scholarly journals Finite Generation of the Extension Algebra Ext(M, M)

1995 ◽  
Vol 59 (3) ◽  
pp. 366-374
Author(s):  
Rainer Schulz
Keyword(s):  
Finitely Generated ◽  
Artin Algebra ◽  
Finite Generation ◽  
Extension Algebra

AbstractFor a module M Over an Artin algebra R, we discuss the question of whether the Yoneda extension algebra Ext(M, M) is finitely generated as an algebra. We give an answer for bounded modules M. (These are modules whose syzygies have direct summands of bounded lengths.)

scholarly journals Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 787-798 ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Diameter Growth ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finite Generation ◽  
Alexandrov Spaces ◽  
Geodesic Spaces ◽  
Special Case ◽  
Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

Finite presentability of generalized Bruck–Reilly ∗-extension of groups

2016 ◽  
Vol 09 (04) ◽  
pp. 1650090 ◽  
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Finite Subset ◽  
Identity Element ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Finite Presentability ◽  
Finitely Presented

In [Finite presentability of Bruck–Reilly extensions of groups, J. Algebra 242 (2001) 20–30], Araujo and Ruškuc studied finite generation and finite presentability of Bruck–Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck–Reilly ∗-extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck–Reilly ∗-extension of a group, [Formula: see text], to be finitely generated and finitely presented. Let [Formula: see text] be a group, [Formula: see text] be morphisms and [Formula: see text] ([Formula: see text] and [Formula: see text] are the [Formula: see text]- and [Formula: see text]-classes, respectively, contains the identity element [Formula: see text] of [Formula: see text]). We prove that [Formula: see text] is finitely generated if and only if there exists a finite subset [Formula: see text] such that [Formula: see text] is generated by [Formula: see text]. We also prove that [Formula: see text] is finitely presented if and only if [Formula: see text] is presented by [Formula: see text], where [Formula: see text] is a finite set and [Formula: see text] [Formula: see text] for some finite set of relations [Formula: see text].

On Algebras Stably Equivalent to an Hereditary Artin Algebra

1978 ◽  
Vol 30 (4) ◽  
pp. 817-829 ◽  
Author(s):  
María Inés Platzeck
Keyword(s):  
Finitely Generated ◽  
Artin Algebra ◽  
Finitely Generated Module ◽  
Artin Ring ◽  
Finitely Generated Modules ◽  
Hereditary Artin Algebra

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

Finite generation of the cohomology rings for the extension algebras of monomial algebras

2022 ◽  
Author(s):  
Hongbo Shi
Keyword(s):  
Cohomology Ring ◽  
Minimal Resolution ◽  
Finitely Generated ◽  
Finite Generation ◽  
Cohomology Rings ◽  
Monomial Algebra ◽  
Monomial Algebras ◽  
Resolution Graph ◽  
Dual Extension

We describe the cohomology ring of a monomial algebra in the language of dimension tree or minimal resolution graph and in this context we study the finite generation of the cohomology rings of the extension algebras, showing among others that the cohomology ring [Formula: see text] is finitely generated [Formula: see text] is [Formula: see text] is, where [Formula: see text] is the dual extension of a monomial algebra [Formula: see text] and [Formula: see text] is the opposite algebra of [Formula: see text].

On the finite generation of high dimensional cohomology ring of virtually torsion-free groups

1998 ◽  
Vol 124 (1) ◽  
pp. 57-72 ◽  
Author(s):  
CHUN-NIP LEE
Keyword(s):  
Automorphism Groups ◽  
Cohomology Ring ◽  
Free Groups ◽  
Outer Automorphism ◽  
Cohomological Dimension ◽  
Group Cohomology ◽  
Mapping Class ◽  
Finitely Generated ◽  
Finite Generation ◽  
A Finite Group

Let Γ be a discrete group and p be a prime. One of the fundamental results in group cohomology is that H*(Γ, [ ]p) is a finitely generated [ ]p-algebra if Γ is a finite group [8, 24]. The purpose of this paper is to study the analogous question when Γ is no longer finite.Recall that Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torion-free subgroup Γ′ of Γ such that Γ′ has finite cohomological dimension over ℤ [4]. By definition vcd Γ is the cohomological dimension of Γ′. It is easy to see that the mod p cohomology ring of a finite vcd-group does not have to be a finitely generated [ ]p-algebra in general. For instance, if Γ is a countably infinite free product of ℤ's, then H1(Γ, [ ]p) is not finite dimensional over [ ]p. The three most important classes of examples of finite vcd-groups in which the mod p cohomology ring is a finitely generated [ ]p-algebra are arithmetic groups [2], mapping class groups [9, 10] and outer automorphism groups of free groups [5]. In each of these examples, the proof of finite generation involves the construction of a specific Γ-complex with appropriate finiteness conditions. These constructions should be regarded as utilizing the geometry underlying these special classes of groups. In contrast, the result we prove will depend only on the algebraic structure of the group Γ.

Simple reflexive modules over Artin algebras

2019 ◽  
Vol 18 (10) ◽  
pp. 1950193
Author(s):  
René Marczinzik
Keyword(s):  
Simple Module ◽  
Positive Answer ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Large Classes ◽  
Gorenstein Algebras ◽  
Artin Algebras ◽  
Reflexive Modules

Let [Formula: see text] be an Artin algebra. It is well known that [Formula: see text] is selfinjective if and only if every finitely generated [Formula: see text]-module is reflexive. In this paper, we pose and motivate the question whether an algebra [Formula: see text] is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.

scholarly journals Rings of Frobenius operators

2014 ◽  
Vol 157 (1) ◽  
pp. 151-167 ◽  
Author(s):  
MORDECHAI KATZMAN ◽  
KARL SCHWEDE ◽  
ANURAG K. SINGH ◽  
WENLIANG ZHANG
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Injective Hull ◽  
Degree Zero ◽  
Prime Characteristic ◽  
Finitely Generated ◽  
Finite Generation

AbstractLet R be a local ring of prime characteristic. We study the ring of Frobenius operators ${\mathcal F}(E)$, where E is the injective hull of the residue field of R. In particular, we examine the finite generation of ${\mathcal F}(E)$ over its degree zero component ${\mathcal F}^0(E)$, and show that ${\mathcal F}(E)$ need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of ${\mathcal F}(E)$ in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.

Wide subcategories of finitely generated Λ-modules

2018 ◽  
Vol 17 (05) ◽  
pp. 1850082 ◽  
Author(s):  
Eduardo N. Marcos ◽  
Octavio Mendoza ◽  
Corina Sáenz ◽  
Valente Santiago
Keyword(s):  
Wide Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Necessary And Sufficient

We explore some properties of wide subcategories of the category [Formula: see text] of finitely generated left [Formula: see text]-modules, for some artin algebra [Formula: see text] In particular we look at wide finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras. Furthermore, for a wide class [Formula: see text] in [Formula: see text] we give necessary and sufficient conditions to see that [Formula: see text] for some projective [Formula: see text]-module [Formula: see text] and finally, a connection with ring epimorphisms is given.

scholarly journals Generation of diagonal acts of some semigroups of transformations and relations

2005 ◽  
Vol 72 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Peter Gallagher ◽  
Nik Ruškuc
Keyword(s):  
Symmetric Group ◽  
Binary Relations ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Infinite Set

The diagonal right (respectively, left) act of a semigroup S is the set S × S on which S acts via (x, y) s = (xs, ys) (respectively, s (x, y) = (sx, sy)); the same set with both actions is the diagonal bi-act. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set A ⊆ S × S such that S × S = AS1 (respectively, S × S = S1A, S × S = SlASl).In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semi-groups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts.

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