scholarly journals Intersection multiplicities over Gorenstein rings

2000 ◽  
Vol 317 (1) ◽  
pp. 155-171 ◽  
Author(s):  
Claudia M. Miller ◽  
Anurag K. Singh
Keyword(s):  
Gorenstein Rings ◽  
Intersection Multiplicities

scholarly journals Endofunctors of singularity categories characterizing Gorenstein rings

2015 ◽  
Vol 143 (9) ◽  
pp. 3777-3779
Author(s):  
Takuma Aihara ◽  
Ryo Takahashi
Keyword(s):  
Gorenstein Rings

Model structures, recollements and duality pairs

2021 ◽  
Author(s):  
Wenjing Chen ◽  
Zhongkui Liu
Keyword(s):  
Projective Modules ◽  
Cotorsion Pair ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Duality Pairs ◽  
Cotorsion Pairs ◽  
Gorenstein Flat ◽  
Model Structures

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

The Poincaré Series Of Stretched Cohen-Macaulay Rings

1980 ◽  
Vol 32 (5) ◽  
pp. 1261-1265 ◽  
Author(s):  
Judith D. Sally
Keyword(s):  
Local Ring ◽  
Poincaré Series ◽  
Local Rings ◽  
Embedding Dimension ◽  
Loop Spaces ◽  
Gorenstein Rings ◽  
Poincare Series ◽  
Image Position

There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré serieswhere k = R/m, has been settled. (For an example of a local ring with non-rational Poincaré series see the recent paper by D. Anick, “Construction of loop spaces and local rings whose Poincaré—Betti series are nonrational”, C. R. Acad. Sc. Paris 290 (1980), 729-732.) In this note, we compute the Poincaré series of a certain family of local Cohen-Macaulay rings and obtain, as a corollary, the rationality of the Poincaré series of d-dimensional local Gorenstein rings (R, m) of embedding dimension at least e + d – 3, where e is the multiplicity of R. It follows that local Gorenstein rings of multiplicity at most five have rational Poincaré series.

A Gorenstein Ring with Larger Dilworth Number than Sperner Number

2000 ◽  
Vol 43 (1) ◽  
pp. 100-104 ◽  
Author(s):  
James S. Okon ◽  
J. Paul Vicknair
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Embedding Dimension ◽  
Gorenstein Rings

AbstractA counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension 3 with larger Dilworth number than Sperner number. The Dilworth number of is computed when A is an unramified principal Artin local ring.

Grothendieck Groups of Bass Orders

1971 ◽  
Vol 23 (1) ◽  
pp. 103-115
Author(s):  
Klaus W. Roggenkamp
Keyword(s):  
Special Class ◽  
Grothendieck Group ◽  
Commutative Rings ◽  
Gorenstein Rings ◽  
Direct Sum ◽  
Maximal Orders ◽  
Grothendieck Groups ◽  
Epimorphic Image ◽  
Carry Over ◽  
Separable Algebras

Commutative Bass rings, which form a special class of Gorenstein rings, have been thoroughly investigated by Bass [1]. The definitions do not carry over to non-commutative rings. However, in case one deals with orders in separable algebras over fields, Bass orders can be defined. Drozd, Kiricenko, and Roïter [3] and Roïter [6] have clarified the structure of Bass orders, and they have classified them. These Bass orders play a key role in the question of the finiteness of the non-isomorphic indecomposable lattices over orders (cf. [2; 8]). We shall use the results of Drozd, Kiricenko, and Roïter [3] to compute the Grothendieck groups of Bass orders locally. Locally, the Grothendieck group of a Bass order (with the exception of one class of Bass orders) is the epimorphic image of the direct sum of the Grothendieck groups of the maximal orders containing it.

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