scholarly journals Some properties of non-commutative regular graded rings

1992 ◽  
Vol 34 (3) ◽  
pp. 277-300 ◽  
Author(s):  
Thierry Levasseur
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Supplementary Condition ◽  
Graded Rings ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Commutative Case ◽  
Finite Global Dimension ◽  
Image Position ◽  
Dimension One

Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a supplementary condition. Before stating it, we recall that the grade of a finitely generated (left or right) module is defined by

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

1996 ◽  
Vol 119 (3) ◽  
pp. 425-445 ◽  
Author(s):  
D. Kirby ◽  
D. Rees
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Noetherian Ring ◽  
Free Module ◽  
Graded Rings ◽  
Finitely Generated ◽  
Graded Ring ◽  
The Subject ◽  
Integral Equivalence ◽  
Image Position

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

2015 ◽  
Vol 67 (1) ◽  
pp. 28-54 ◽  
Author(s):  
Javad Asadollahi ◽  
Rasool Hafezi ◽  
Razieh Vahed
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Derived Categories ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Finite Global Dimension

AbstractWe study bounded derived categories of the category of representations of infinite quivers over a ring R. In case R is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

The Artin–Rees equations

1987 ◽  
Vol 102 (3) ◽  
pp. 385-387
Author(s):  
A. Caruth
Keyword(s):  
Prime Ideal ◽  
Unique Solution ◽  
Noetherian Ring ◽  
Identity Element ◽  
Solution Set ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Maximal Force ◽  
The Family ◽  
Image Position

Let R denote a commutative Noetherian ring with an identity element and N a finitely generated R -module. When K is a submodule of N and A an ideal of R the Artin–Rees lemma states that there is an integer q ≥ 0 such that AnN ∩ K = An−q(AqN ∩ K) for all n ≥ q (Rees[4]; Northcott [3], theorem 20, p. 210; Atiyah and Macdonald [1], proposition 10·9, p. 107; Nagata [2], theorem (3·7), p. 9). The above equation belongs to the family of module equations involving A and K which is considered below. We characterize, in terms of A and K, the set of submodules X of N for which there is an integer q = q(X) ≥ 0 satisfying the equationEquation (1), which we call the Artin–Rees equation related to A and K, gets its maximal force when X is largest and we determine the best possible solution in this sense. Notice that for any submodule X satisfying (1), X ⊆ K:NAn for all n ≥ q(X). Since N is a Noetherian R-module ([3], proposition 1 (corollary), p. 177), there is an integer t ≥ 1 such that K:NAt = K:NAt+n for all n ≥ 0. We define M = K:NAt and prove, in Theorem 1, that X = Q satisfies equation (1), for a suitable integer q(Q) ≥ 0, if and only if K ⊆ Q:NAυ ⊆ M for some integer υ ≥ 0. In topological terms, the A-adic topology of K coincides with the topology induced by the A-adic topology of N on the subspace Q if the inequality K ⊆ Q:NAυ ⊆ M is satisfied. It follows that the solution set of equation (1) includes every submodule of N of the form An−rK when n ≥ r = q(K) as well as every submodule lying between K and M. Hence, X = M is the strongest solution, in the sense that M is the largest such submodule contained in An−s (AsN ∩ K): NAn for all n ≥ s = q(M). Recall that M is strictly larger than K if and only if A is contained in at least one prime ideal of R belonging to K ([3], theorem 14 (corollary 1), p. 193). Thus, equation (1) has a unique solution (necessarily X = K) if and only if A is not contained in any prime ideal of R belonging to any solution.

scholarly journals The finiteness of the Gorenstein dimension for Artin algebras

2019 ◽  
Vol 18 (06) ◽  
pp. 1950112
Author(s):  
René Marczinzik
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Artin Algebra ◽  
Artinian Rings ◽  
Gorenstein Dimension ◽  
Finite Global Dimension ◽  
Artin Algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

Pro- Iwahori–Hecke algebras are Gorenstein

2013 ◽  
Vol 13 (4) ◽  
pp. 753-809 ◽  
Author(s):  
Rachel Ollivier ◽  
Peter Schneider
Keyword(s):  
Upper Bound ◽  
Reductive Group ◽  
Global Dimension ◽  
Arbitrary Field ◽  
Graded Rings ◽  
Injective Dimension ◽  
Locally Compact ◽  
Invariant Vectors ◽  
Residue Characteristic ◽  
Iwahori Subgroup

AbstractLet$\mathfrak{F}$be a locally compact nonarchimedean field with residue characteristic$p$, and let$\mathrm{G} $be the group of$\mathfrak{F}$-rational points of a connected split reductive group over$\mathfrak{F}$. For$k$an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke$k$-algebra${\mathrm{H} }^{\prime } $and of the pro-$p$Iwahori–Hecke$k$-algebra$\mathrm{H} $of$\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of$\mathrm{G} $. If$\mathrm{G} $is semisimple, we also show that this upper bound is sharp, that both$\mathrm{H} $and${\mathrm{H} }^{\prime } $are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of$\mathrm{H} $(respectively${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to$\mathrm{H} $and${\mathrm{H} }^{\prime } $.When$k$has characteristic$p$, we prove that in ‘most’ cases$\mathrm{H} $and${\mathrm{H} }^{\prime } $have infinite global dimension. In particular, we deduce that the category of smooth$k$-representations of$\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$generated by their invariant vectors under the pro-$p$Iwahori subgroup has infinite global dimension (at least if$k$is algebraically closed).

scholarly journals Behavior of the Auslander condition with respect to regradings

2019 ◽  
Vol 18 (09) ◽  
pp. 1950168 ◽  
Author(s):  
G.-S. Zhou ◽  
Y. Shen ◽  
D.-M. Lu
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Noetherian Ring ◽  
Invertible Element ◽  
Quotient Ring ◽  
Global Dimension ◽  
Injective Dimension ◽  
Graded Ring ◽  
Ideal Formula ◽  
The Ideal

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen–Macaulay property from the same perspective as that for the Auslander condition. A key step of our approach is to establish homological relations between a graded ring [Formula: see text], its quotient ring modulo the ideal [Formula: see text] and its localization ring with respect to the Ore set [Formula: see text], where [Formula: see text] is a homogeneous regular normal non-invertible element of [Formula: see text].

Remarks on Op and Towber Rings

1970 ◽  
Vol 22 (6) ◽  
pp. 1109-1117 ◽  
Author(s):  
David Lissner ◽  
Anthony Geramita
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Finitely Generated ◽  
Outer Product ◽  
Dimension 2

In this paper all rings considered have identity and are commutative, and all modules are finitely generated. We shall make liberal use of the definitions and notation established in [6; 7].Towber observed in [9] that a local Outer Product ring (OP-ring) must have v-dimension ≦ 2, and so a local OP-ring is either regular of global dimension ≦ 2 or it has infinite global dimension. Since the global dimension of a noetherian ring is the supremum of the global dimensions of its localizations, we immediately obtain the following result.THEOREM 1.1. The global dimension of a noetherian OP-ring is either∞ or ≦ 2.

scholarly journals Modules over polycyclic groups have many irreducible images

1981 ◽  
Vol 22 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Kenneth A. Brown
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Left Module ◽  
Image Position ◽  
Polycyclic Groups ◽  
Left Annihilator ◽  
The Ideal

Recall that a Noetherian ring R is a Hilbert ring if the Jacobson radical of every factor ring of R is nilpotent. As one of the main results of [13], J. E. Roseblade proved that if J is a commutative Hilbert ring and G is a polycyclic-by-finite group then JG is a Hilbert ring. The main theorem of this paper is a generalisation of this result in the case where all the field images of J are absolute fields—we shall say that J is absolutely Hilbert. The result is stated in terms of the (Gabriel–Rentschler–) Krull dimension; the definition and basic properties of this may be found in [5]. Let M be a finitely generated right module over the ring R. We write AnnR(M) (or just Ann(M)) for the ideal {r ∈ R: Mr = 0}, the annihilator of M in R. If M is also a left module, its left annihilator will be denoted l-AnnR(M). If R is a group ring JG, put

scholarly journals Localizations of injective modules

1985 ◽  
Vol 28 (3) ◽  
pp. 289-299 ◽  
Author(s):  
K. R. Goodearl ◽  
D. A. Jordan
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Simple Module ◽  
Global Dimension ◽  
Injective Hull ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Noetherian Domain ◽  
Left And Right

The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

scholarly journals Injective homogeneity and the Auslander–Gorenstein property

1995 ◽  
Vol 37 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Zhong Yi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Global Dimension ◽  
Noetherian Rings ◽  
Connected Component ◽  
Injective Dimension ◽  
The Right

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

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