All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
12
votes
1
answer
922
views
Does this algebra have finite global dimension ? (Human vs computer)
Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...
- 1,244
4
votes
1
answer
303
views
Algebras derived equivalent to quasi-hereditary algebras
Let an algebra always be finite dimensional over a field and connected.
It is well known that a quasi-hereditary algebra with $n$ simple modules has global dimension at most $2n-2$.
Questions:
1. ...
- 35.2k
4
votes
0
answers
84
views
Finitistic dimension via a bimodule
Let $A$ be a connected finite dimensional basic algebra.
Question: Is there an indecomposable $A$-bimodule $W$ such that the finitistic dimension of $A$ is equal to the right projective dimension ...
- 26.5k
13
votes
1
answer
745
views
Combinatorial inequality for dominant dimension
In the following I present a conjecture on Nakayama algebras that I have for nearly 2 years now. Since I was not able to solve it and it can be stated purely combinatorically, I thought it might be ...
- 85.6k
5
votes
1
answer
243
views
Questions on group and Nakayama algebras from a book
Recall that a Nakayama algebra (also called serial algebras in the literature) is an algebra such that every indecomposable module has a unique composition series.
In the book "Classical artinian ...
- 35.2k
3
votes
1
answer
153
views
Finitistic dimension via reflexive modules
Recall that an $A$-module $M$ is reflexive in case the natural evaluation map $M \rightarrow M^{**}$ is an isomorphism, where $M^{*}=Hom_A(M,A)$.
Question:
Given a finite dimensional algebra $A$ ...
- 35.2k
5
votes
1
answer
353
views
Existence of non-trivial reflexive modules
Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f_M:M \rightarrow M^{**}$ (where $M^{*}=Hom_R(M,R)$) is an isomorphism, where $f_M(m)=g$ with $g(h)=h(m)$, ...
CommunityBot
- 1
3
votes
0
answers
67
views
Upper bound for embedding of submodules of projective modules
Assume we have a finite dimensional algebra $A$ with the following property:
Every indecomposable submodule of a projective module embedds into $A^n$ for a fixed $n$.
Is there a good method to ...
- 26.5k
4
votes
1
answer
463
views
Global dimension of a graded algebra
Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...
- 3,686
4
votes
1
answer
685
views
Quadratic algebras and Koszul algebras
Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$.
In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...
CommunityBot
- 1
2
votes
0
answers
83
views
Length 2 modules over finite dimensional algebras
Given a finite dimensional algebra $A$ over an infinite field and two simple modules $S,T$.
Question 1: Is there a useful (homological/computational) crtierion to decide when there are infinitely ...
- 26.5k
3
votes
1
answer
189
views
Question on $\operatorname{Ext}$ in a local Frobenius algebra
Let $A$ be a finite dimensional local Frobenius algebra with simple module $k$ and an indecomposable non-projective module $M$ (that is also finite dimensional).
Question:
Is there an example of ...
- 35.2k
3
votes
2
answers
214
views
History of an open problem on partial tilting modules
The following is an open problem:
Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting ...
CommunityBot
- 1
2
votes
0
answers
164
views
On a conjecture about tilting modules
There is the following conjecture on tilting modules (see also History of an open problem on partial tilting modules):
Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is ...
CommunityBot
- 1
8
votes
0
answers
140
views
$n$-fold tensor products of $D(A)$ for finite dimensional algebras
Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected).
Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
- 26.5k
1
vote
0
answers
141
views
Question on vanishing Hochschild cohomology
Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$.
Question:
Is there a finite dimensional selfinjective ...
- 26.5k
1
vote
1
answer
127
views
Gorenstein projective modules of a certain triangular matrix algebra
Let $B$ be a finite dimensional selfinjective algebra over a field $k$ with a finite dimensional non-projective $B$-module $M$ and $$A=\pmatrix{k&M\\0&B}.$$
A module $N$ over an algebra $C$ ...
- 35.2k
6
votes
1
answer
420
views
Calculating the Ext-algebra with a computer
Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension.
Let $B$ be the Ext algebra of $M$, that is $B:=\...
- 1,244
5
votes
0
answers
105
views
Derived invariant acyclic algebras
Call a connected quiver algebra $A=KQ/I$ (finite quiver Q and admissible ideal I) derived-invariant in case every quiver algebra derived equivalent to $A$ is even isomorphic to $A$.
For example local ...
- 26.5k
3
votes
0
answers
71
views
$\Omega^2(S) \cong \tau(S)$ for simple modules
Let $A$ be an Artin algebra with $\Omega^2(S) \cong \tau(S)$ for each simple module $S$. Is $A$ a quasi-Frobenius (=selfinjective) algebra?
Here $\tau$ denotes the Auslander-Reiten translate, which is ...
- 26.5k
6
votes
2
answers
768
views
Auslander-Reiten theory for Gorenstein algebras
In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
- 1,244
2
votes
0
answers
55
views
$Ext_{A^e}^i(D(A),A)$ for finite dimensional algebras
Let $A$ be a finite dimensional non-semisimple algebra over a field $K$ with enveloping algebra $A^e=A^{op} \otimes_K A$. Let $D(A)=Hom_K(A,K)$.
Question:
Is there always a positive integer $i>...
- 26.5k
9
votes
1
answer
593
views
Hochschild homology with coefficients in a certain bimodule
Let $A$ be a finite-dimensional $k$-algebra and $U$ and $V$ two finite-dimensional projective $A$-modules (maybe neither the finiteness nor projectivity has to play a role, but these requirements are ...
- 1,382
4
votes
1
answer
179
views
Reference for isomorphism $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$
Let $A$ be a finite dimensional algebra and $X,Y$ indecomposable modules and $n \geq 2$.
We know that $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$.
Now such an isomorphism should be given by ...
- 52.6k
4
votes
2
answers
337
views
When is $\Omega^1$ an equivalence?
Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...
- 35.2k
5
votes
0
answers
88
views
Cluster-tilting object for a local non-selfinjective algebra
Let $A$ be a non-selfinjective (which is equivalent to non-Gorenstein) local finite dimensional algebra.
Is there a known example of such an $A$ having a cluster-tilting object?
Id be surprised to ...
- 26.5k
2
votes
0
answers
43
views
Projective dimensions of the terms in a minimal injective resolution of the regular module
Let $A$ be a finite dimensional algebra with finite global dimension and with minimal injective coresolution $I_i$ of the regular module $A$.
The study of the projective dimensions of the $I_i$ is an ...
CommunityBot
- 1
4
votes
0
answers
90
views
Number of hereditary modules of a hereditary algebra
Let $Q$ always denote a Dynkin quiver.
Given a connected path algebra $A=kQ$ and a module $M$, is there a useful criterion on $M$ when $End_A(M)$ is again a connected quiver algebra?
Call a module ...
- 26.5k
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
3
votes
0
answers
213
views
Cohen-Macaulay Artin algebras
In http://link.springer.com/chapter/10.1007%2F978-3-0348-8658-1_8#page-1
Auslander and Reiten introduced Cohen-Macaulay Artin algebras as a generalisation of Gorenstein algebras. Let X be the full ...
- 26.5k
5
votes
1
answer
151
views
Derived equivalences of Artin algebras with finitistic dimension zero
Let $A$ be an Artin algebra of finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have finitistic dimension zero?
In case this is true, this might generalise the ...
- 35.2k
1
vote
0
answers
47
views
Piecewise hereditary algebras of Dynkin type that are QF-3
Is there an easy classification of piecewise hereditary (which means derived equivalent to a hereditary algebra) algebras of Dynkin type
that are Quasi-Frobenius-3 (meaning that the injective envelope ...
- 26.5k
4
votes
0
answers
81
views
Sum of all projective dimensions of simple modules
Let $X_{n,t}$ be the set of all finite dimensional algebras (we can assume they are given by a connected quiver and admissible relations) that have global dimension equal to $n$ and $t$ simple modules....
- 26.5k
4
votes
0
answers
135
views
Question on syzygies
Given a finite dimensional algebra $A$ over a field $K$ with $\Omega^i(D(A)) \cong \Omega^{i+1}(D(A))$ for some $i \geq 1$, where $D(A)=\operatorname{Hom}_K(A,K)$.
Do we then also have $\Omega^{-i}(A)...
- 4,713
5
votes
0
answers
253
views
Tannakian theory for Lie algebras
Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
10
votes
1
answer
400
views
Derived equivalences of Dyck paths
Call two Dyck paths $D_1$ and $D_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top ...
- 26.5k
3
votes
0
answers
136
views
Can a acyclic quiver algebra be derived equivalent to a non-acyclic quiver algebra?
Can a quiver algebra with acyclic quiver be derived equivalent to a quiver algebra with non-acyclic quiver?
(I moved this question from another thread Derived equivalences of Dyck paths , where the ...
- 26.5k
1
vote
0
answers
117
views
Derived equivalences and the Coxeter polynomial
Let $A$ be a quiver algebra with an acyclic quiver and primitive idempotents $e_i$.
The Cartan matrix $C_A$ of $A$ is defined as the matrix with entries $dim(e_i A e_j)$ and the Coxeter matrix $\phi_A$...
CommunityBot
- 1
2
votes
0
answers
201
views
Homological conjecture for finite dimensional algebras
In the theory of finite dimensional algebras there are many homological conjectures. When working over an algebraically closed field it is well known that any such algebra is Morita equivalent to a ...
- 356
2
votes
1
answer
206
views
Extensions of lattices
Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\...
- 23
3
votes
1
answer
211
views
Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras
For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we ...
- 52.9k
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
5
votes
0
answers
380
views
A tensor product for dg-categories
For a finite group denote by $\mathbf{Ch}^G$ the dg-category of $G$-representations in chain complexes over a field.
Is there a tensor product $\otimes$ of dg-categories (similar to the Deligne ...
- 1,382
3
votes
0
answers
68
views
Derived invariant algebras and cluster tilting objects
This question is perhaps a little stupid as I have barely any evidence for it, but another thread just reminded me about it so maybe someone has an idea or counterexample for this.
Let $A$ be a ...
- 26.5k
0
votes
0
answers
116
views
A question on the paper "The classification of algebras by dominant dimension"
I'm reading the paper "the classification of algebras by dominant dimension" by Bruno J.Mueller, the link is here http://cms.math.ca/10.4153/CJM-1968-037-9.
In the proof of lemma 3 on page 402, there ...
- 15.2k
2
votes
0
answers
74
views
Units in the (stable) center of a Frobenius algebra [duplicate]
Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
- 15.2k
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
3
votes
1
answer
163
views
Identity for $Ext^1$ for special algebras
Let $A$ be a finite dimensional algebra and assume all modules are also finite dimensional. A module $M$ is said to have dominant dimension at least $n$ in case the term $I_i$ for $i=0,1,...,n-1$ are ...
- 26.5k
2
votes
0
answers
207
views
Does the first Tachikawa conjecture imply the Nakayama conjecture?
Let $A$ be a non-selfinjective algebra with dominant dimension at least one and $Af$ a minimal faithful projective-injective left $A$-module (but we use right modules in the following).
The Nakayama ...
- 26.5k
3
votes
1
answer
258
views
Invertible bimodules which are isomorphic in the stable module category
I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ ...
- 3,176