scholarly journals Cycle-Finite Module Categories

2013 ◽  
pp. 209-252 ◽  
Author(s):  
Piotr Malicki ◽  
José A. de la Peña ◽  
Andrzej Skowroński
Keyword(s):  
Finite Module

scholarly journals On algebra with universal finite module of differentials

1981 ◽  
Vol 83 ◽  
pp. 107-121 ◽  
Author(s):  
Norio Yamauchi
Keyword(s):  
Local Ring ◽  
Positive Characteristic ◽  
Regularity Theorem ◽  
Local Case ◽  
Finite Module ◽  
Characteristic Case ◽  
Differential Modules ◽  
Module Of Differentials

Let k be a field and A a noetherian k-algebra. In this note, we shall study the universal finite module of differentials of A over k, which is denoted by Dk(A). When the characteristic of k is zero, detailed results have been obtained by Scheja and Storch [8]. So we shall treat the positive characteristic case. In § 1, we shall study differential modules of a local ring over subfields. We obtain a criterion of regularity (Theorem (1.14)). In § 2, we shall study the formal fibres and regular locus of A with Dk(A). Our main result is Theorem (2.1) which shows that, if Dk(A) exists, then A is a universally catenary G-ring under a certain assumption. In the local case, this is a generalization of Matsumura’s theorem ([5] Theorem 15), where regularity of A is assumed.

scholarly journals Local Homology, Cohen–Macaulayness and Cohen–Macaulayfications

2008 ◽  
Vol 15 (01) ◽  
pp. 63-68
Author(s):  
Michael Hellus
Keyword(s):  
Local Ring ◽  
Finite Length ◽  
Necessary Condition ◽  
Direct Generalization ◽  
Local Homology ◽  
Finiteness Result ◽  
Finite Module ◽  
Noetherian Dimension

Let (R,𝔪) be a local ring, X an artinian R-module of noetherian dimension d; let x1,…,xd ∈ 𝔪 be such that 0:X (x1,…,xd)R has finite length. We show by an example that [Formula: see text] is not finite as an R-module in general; it is finite if we assume R is complete. This answers a question posed by Tang. As a first application of the latter finiteness result, we give a necessary condition for a finite module to be Cohen–Macaulay; secondly we propose a notion of Cohen–Macaulayfication and prove its uniqueness; finally we show that this new notion of Cohen–Macaulayfication is a direct generalization of a notion of Cohen–Macaulayfication introduced by Goto.

Affine algebras of Gelfand-Kirillov dimension one are PI

1985 ◽  
Vol 97 (3) ◽  
pp. 407-414 ◽  
Author(s):  
L. W. Small ◽  
J. T. Stafford ◽  
R. B. Warfield
Keyword(s):  
Polynomial Identity ◽  
Prime Radical ◽  
Finitely Generated ◽  
Finite Module ◽  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension ◽  
Algebra Over A Field

The aim of this paper is to prove:Theorem.Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

scholarly journals Artinian and Non-Artinian Local Cohomology Modules

2011 ◽  
Vol 54 (4) ◽  
pp. 619-629 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Alireza Vahidi
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Maximal Element ◽  
Local Cohomology ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Finite Module ◽  
Bass Numbers ◽  
Cohomology Modules

AbstractLet M be a finite module over a commutative noetherian ring R. For ideals a and b of R, the relations between cohomological dimensions of M with respect to a, b, a ⋂ b and a + b are studied. When R is local, it is shown that M is generalized Cohen–Macaulay if there exists an ideal a such that all local cohomology modules of M with respect to a have finite lengths. Also, when r is an integer such that 0 ≤ r < dimR(M), any maximal element q of the non-empty set of ideals ﹛a : (M) is not artinian for some i, i ≥ r} is a prime ideal, and all Bass numbers of (M) are finite for all i ≥ r.

scholarly journals On the Computation of Certain Homotopical Functors

1998 ◽  
Vol 1 ◽  
pp. 25-41 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Integral Homology ◽  
Finitely Presented Groups ◽  
The Third ◽  
Finite Module ◽  
Nonabelian Tensor ◽  
A Finite Group ◽  
Finitely Presented

AbstractThis paper provides details of a Magma computer program for calculating various homotopy-theoretic functors, defined on finitely presented groups. A copy of the program is included as an Add-On. The program can be used to compute: the nonabelian tensor product of two finite groups, the first homology of a finite group with coefficients in the arbirary finite module, the second integral homology of a finite group relative to its normal subgroup, the third homology of the finite p-group with coefficients in Zp, Baer invariants of a finite group, and the capability and terminality of a finite group. Various other related constructions can also be computed.

scholarly journals Dual of the Auslander-Bridger formula and GF-perfectness

2007 ◽  
Vol 101 (1) ◽  
pp. 5 ◽  
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Local Cohomology Module ◽  
Injective Dimension ◽  
Gorenstein Dimension ◽  
Finite Module ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective

Ext-finite modules were introduced and studied by Enochs and Jenda. We prove under some conditions that the depth of a local ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of an Ext-finite module of finite Gorenstein injective dimension. Let $(R,\mathfrak m)$ be a local ring. We say that an $R$-module $M$ with $\dim_R M=n$ is a Grothendieck module if the $n$-th local cohomology module of $M$ with respect to $\mathfrak m$, $\mathrm{H}_{\mathfrak m}^n (M)$, is non-zero. We prove the Bass formula for this kind of modules of finite Gorenstein injective dimension and of maximal Krull dimension. These results are dual versions of the Auslander-Bridger formula for the Gorenstein dimension. We also introduce GF-perfect modules as an extension of quasi-perfect modules introduced by Foxby.

scholarly journals An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras

2018 ◽  
Vol 2020 (13) ◽  
pp. 4103-4143 ◽  
Author(s):  
Dražen Adamović ◽  
Victor G Kac ◽  
Pierluigi Möseneder Frajria ◽  
Paolo Papi ◽  
Ozren Perše
Keyword(s):  
Maximal Ideal ◽  
Vertex Algebra ◽  
Lie Superalgebras ◽  
Vertex Algebras ◽  
Locally Finite ◽  
Complete Reducibility ◽  
Finite Module ◽  
Nontrivial Ideal ◽  
Conformal Embeddings ◽  
Reducibility Result

Abstract We discover a large class of simple affine vertex algebras $V_{k} ({\mathfrak{g}})$, associated to basic Lie superalgebras ${\mathfrak{g}}$ at non-admissible collapsing levels $k$, having exactly one irreducible ${\mathfrak{g}}$-locally finite module in the category ${\mathcal O}$. In the case when ${\mathfrak{g}}$ is a Lie algebra, we prove a complete reducibility result for $V_k({\mathfrak{g}})$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k ({\mathfrak{g}})$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one nontrivial ideal.

Gröbner bases for quadratic algebras of skew type

2012 ◽  
Vol 55 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Regular Language ◽  
Normal Forms ◽  
Free Monoid ◽  
Lexicographic Order ◽  
Semigroup Algebra ◽  
Baxter Equation ◽  
Finitely Generated ◽  
Quadratic Algebras ◽  
Defining Relations ◽  
Finite Module

AbstractNon-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and $\binom{n}{2}$ quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

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