All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
2
votes
0
answers
196
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Commutative local rings which satisfy Krull-Remak-Schmidt
Question 1: Can the class of local (always noetherian and commutative) rings be classified for which the Krull-Remak-Schmidt theorem (KRS) holds for finitely generated modules? They contain for ...
- 26.5k
4
votes
0
answers
114
views
Classification of 2-periodic triangulated categories
Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$.
Question 1: Is there a ...
- 26.5k
5
votes
2
answers
479
views
How to define cohomology of algebraic structures?
I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain
\begin{align*}
\cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
- 12.9k
1
vote
0
answers
115
views
Prove that $B$ is a directing module
Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
- 839
5
votes
1
answer
400
views
Injective modules
Let $A$ be a finite dimensional $k$-algebra and let $I$ be an injective module。My question is whether $I$ is a direct sum of finite-dimensional injective modules。
- 63.9k
2
votes
0
answers
84
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Representation finite Hopf algebras up to stable equivalence
It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra.
Question: Is it true that every representation-finite Hopf algebra is stable ...
- 66
1
vote
1
answer
218
views
A result of Schofield in the case of quivers with relations
Let $Q$ be a quiver without oriented cycles. A result of Schofield says that, for dimension vectors $\alpha$ and $\beta$ of $Q$, $\beta\hookrightarrow\alpha$ iff $\operatorname{ext}(\beta, \alpha-\...
- 839
5
votes
1
answer
187
views
Periodic objects in Frobenius categories
Let $A$ be a finite dimensional Gorenstein algebra and $C$ the stable module category of $A$.
Question: Does there always exist an indecomposable periodic object $X$ in C, that is an object with $\...
- 35.2k
3
votes
0
answers
96
views
Criterion for representation-finite algebras
Let $A=KQ/I$ a quiver algebra with acyclic $Q$.
Question: Is $A$ representation-finite if and only if $\tau^{-n}(A)=0$ for some $n \geq 1$?
Here $\tau$ is the Auslander-Reiten translate of $A$.
This ...
- 26.5k
5
votes
1
answer
207
views
Finite lattices that are Koszul
Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$.
It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
- 26.5k
2
votes
2
answers
140
views
Finding exceptional regular representations of $\tilde{D}_4$ efficiently
Let $A$ be the path algebra of the quiver $\tilde{D}_4$. I would like to find its exceptional regular representations with as little computation as possible.
Of course, we can compute the whole ...
- 6,292
2
votes
0
answers
70
views
Rigid modules for hereditary algebras
Let $A=KQ$ be a path algebra of a connected quiver. (K algebraically closed if it helps)
Question: Is there an explicit classification of all indecomposable $A$-modules $M$ that are rigid, that is $...
- 26.5k
3
votes
0
answers
74
views
Is any $n$-angulated category a $(n-2)$-cluster tilting subcategory of some triangulated category?
Geiss, Keller and Oppermann told us in "n-angulated categories" that some $(n-2)$-cluster tilting subcategory of a triangulated category is a $n$-angulated category.
$\require{wasysym}$
...
- 603
9
votes
0
answers
366
views
A characterisation of symmetric algebras using Hochschild (co)homology
A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
3
votes
0
answers
238
views
What do the indecomposable objects of the homotopy category of chain complexes look like?
I am trying to understand indecomposable objects in the homotopy category of chain complexes. Let $\mathcal{A}$ be an abelian category. Denote by $C(\mathcal{A})$ the category whose objects are chain ...
- 41
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
0
answers
112
views
Finite global dimension via the Cartan determinant
Let $A=T(KQ)$ be the trivial extension algebra of a path algebra of Dynkin type $KQ$.
The indecomposable module of $A$ correspond to the roots of $Q$ (and not just the positive roots as for $KQ$).
Let ...
- 26.5k
4
votes
0
answers
227
views
Orbits of group representation over $\mathbb{F}_2$
Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
- 161
2
votes
1
answer
244
views
Computing (relative) cohomology classes on quotient (vector) space via Hodge theorem
I am working on a graded vector space $V = \bigoplus_{i\in \mathbb{N}}V_i$ (which is a parabolic Verma module in the sense of [1], but let's ignore such specifics) with a positive definite inner ...
CommunityBot
- 1
5
votes
0
answers
190
views
On the not so clear relationship between torsion theories and localization for a newcomer
Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so ...
- 83
10
votes
1
answer
1k
views
What's the relationship between spherical twist functors and tilting?
I've been reading about connections between Coxeter groups and preprojective algebras, and I keep running into two operations on the derived categories of preprojective algebras which seem very ...
CommunityBot
- 1
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 ...
- 63.9k
4
votes
1
answer
290
views
An inequality for Ext
$\DeclareMathOperator\Ext{Ext}$Consider the following statement for a $K$-algebra $A$:
$\dim(\Ext^1 (M, N )) \ge \min( \dim(\Ext^1(M, M )), \dim(\Ext^1 (N, N )) ),$
for all finite dimensional ...
CommunityBot
- 1
3
votes
1
answer
177
views
Quiver and relations for ADE singularities in dimension one
Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...
- 26.5k
10
votes
1
answer
396
views
Generalising the union-closed sets conjecture from lattice to a larger class of posets
(edit: I decided to simplify the question and only pose it for bounded posets first)
The Union-closed sets conjecture is equivalent for lattices P to:
There exists a join-irreducible element $a$ with ...
- 63.9k
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 ...
- 356
4
votes
2
answers
299
views
Relation of the first Hochschild cohomology and the outer automorphism group
Let $R$ be a ring.
Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?
(It is not true, by the two answers. Is it ...
CommunityBot
- 1
4
votes
0
answers
196
views
Quillen–Suslin theorem in a more general context
Let $A$ be a finite dimensional local Frobenius algebra that is Koszul.
Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...
- 26.5k
9
votes
1
answer
395
views
Which finite posets are Koszul self-dual?
Let $P$ be a finite connected poset with incidence algebra $A_P$.
For the definition and results on Koszul algebras for incidence algebras, see for example here
Question: Which posets have the ...
- 1,554
12
votes
0
answers
402
views
Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
- 1,161
6
votes
1
answer
244
views
What is a Serre-smooth algebra?
Let $A$ be an $R$-algebra.
In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction.
But no formal ...
- 4,277
3
votes
0
answers
80
views
Coxeter polynomials of graphs
Let $Q$ be a finite connected and directed graph with $n$ points.
Assume $Q$ is acyclic as a directed graph.
Let $C=C_Q$ be the Cartan matrix of $Q$, that is the matrix with entries $c_{i,j}$ being ...
- 26.5k
3
votes
1
answer
565
views
Finiteness of cohomology group
Suppose $G$ is a finite Galois group, and $M$ is an infinite $G$-module. When can I say that $H^1(G, M)$ is finite?
I know this not true in general. Is it true under certain assumptions on $M$?
To be ...
- 143
3
votes
0
answers
102
views
Frobenius algebras associated to posets and coalgebra structures
Let $P$ be a finite poset that we assume for simplicity to be bounded (that is it has a global maximum M and minimum m).
Let k be a field, then the classical incidence algebra $kP$ has $k$-vector ...
- 26.5k
1
vote
0
answers
144
views
A question concerning extension groups between simple modules
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$Let A be a finite dimensional algebra over some field k with exactly m simple modules up to isomorphism. Let S be a simple left A-module. ...
- 775
4
votes
1
answer
158
views
Can we always choose 2 nonisomorphic simple modules to satisfy the following nonvanishing extension conditions?
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$Let A be a finite dimensional algebra over some field k and S a nonprojective simple left A-module. Suppose the projective dimension $\pd_A(S)$...
- 63.9k
2
votes
0
answers
91
views
When does a stable endomorphism ring have injective dimension at most one?
tLet $A$ be a Frobenius algebra (we can assume that $A$ is given by quiver and relations) and let $M$ be a basic $A$-module without projective direct summands (we can assume we know the decomposition ...
- 26.5k
4
votes
1
answer
655
views
Are all algebras Igusa-Todorov?
A finite dimensional algebra A is called (n-)Igusa-Todorov in case there exists a module V such that for any module M there is an exact sequence:
$0 \rightarrow V_2 \rightarrow V_1 \rightarrow \Omega^{...
- 26.5k
4
votes
0
answers
231
views
How far is the sphere spectrum to be tilting/silting
In the last years the following definition of a tilting/silting object in an arbitrary triangulated category with coproducts emerged:
Let $\mathcal D$ be a triangulated category with coproducts and ...
- 666
6
votes
1
answer
203
views
Question on a subcategory being extension-closed
In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\...
- 26.5k
5
votes
0
answers
83
views
It there an algebra of the form $B_T$ with global dimension 3?
Let $A$ be the (symmetric Frobenius) algebra $A=K[x]/(x^3) \otimes_K K[x]/(x^3)$ over a field $K$, which is isomorphic to the group algebra of $C_3 \times C_3$, with $C_3$ cyclic of order 3, when $K$ ...
- 7,893
2
votes
0
answers
135
views
How to compute the derived functor of bounded derived categories of hereditary algebra?
Let $\Lambda$ be
a finite dimensional algebra given by the quiver
$$\cdot\leftarrow\cdot\leftarrow\cdot\rightarrow\cdot.$$
It can be view as an triangulated matrix algebra.
$$\Lambda={A\ \ M\choose0\ ...
- 63.9k
4
votes
0
answers
62
views
Deforming relations for a symmetric Frobenius algebra to obtain an algebra of finite global dimension
By doing computer experiments with the GAP-package QPA I found a (connected non-semisimple) symmetric quiver algebra $A=KQ/I$ with admissible relations $I$ containing relations of the form $w_1 - w_2$ ...
- 26.5k
4
votes
1
answer
104
views
Which Auslander algebras satisfy $Ext_B^1(D(B),B)=0$?
Let $B$ be the Auslander algebra of a representation-finite algebra $A$.
Question: When do we have $Ext_B^1(D(B),B)=0$? Can this be expressed in terms of nice properties of $A$?
This is for example ...
- 66
3
votes
1
answer
118
views
Weakly symmetric Frobenius algebras
Let $A$ be a finite dimensional Frobenius algebra and $e$ and idempotent of $A$.
It is well known that the algebra $eAe$ does not have to be a Frobenius algebra. But if $A$ is additionally symmetric, ...
- 1,690
10
votes
1
answer
307
views
Rings where all indecomposable projective modules are finitely generated
Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.
Question 1: Is there a nice equivalent ...
- 38.6k
7
votes
1
answer
495
views
Serre functor on the category $Perf(A)$, $A$ - k-algebra
Consider a finite-dimensional $k$-algebra $A$ of finite global dimension. Then it is known that the Serre functor on $D^b(mod-A)$ exists and is given by the Nakayama functor. The proof goes something ...
- 26.5k
5
votes
0
answers
142
views
A practical way to check whether a module is periodic
A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
- 26.5k
5
votes
2
answers
360
views
Injective dimension of the Jacobson radical and global dimension
Given a finite dimensional algebra $A$ with Jacobson radical $J$.
Is the global dimension of $A$ equal to the injective dimension of $J$?
Together with Xiao-Wu Chen and Srikanth Iyengar we proved this ...
- 26.5k
12
votes
0
answers
516
views
Does $\mathrm{Ext}^1(M,M) \neq 0$ imply $\mathrm{Ext}^2(M,M) \neq 0$?
$\DeclareMathOperator{\Ext}{\operatorname{Ext}}$The first question is about group algebras:
Question 1: Let $A=kG$ be a group algebra (with $G$ finite) and let $M$ be an indecomposable $A$-module. ...
- 63.9k