All Questions
23 questions
3
votes
0
answers
107
views
Dimension of hom spaces between indecomposable modules
Undergraduate-Level Background
Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A$...
- 5,230
3
votes
1
answer
301
views
If a bimodule is "generated" by single elements, must the elements be conjugate?
Let $A$ and $B$ be Artin $k$-algebras for a commutative artinian ring $k$ (e.g. $A$ and $B$ are finite dimensional $k$-algebras for a field $k$). Let $M$ be an $A$-$B$-bimodule of finite length over $...
- 696
3
votes
1
answer
339
views
If the Hom-space of finite length modules is generated by single elements, must the elements be conjugate?
Let $A$ be an Artin $k$-algebra for a commutative artinian ring $k$ (e.g. $A$ is a finite dimensional algebra over a field $k$). Let $X,Y$ be finite length left $A$-modules. If $\text{Hom}_A(X,Y)$ is ...
- 696
2
votes
2
answers
139
views
Infinite radical ideal cubed equals zero for tame hereditary Artin algebras
Let $A$ be a tame hereditary Artin algbera and mod$A$ the category of finitely generated (left) $A$-modules. Further, let rad$_A$ be the radical ideal of mod$A$, which is the smallest ideal containing ...
- 696
4
votes
1
answer
198
views
Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?
All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...
- 696
4
votes
1
answer
107
views
For a finite dimensional $k$-Algebra $A$ does infinite representation type imply $(\text{rad}_A^{\omega})^2 \neq 0$?
Let $k$ be a field, $A$ a finite dimensional $k$-Algebra, $\text{mod}\,A$ the category of finite dimensional left $A$-modules and $\text{rad}_A$ the collection of radical morphisms in $\text{mod}\,A$. ...
- 696
7
votes
0
answers
275
views
Split epimorphism of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...
- 696
3
votes
1
answer
240
views
Split monomorphisms of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for ...
- 696
5
votes
1
answer
235
views
Concept of an exact ideal of a module category
Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
- 696
3
votes
1
answer
244
views
Left module which cannot be made into a bimodule?
Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a ...
1
vote
0
answers
77
views
n-Gorenstein algebras and tilting modules
Let $\Lambda$ be an artin algebra over a commutative artinian ring $R$. $\Lambda$ is said to be $n$-Gorenstein, for some natural number $n$, provided it have finite self-injective dimension at most $n$...
- 775
3
votes
1
answer
252
views
Higher Extension Group Question
Suppose we have an associative unital ring $R$, and we have an $R$-module $M$ with a length 3 socle filtration, i.e. write
$$soc(M) \text{ for the socle of } M,$$
$$soc^2(M) \text{ for the preimage ...
- 1,077
1
vote
1
answer
149
views
Finding modules to check for finite global dimension
Given a finite dimensional algebra $A$ and a generator-cogenerator $M$ and let $B:=End_A(M)$. $B$ has finite global dimension iff every $A$-module has finite $add(M)$-resolution dimension, which is ...
- 26.5k
1
vote
0
answers
361
views
Property of the syzygy functor of $\operatorname{\underline{mod}} A$
Let $A$ be an artin algebra. We denote by $\operatorname{mod}A$ the category of finitely generated left $A$-modules. We denote by $[A]$ the ideal of morphisms between $A$-modules which factor through ...
- 775
2
votes
1
answer
119
views
How to get that $\Omega^2_{\Lambda}(N) \cong \textrm{Hom}_A(M,Y)$?
Let $A$ be an algebra over a field k. A module $_AM$ is called a generator if $\textrm{add}(A) \subseteq \textrm{add}(M)$, a cogenerator if $\textrm{add}\big(D(A)\big) \subseteq \textrm{add}(M)$. $M$ ...
- 775
4
votes
1
answer
241
views
Questions in the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences"
I am reading the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences", the link is here:https://pub.uni-bielefeld.de/publication/1780235.
In the paper, $A$ ...
- 775
6
votes
1
answer
361
views
How to check whether a module is an n-th syzygy
Given a finite dimensional algebra $A$, define $\Omega^{n}(mod-A)$ (modules here are always finite dimensional) to be the full subcategory of projective modules or modules $M$ such that $M \cong \...
- 26.5k
2
votes
0
answers
105
views
Why $T'$ dosen't have projective direct summand?
Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let ...
- 775
2
votes
1
answer
450
views
The definitions of a generator module?
Recently I have seen two definition of a generator module:
1) A generator for a category $C$ is an object $G$ such that for any two parallel morphisms $f,g:X \rightarrow Y$ with $f \neq g$, then ...
- 775
2
votes
0
answers
203
views
Could Partial Tiltings be studied as Almost Complete Tiltings?
The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end.
Here $k$ denotes an algebraically closed field,...
- 493
2
votes
1
answer
395
views
Projectivity of torsion-free modules over integral group rings
Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...
- 2,998
7
votes
1
answer
180
views
Complexity of rational $\mathrm{GL}_{n(r)}$-modules
Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
- 768
1
vote
0
answers
238
views
Is this a pure monomorphism?
Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...
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