Graded and Filtered Rings and Modules

Năstăsescu, C.; Oystaeyen, Freddy Van

doi:10.1007/bfb0067331

Cited by 116 publications

(69 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(d) Recall that the global dimension of a connected graded noetherian algebra equals the projective dimension of its trivial module (e.g., [23,Chapter 1,Corollary 8.7]). For A 1 and A 2 , this means…”

Section: Proof Of Theorem 01mentioning

confidence: 99%

Homological properties of quantized coordinate rings of semisimple groups

Goodearl

Zhang

2007

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. We prove that the generic quantized coordinate ring Oq(G) is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group G. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a new set of positive cases for a question of Brown and the first author. IntroductionA guiding principle in the study of quantized coordinate rings has been that these algebras should enjoy noncommutative versions of the algebraic properties of their classical analogs. Moreover, based on the types of properties that have been established, one also conjectures that quantized coordinate rings should enjoy properties similar to the enveloping algebras of solvable Lie algebras. A property of the latter type is the catenary condition (namely, that all saturated chains of prime ideals between any two fixed primes should have the same length), which was established for a number of quantized coordinate rings by Lenagan and the first author [11]. However, among the quantized coordinate rings O q (G) for semisimple Lie groups G, catenarity has remained an open question except for the case G = SL n [11, Theorem 4.5]. The first goal of the present paper is to establish catenarity for (the C-form of) all the algebras O q (G).Following the principle indicated above, one expects the quantized coordinate rings of Lie groups to be homologically nice; in the noncommutative world, this means one looks for the Auslander-regular and Cohen-Macaulay conditions. In fact, Gabber's method for proving catenarity in enveloping algebras of solvable Lie algebras relies crucially on these properties, as does Lenagan and the first author's adaptation to quantum algebras. These conditions were verified for O q (SL n ) by Levasseur and Stafford, but remained open for arbitrary O q (G), although Brown and the first author were able to show that O q (G) has finite global dimension [6, Proposition 2.7]. Our second goal here is to establish the Auslander-regular and Cohen-Macaulay conditions for general O q (G).Let G be a connected semisimple Lie group over C. By O q (G), we mean the subalgebra of the Hopf dual of U q (g), where g is the Lie algebra of G, generated by the coordinate functions of the type 1 highest weight modules whose weights

show abstract

Section: Proof Of Theorem 01mentioning

confidence: 99%

Homological properties of quantized coordinate rings of semisimple groups

Goodearl

Zhang

2007

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Algebras graded over a small category generalise the widely known group graded algebras (see [16,12,8,15,14]), the recently introduced groupoid graded algebras (see [10,11,9]), and Z-algebras, used in the theory of operads (see [17]). …”

Section: Introductionmentioning

confidence: 99%

Semiperfect Category-Graded Algebras

Yudin

2010

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…These results are used to show that over a graded left artinian ring every simple module, every projective module, and every injective left module is isomorphic to a graded module, as is every direct summand of a finitely generated graded left module. Also they yield information about the relative structure of a graded ring and its initial subring, for example, a finitely graded ring is semiprimary if and only if so is its initial subring.Recall (see [3,6]) that a (Z-) graded ring is a ring R together with an abelian group decomposition R = (BzRn such that R"Rm C Rn+m (n, m E Z), and that R0 is a unital subring called the initial subring of R. We let J = J(R) denote the radical of R and J0 = J(R0). Our first result, generalizing [4, Theorem 3.1], requires no more than this.…”

mentioning

confidence: 99%

“…Recall (see [3,6]) that a (Z-) graded ring is a ring R together with an abelian group decomposition R = (BzRn such that R"Rm C Rn+m (n, m E Z), and that R0 is a unital subring called the initial subring of R. We let J = J(R) denote the radical of R and J0 = J(R0). Our first result, generalizing [4, Theorem 3.1], requires no more than this.…”

mentioning

confidence: 99%

“…A left F-module AÍ is graded [3,6] According to [6, Lemma 3.3.6, p. 9] if any two maps in an equation f-gh of F-maps between graded F-modules are of degree zero, then the third can be replaced by a degree zero homomorphism. It follows that any homogeneous F-direct Summand of a graded F-module has at least one homogeneous complement, and that graded projective (injective) (semisimple) F-modules are projective (injective) (semisimple) in F-Gr.…”

mentioning

confidence: 99%

See 1 more Smart Citation

On Graded Rings with Finiteness Conditions

Camillo¹,

Fuller²

1982

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

Abstract. It is proved that a graded ring that is finitely graded modulo its radical is local if its initial subring is local, and that a graded artinian ring is finitely generated over its initial subring which is also artinian. These results extend theorems of Gordon and Green on artin algebras. Other results relating the structure of a graded ring to that of its initial subring are also presented.In this note we provide simple proofs of extensions of the principal results in R. Gordon and E. Green's recent paper [4] on graded artin algebras. First we prove that any finitely graded (modulo the radical) ring with local initial subring is itself local. From this it follows that over any graded ring a finitely graded module with a composition series is an indecomposable module if and only if it is indecomposable as a graded module. Then we prove that a graded ring is left artinian if and only if it has a composition series as a left module over its initial subring. These results are used to show that over a graded left artinian ring every simple module, every projective module, and every injective left module is isomorphic to a graded module, as is every direct summand of a finitely generated graded left module. Also they yield information about the relative structure of a graded ring and its initial subring, for example, a finitely graded ring is semiprimary if and only if so is its initial subring.Recall (see [3,6]) that a (Z-) graded ring is a ring R together with an abelian group decomposition R = (BzRn such that R"Rm C Rn+m (n, m E Z), and that R0 is a unital subring called the initial subring of R. We let J = J(R) denote the radical of R and J0 = J(R0). Our first result, generalizing [4, Theorem 3.1], requires no more than this.1. Theorem. Let R= (BzRnbe a graded ring. If R0 is local and Rn ÇJ for all but finitely many n G Z, then R is a local ring.Proof. By hypothesis there is an N > 0 such that R" CJ whenever | n \ > N. Thus each Rn with n ¥= 0 consists of zero divisors modulo /, and it follows that R"R" contains no units. Therefore, since R0 is local, R"R-n C J0 whenever n ¥= 0. Let k > 0 and suppose, inductively, that R" QJ whenever \n\> k. Let bk E Rk, a = 2 a" E R, and x -1 -bk a. To see that bk E J we show that x is in vertible. Since

show abstract

Graded and Filtered Rings and Modules

Cited by 116 publications

References 0 publications

Homological properties of quantized coordinate rings of semisimple groups

Homological properties of quantized coordinate rings of semisimple groups

Semiperfect Category-Graded Algebras

On Graded Rings with Finiteness Conditions

Contact Info

Product

Resources

About