All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
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
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
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
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 ...
- 1,071
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
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
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
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
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 ...
- 26.5k
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
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 ...
- 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 ...
- 26.5k
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
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 ...
- 26.5k
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 $\pd_A(S)$...
- 775
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
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
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
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 $K$ ...
- 26.5k
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
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\ ...
- 99
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 ...
- 453
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 ...
- 121
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 ...
- 26.5k
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, ...
- 26.5k
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 ...
- 26.5k
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 ...
- 1,071
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 ...
- 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
6
votes
3
answers
368
views
Is there a finite dimensional algebra with left finitistic dimension different from its right finitistic dimension?
Let $\Lambda$ be finite dimensional algebra over a field $k$. The (left) finitistic dimension of a finite dimensional algebra is defined as
$$\operatorname{findim}(\Lambda)=\sup\{\operatorname{pd}M | ...
- 497
1
vote
0
answers
108
views
When is a Koszul algebra derived equivalent to its dual
Let $A$ be a finite dimensional Koszul algebra of finite global dimension.
Question: When is $A$ derived equivalent to its Koszul dual algebra?
I wonder whether there is an exact condition to ...
- 26.5k
4
votes
0
answers
83
views
Number of K-generators of an algebra and type $D_n$-parking functions
Let $A$ be a representation-finite quiver algebra.
When $A$ has $n$ simple modules a basic module $M$ with $n$ indecomposable summands $M_i$ is called a K-generator when the $M_i$ generate $K_0(A)$, ...
- 26.5k
2
votes
0
answers
116
views
Functors with adjoints
I want to find a functor between abelian categories, which is faithful but not full. And this functor has left and right adjoint. I want to know a nontrivial example,which is not inducecd by a ring ...
- 169
4
votes
0
answers
121
views
Perfect modules for the Beilinson algebra
The Beilinson algebra $A=A_n$ is a finite dimensional quiver algebra that is derived equivalent to the category of coherent sheaves of $\mathcal{P}^n$. See for example https://link.springer.com/...
- 26.5k
6
votes
1
answer
186
views
Endomorphism ring of trivial source modules for abelian p-groups
Bernhard Böhmler (who is also on MO) and myself had the following idea:
Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
- 26.5k
3
votes
0
answers
48
views
Questions on piecewise hereditary algebras
Let $A$ be a finite dimensional quiver algebra over a field $k$ that is quasi-tilted and representation-finite (this implies that $A$ is a tilted algebra). Assume that the Coxeter polynomial of $A$ is ...
- 26.5k
2
votes
0
answers
77
views
Bimodule Ext for Dynkin path algebras
Let $A=kQ$ be a path algebra of Dynkin type $Q$ and $B=A^{op} \otimes_k A$ the enveloping algebra of $A$. Note that $mod-B$ is just the category of $A$-bimodule and $A$ is a $B$-module.
For a B-module ...
- 26.5k
6
votes
1
answer
281
views
An identity for Ext for rings
Let $A$ be a two-sided noetherian ring (which we should assume to be Gorenstein first so that everything is well defined, otherwise it is only well defined up to a conjecture, which states that every ...
- 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 ...
- 26.5k
3
votes
0
answers
99
views
The union-closed sets conjecture for finite dimensional algebras
Say a finite dimensional algebra $A$ satisfies the right UC-condition if there exists an indecomposable projective module $P$ of $A$ such that $\operatorname{injdim}(\operatorname{top}(P))=1$ and $P$ ...
- 26.5k
7
votes
0
answers
355
views
A homological algebra approach to the Union-closed sets conjecture
I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
- 26.5k
5
votes
0
answers
116
views
An intelligent ant living on a symmetric quiver algebra - Does it have a way to find out whether it lives on a trivial extension?
For a given algebra $B$ over a field $K$ the trivial extension $T(B)$ of $B$ is defined as follows:
The underlying vectorspace is $T(B)=B \oplus D(B)$ where $D(B)=Hom_K(B,K)$ and the multiplication is ...
- 26.5k
5
votes
0
answers
76
views
Reference on two numbers associated to a module of finite homological dimension
Let $A$ be a finite dimensional algebra over a field $K$ with a module $M$ which has finite projective dimension and finite injective dimension.
Let $n \geq 1$.
Let $(P_i)$ be a minimal projective ...
- 26.5k