All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
5
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0
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114
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Extreme no loop conjecture for group algebras
Let $A=KG$ be a group algebra for a finite group $G$. Let $S$ be a simple $A$-module. The extreme no loop conjecture predicts that $Ext_A^1(S,S) \neq 0$ implies $Ext_A^i(S,S) \neq 0$ for infinitely ...
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2
votes
1
answer
97
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When is $N^{*} \otimes_K M$ projective for a local Hopf algebra?
Given a finite dimensional local Hopf algebra $A$ over a field $K$ and two finite dimensional indecomposable modules $N$ and $M$. Is it known when the module $N^{*} \otimes_K M$ is projective? Can ...
- 26.5k
4
votes
0
answers
69
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Auslander-Solberg algebras from non-rigid modules
Let $A$ be a Nakayama algebra and $M$ be the direct sum of all indecomposable $A$-modules $N$ with $Ext_A^1(N,N) \neq 0$.
The following is suggested by computer experiments with QPA:
Question: Is ...
- 26.5k
2
votes
1
answer
169
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Combinatorial problem on periodic dyck paths from homological algebra
edit: I added conjecture 2 that looks much more accessible.
Here is the elementary combinatorial translation of the problem (read below for the homological background):
Let $n \geq 2$.
A Nakayama ...
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2
votes
0
answers
51
views
Selfinjective algebras with loops
Given a selfinjective finite dimensional algebra $A$ with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$.
Question:
Is A derived equivalent to an algebra with a loop in the quiver in ...
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2
votes
0
answers
53
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Strong no loop conjecture for uniserial modules
Let $A$ be a an Artin algebra. The strong no loop conjecture states that a simple $A$-module with $Ext_A^1(S,S) \neq 0$ has infinite projective dimension.
This conjecture was recently proved for ...
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2
votes
0
answers
45
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On monomial and $\Omega^d$-finite algebras
Let $Q$ be a finite quiver and $I$ a monomial admissible ideal of the path algebra $KQ$ for a field $K$. Then an algebra $A=KQ/I$ is called a monomial algebra.
It is well known that monomial algebras ...
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1
vote
0
answers
55
views
$\mathrm{Ext}^1$-ordering on ${}^IW^{\Sigma_\mu}$
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$.
Let $W$ be the associated Weyl group and let $\Phi$ be its root system.
We write $\Phi^+...
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4
votes
1
answer
149
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Testing whether a module generates $K_0(\mbox{mod-}A)$
Given a representation-finite (connected) quiver algebra $A$ and a module $M$.
Is there a good way to test whether the set $\{ [N] \mid N \in \mathrm{add}(M) \}$ generates $K_0(\mbox{mod-}A)$?
Can ...
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4
votes
0
answers
82
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On strongly simply connected quiver algebras
Let $A$ be a representation-finite quiver algebra. In this case $A$ is simply connected if and only if its first Hochschild cohomology vanishes by a result of Buchweitz and Liu. $A$ is called strongly ...
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2
votes
0
answers
69
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Stable m-Calabi Yau property for Frobenius categories
Let $C$ be a Frobenius category. The stable category $\underline{C}$ is called $m$-Calabi Yau in case it is Hom-finite and there is a functorial duality
$D \underline{Hom}(X,Y)=\underline{Hom}(Y,\...
- 26.5k
2
votes
0
answers
73
views
Equivalence from a tilting module
Let $A$ be a finite dimensional algebra. For a subcategory $C$ of $mod-A$ let $\overline{C}$ be the objects $X \in mod-A$ such that there exists an exact sequence $0 \rightarrow C_n \rightarrow ... \...
- 26.5k
2
votes
0
answers
56
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Characterisation of representation-directed algebras
A representation-directed algebra $A$ over a field $K$ has the property that for every indecomposable module $M$ we have $\mathrm{End}_A(M) \cong K$ and $\mathrm{Ext}_A^i(M,M)=0$ for all $i>0$.
...
- 63.9k
16
votes
2
answers
694
views
How complicated can a finite double complex over a field be?
A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is ...
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4
votes
0
answers
88
views
Minimal injective coresolution in the stable Auslander algebra
Let $A$ be a finite dimensional (connected) quiver algebra. Let $T(A)$ denote the full subcategory of coherent functors from $mod-A$ to $Ab$ that vanish on projective objects. $T(A)$ is equivalent to ...
- 26.5k
5
votes
0
answers
140
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Open problems about Morita and derived invariants
Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...
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3
votes
0
answers
48
views
Endomorphism ring of a generator-cogenerator over acyclic algebras
Let $A$ be an acyclic quiver algebra, $M$ a generator-cogenerator and $B=End_A(M)$.
Questions:
Does $B$ have finite global dimension?
Does $B$ have finite global dimension in case $M=A \oplus D(A)$?
...
- 26.5k
4
votes
0
answers
58
views
Interpretation of stable Hom in Nakayama algebras
Let $A$ be a Nakayama algebra with Kupisch series $[c_0,c_1,...,c_{n-1}]$ and Jacobson radical $J$ (given by quiver and relations). As is well known every indecomposable $A$-module is of the form $e_i ...
- 26.5k
9
votes
0
answers
123
views
Cartan determinant of stable categories
Let $A$ be a finite dimensional algebra with finitely many indecomposable non-projective modules $M_1, M_2,...,M_n$.
Let $a_{i,j}:=\dim(\underline{Hom_A}(M_j,M_i))$ (the dimension of the stable Homs ...
- 26.5k
5
votes
0
answers
91
views
Bound on the sum of projective and injective dimension
Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category.
In proposition 1.2. of https://link.springer.com/article/10....
- 26.5k
6
votes
1
answer
204
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On the injective dimension of $A$ as a bimodule
(Note: I modified the question and instead of looking at simple modules in the question, I look at all indecomposable modules.)
Let $A$ be a finite dimensional algebra over a field $K$ given by an ...
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8
votes
2
answers
1k
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When is the exterior algebra a Hopf algebra?
I have several questions on the exterior algebra of a vector space:
Q1:When has the exterior algebra A (viewed just as an algebra, not considered as a graded algebra) of an $n$-dimensional vector ...
- 1,028
5
votes
1
answer
186
views
(Stable) Auslander algebras in a specific example
Let $k$ be a field and $Q$ be the quiver with two vertices 1 and 2 and three arrows:
$a$ from 2 to 1, $b$ from 2 to 1 and $c$ from 2 to 2.
Let $I_1=\langle ab-c^2,ba\rangle$ and $I_2=\langle ab-c^2,c^...
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5
votes
0
answers
125
views
Stable equivalence and stable Auslander algebras
Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules.
Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...
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3
votes
0
answers
427
views
When is the stable category abelian
For which Artin algebras $A$ is the stable module (that is the module category modulo projectives) category of finitely generated modules abelian?
If you like you may take rings that are not Artin ...
- 26.5k
2
votes
1
answer
203
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Proving indecomposability of special modules
I'm reading the following paper: http://math0.bnu.edu.cn/~huwei/paper/Holm-Hu-1.pdf
On page 795 and 796 there are the definitions (in a diagrammatical way) of some $A_n$ modules, whereupon $A_n:=k[x,...
- 1,755
2
votes
0
answers
60
views
$\Omega$-periodic modules in selfinjective algebras
Given a representation infinite (connected) selfinjective algebra $A$ with an indecomposable $\Omega$-periodic module $M$.
Does $A$ then have infinitely many indecomposable $\Omega$-periodic ...
- 26.5k
2
votes
0
answers
65
views
Constructing stable equivalences for finite dimensional algebras
Given a finite dimensional (non-selfinjective) algebra $A$.
Is there a method (for example using QPA) to construct algebras stable equivalent to $A$?
Such a thing is easily possible for derived ...
- 26.5k
4
votes
0
answers
85
views
Deciding whether two algebras are derived equivalent
Given two finite dimensional quiver algebras $A$ and $B$ (over a nice field in case that helps, for example a finite field).
Question: Can an there be a finite algorithm that decides whether $A$ ...
- 26.5k
4
votes
0
answers
43
views
Cartan determinants of minimal Auslander-Gorenstein algebras
Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite dominant dimension ($\geq 2$) equal to the Goreinstein dimension in https://www.sciencedirect.com/science/...
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4
votes
4
answers
596
views
Homology of solvable (nilpotent) Lie algebras
Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for ...
- 3,545
3
votes
0
answers
106
views
Bounds for the finitistic dimension
The finitistic dimension of an algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension.
For finite dimensional algebras $A$ with radical cube ...
- 26.5k
4
votes
0
answers
71
views
Koszul and quadratic algebras with Gorenstein dimension 2
In proposition 2.19. of http://inmabb.criba.edu.ar/revuma/pdf/v48n2/v48n2a05.pdf it was mentioned that a finite dimensional algebra of global dimension 2 is quadratic if and only if it is Koszul.
...
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3
votes
0
answers
61
views
Number of algebras stably equivalent to a given algebra
For $n \geq 2$ let $B_n$ be the algebra of upper triangular matrices over a field $K$.
Recall that two algebras are said to be stably equivalent in case their module categories modulo projectives are ...
- 26.5k
3
votes
0
answers
96
views
When is the category of complexes of finite type?
For a ring $R$ define the category of complexes of length $n \geq 2$ as the category $C_n(R)$ with objects the complexes of the form $0 \rightarrow X_n \rightarrow \cdots X_1 \rightarrow 0$ with the ...
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4
votes
2
answers
453
views
Global dimension of the tensor algebra
Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$.
Question 1: Is there a simple proof that $...
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7
votes
1
answer
462
views
On a problem for determinants associated to Cartan matrices of certain algebras
This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
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7
votes
0
answers
266
views
Closed formula for some dimension
This question has a background from representation theory/homological algebra, but I state everything in elementary terms here:
Call an n-CNak an n-tupel $[c_0,c_1,...,c_{n-1}]$ where $c_0=c_1=...=c_{...
- 13.4k
6
votes
1
answer
689
views
Tensor product of bimodules
Im not sure whether this question is appropriate for MO, but I do not have much experience with bimodules (or I forgot many things).
Let $A$ be a finite dimensional (connected) algebra over a field $...
- 1,244
11
votes
2
answers
558
views
Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
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2
votes
0
answers
69
views
Do the values of the global dimension constitute an interval?
Let $Q$ be a fixed finite connected quiver and $k$ a fixed field. Set $Z_Q:= \{ gldim(kQ/I) < \infty | I $ an admissible ideal $\}$.
Question: Is $Z_Q$ an intervall?
This is true for example in ...
- 30.3k
6
votes
2
answers
269
views
Derived invariance of the Cartan determinant
The Cartan matrix $C$ of a finite quiver algebra $A$ with points $e_i$ is defined as the matrix having entries $c_{i,j}=\dim(e_i A e_j)$. The Cartan determinant is defined as the determinant of the ...
- 35.2k
4
votes
0
answers
98
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Bound for the global dimension of higher Auslander algebras
Let algebras be finite dimensional and connected.
Recall that an algebra $A$ is called a higher Auslander algebra in case it the dominant dimension coincides with the global dimension and both ...
- 26.5k
7
votes
1
answer
370
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Gorenstein symmetric conjecture for arbitrary rings
The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension ...
- 35.2k
4
votes
1
answer
159
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Question on $n$-regular modules
Let $A$ be finite dimensional connected algebra. A simple module $S$ is called $n$-regular in case $pd(S)=n$, $Ext_A^i(S,A)=0$ for $i=0,1,...,n-1$ and $Ext_A^n(S,A)$ being a simple $A$-left module. ...
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4
votes
0
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66
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Periodic modules in Frobenius algebras
Let $A$ be a finite dimensional Frobenius algebra and assume there exists an indecomposable periodic module $M$, that is $\Omega^n(M) \cong M$ for some $n$.
Question: Does this imply that there is ...
- 26.5k
1
vote
0
answers
57
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Modules with arbitrary large complexity
Let $A$ be a finite dimensional algebra with a module $M$ who has minimal injective coresolution $(I_i)$. Define the complexity $cx(M)$ as $cx(M):= \inf \{ t \geq 0 | \dim(I_i) \leq a i^{t-1}$ for all ...
- 26.5k
2
votes
0
answers
29
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Complexity of the regular module
Let $A$ be a finite dimensional algebra with a module $M$ who has minimal injective coresolution $(I_i)$. Define the complexity $cx(M)$ as $cx(M):= \inf \{ t \geq 0 | \dim(I_i) \leq a i^{t-1}$ for all ...
- 26.5k
3
votes
0
answers
156
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Weak generators of the right-bounded derived category of a finite-dimensional algebra
The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...
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3
votes
0
answers
134
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Proving that the exterior algebra is symmetric via the polynomial ring
Recall that a finite dimensional algebra $A$ over a field $K$ is called Frobenius in case $A \cong D(A)$ as right modules, and it is called symmetric in case $A \cong D(A)$ as bimodules (where $D=...
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