenstein-projective Modules

Algebra Colloquium ◽

10.1142/s1005386718000275 ◽

2018 ◽

Vol 25 (03) ◽

pp. 377-386

Author(s):

Miantao Liu ◽

Ruixin Li ◽

Nan Gao

Keyword(s):

Matrix Algebra ◽

Triangular Matrix ◽

Projective Modules ◽

Lower Triangular Matrix ◽

Categorical Equivalence ◽

Triangular Matrix Algebra ◽

Gorenstein Projective Modules ◽

Frobenius Category ◽

Lower Triangular ◽

Auslander Algebra

Let Λ be an algebra of finite Cohen-Macaulay type and Γ its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor(Λ-Gproj) of Gorenstein-projective Λ-modules in terms of the module category Γ-mod by a categorical equivalence. Based on this, we obtain that some factor category of the epimorphism category Epi(Λ-Gproj) is a Frobenius category, and also, we clarify the relations among Mor(Λ-Gproj), Mor(T2Λ-Gproj) and Mor(Δ-Gproj), where T2(Λ) and Δ are respectively the lower triangular matrix algebra and the Morita ring closely related to Λ.

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Noncommutative G-semihereditary rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500147 ◽

2018 ◽

Vol 17 (01) ◽

pp. 1850014 ◽

Cited By ~ 2

Author(s):

Jian Wang ◽

Yunxia Li ◽

Jiangsheng Hu

Keyword(s):

Associative Ring ◽

Sufficient Condition ◽

Projective Modules ◽

The Third ◽

Flat Modules ◽

Gorenstein Projective Modules ◽

Direct Limits ◽

Gorenstein Flat ◽

Finitely Presented ◽

Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

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Gorenstein Projective Modules over Triangular Matrix Rings

Algebra Colloquium ◽

10.1142/s1005386716000122 ◽

2016 ◽

Vol 23 (01) ◽

pp. 97-104 ◽

Cited By ~ 1

Author(s):

H. Eshraghi ◽

R. Hafezi ◽

Sh. Salarian ◽

Z. W. Li

Keyword(s):

Triangular Matrix ◽

Projective Modules ◽

Matrix Rings ◽

Matrix Algebras ◽

Gorenstein Projective Modules ◽

Artin Algebras ◽

Acyclic Complexes ◽

Matrix Extension

Let R and S be Artin algebras and Γ be their triangular matrix extension via a bimodule SMR. We study totally acyclic complexes of projective Γ-modules and obtain a complete description of Gorenstein projective Γ-modules. We then construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras.

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