The existence of n-Auslander–Reiten sequences via determined morphisms

2021 ◽  
Author(s):  
Zongyang Xie ◽  
Zhongkui Liu ◽  
Xiaoyan Yang
Keyword(s):  
Artinian Ring ◽  
Finitely Generated ◽  
Stable Category ◽  
Equivalent Characterization ◽  
Stable Categories

Let [Formula: see text] be a commutative artinian ring and [Formula: see text] a small Ext-finite Krull–Schmidt [Formula: see text]-abelian [Formula: see text]-category with enough projectives and injectives. We introduce two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text] in terms of the representable functors from the stable category of [Formula: see text] to category of finitely generated [Formula: see text]-modules. Moreover, we define two additive functors [Formula: see text] and [Formula: see text], which are mutually quasi-inverse equivalences between the stable categories of this two full subcategories. We give an equivalent characterization on the existence of [Formula: see text]-Auslander–Reiten sequences using determined morphisms.

scholarly journals Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components

1998 ◽  
Vol 40 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Zygmunt Pogorzały
Keyword(s):  
Unit Element ◽  
Global Dimension ◽  
Closed Field ◽  
Modular Representation Theory ◽  
Stable Category ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Stable Categories

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.

Commutative rings with infinitely many maximal subrings

2014 ◽  
Vol 13 (07) ◽  
pp. 1450037 ◽  
Author(s):  
Alborz Azarang ◽  
Greg Oman
Keyword(s):  
Direct Product ◽  
Noetherian Ring ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Semilocal Ring ◽  
Finitely Generated ◽  
Fixed Ring ◽  
Irreducible Elements ◽  
Semisimple Ring ◽  
Nonzero Characteristic

It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

scholarly journals Combinatorial study of stable categories of graded Cohen–Macaulay modules over skew quadric hypersurfaces

2021 ◽  
Author(s):  
Akihiro Higashitani ◽  
Kenta Ueyama
Keyword(s):  
Representation Theory ◽  
Polynomial Algebra ◽  
Derived Category ◽  
Stable Category ◽  
Irreducible Components ◽  
Point Scheme ◽  
Skew Polynomial ◽  
Stable Categories

AbstractIn this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let S be a graded ($$\pm 1$$ ± 1 )-skew polynomial algebra in n variables of degree 1 and $$f =x_1^2 + \cdots +x_n^2 \in S$$ f = x 1 2 + ⋯ + x n 2 ∈ S . We prove that the stable category $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed using the four graphical operations. As a consequence, $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) is equivalent to the derived category $$\mathsf {D}^{\mathsf {b}}({\mathsf {mod}}\,k^{2^r})$$ D b ( mod k 2 r ) , and this r is obtained as the nullity of a certain matrix over $${\mathbb F}_2$$ F 2 . Using the properties of Stanley–Reisner ideals, we also show that the number of irreducible components of the point scheme of S that are isomorphic to $${\mathbb P}^1$$ P 1 is less than or equal to $$\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) $$ r + 1 2 .

A CONSTRUCTION OF RIGHT ARTINIAN RIGHT SERIAL RINGS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350057
Author(s):  
SURJEET SINGH
Keyword(s):  
Homomorphic Image ◽  
Artinian Ring ◽  
Finitely Generated ◽  
Serial Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Universal Construction ◽  
Generating System

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).

Ghosts and Strong Ghosts in the Stable Category

2016 ◽  
Vol 59 (4) ◽  
pp. 682-692
Author(s):  
Jon F. Carlson ◽  
Sunil K. Chebolu ◽  
Ján Mináč
Keyword(s):  
Finite Group ◽  
Strong Form ◽  
Finite Range ◽  
Finitely Generated ◽  
Stable Category ◽  
Tate Cohomology ◽  
Trivial Module ◽  
A Finite Group ◽  
Thick Subcategory ◽  
Generating Hypothesis

AbstractSuppose that G is a finite group and k is a field of characteristic p > 0. A ghost map is a map in the stable category of finitely generated kG-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow p-subgroup of G is cyclic of order 2 or 3. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd’s generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps that mimic ghosts in high degrees.

scholarly journals Weakly Projective and Weakly Injective Modules

1994 ◽  
Vol 46 (5) ◽  
pp. 971-981 ◽  
Author(s):  
S. K. Jain ◽  
S. R. López-Permouth ◽  
K. Oshiro ◽  
M. A. Saleh
Keyword(s):  
Positive Integer ◽  
Artinian Ring ◽  
Projective Cover ◽  
Finitely Generated ◽  
Matrix Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Qf Ring ◽  
Finitely Generated Modules

AbstractA module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:(1) R is a left perfect and every weakly projective right R-module is weakly injective.(2) R is a direct sum of matrix rings over local QF-rings.(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.(4) R is right artinian ring and every weakly injective right R-module is weakly projective.(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

scholarly journals Azumaya’s Canonical Module and Completions of Algebras

1973 ◽  
Vol 49 ◽  
pp. 9-19 ◽  
Author(s):  
James Osterburg
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Artinian Ring ◽  
Injective Hull ◽  
Finitely Generated ◽  
Canonical Module

We are concerned with an algebra S over a commutative ring. Precisely S is a non-commutative ring with identity which is also a finitely generated unital R module such that r(xy) = (rx)y = x(ry) for r in R and x, y ∈ S. In section one, we assume A is a commutative, Artinian ring. Following Goro Azumaya (see (1, p. 273)), we define the canonical module F of A to be the injective hull of A modulo the Jacobson radical of A i.e. F = I(A/J(A)).

scholarly journals A RESULT ON SEMI-ARTINIAN RINGS

2003 ◽  
Vol 46 (1) ◽  
pp. 63-66 ◽  
Author(s):  
Hai Quang Dinh ◽  
Patrick F. Smith
Keyword(s):  
Artinian Ring ◽  
Partial Solution ◽  
Subject Classification ◽  
Finitely Generated ◽  
Artinian Rings

AbstractIt was shown by Huynh and Rizvi that a ring $R$ is semisimple artinian if and only if every continuous right $R$-module is injective. However, a characterization of rings, over which every finitely generated continuous right module is injective, has been left open. In this note we give a partial solution for this question. Namely, we show that for a right semi-artinian ring $R$, every finitely generated continuous right $R$-module is injective if and only if all simple right $R$-modules are injective.AMS 2000 Mathematics subject classification: Primary 16D50. Secondary 16P20; 16P60

scholarly journals Boundary Convex Cocompactness and Stability of Subgroups of Finitely Generated Groups

2017 ◽  
Vol 2019 (6) ◽  
pp. 1699-1724
Author(s):  
Matthew Cordes ◽  
Matthew Gentry Durham
Keyword(s):  
Kleinian Groups ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Limit Set ◽  
Class Groups ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Equivalent Characterization ◽  
Convex Cocompact

Abstract A Kleinian group $\Gamma &lt; \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $\Gamma$ in $\mathbb H^3$ is quasiconvex or, equivalently, $\Gamma$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups. Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.

scholarly journals Interlacing methods and large indecomposables

1970 ◽  
Vol 3 (3) ◽  
pp. 337-348 ◽  
Author(s):  
S. E. Dickson ◽  
G. M. Kelly
Keyword(s):  
Artinian Ring ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Number Of Generators ◽  
Finite Dimensional ◽  
Factor Module ◽  
Algebra Over A Field

The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module of a direct sum of copies (finite or infinite) of an original module M. The conditions given in a previous paper by the first author in order that the interlacing module (using finitely many copies) be indecomposable are here greatly weakened, and we further allow the number of copies of the original to be infinite. R. Colby has shown that if R is a left artinian ring, the existence of a bound on the number of generators required for any indecomposable finitely-generated left R-module implies that R has a distributive lattice of two-sided ideals. This result is extended to rings whose identity is a sum of orthogonal local idempotents.For these rings the same distributivity is proved in case every indecomposable interlacing module of the above type which begins with an indecomposable projective M is finitely-generated. A consequence is that any finite-dimensional algebra over a field having infinitely many two-sided ideals has infinite-dimensional indecomposables.

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