All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
2
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Is a triangulated category admitting a tilting object triangle equivalent to the unbounded derived category of the endomorphism ring of this object?
Let $\mathcal{T}$ be a triangulated category. We call an object $G$ tilting if
$G$ is compact, that is, $\mathrm{Hom}_{\mathcal{T}}(G, -)$ preserves all set-indexed coproducts;
$G$ is a generator, ...
- 21
2
votes
0
answers
45
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$K_0$-basis modules with a unique extension related to parking functions
Let $B=B_n$ be the quiver algebra of type $A_n$ (with some orietnation) with $n$ points.
A $B$-module $M$ is a basis of the Grothendieck group $K_0$ if $M$ has $n$ indecomposable direct summands and ...
- 26.5k
2
votes
0
answers
55
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Depth and codepth of an algebra
Let $A$ be a finite dimensional $K$-algebra over a field $K$ and
$0 \rightarrow A \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots$
a minimal injective coresolution of the regular module $A$.
The ...
- 26.5k
2
votes
0
answers
67
views
Bounds for sum of the homological dimensions in the incidence algebra of a Boolean lattice
Let $A$ be a finite dimensional algebra.
Define $\varphi_A:= \sup \{ \operatorname{pd} M + \operatorname{id} M \mid M \in \operatorname{ind}(A) \}$, where $\operatorname{pd} M$ denotes the projective ...
- 26.5k
2
votes
0
answers
86
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Example of a triangular string algebra that is rep infinite, but $\tau$-tilting finite
Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. Hence, $A$ can be realized as the path algebra of a bound quiver $(Q,I)$, where $I\subseteq\mathbb{K}Q$ is an ...
- 839
2
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77
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Equivalence of two descriptions of differentials of Koszul complex
My question comes from learning the paper [BGS96] Koszul duality patterns in representation theory by Beilinson, Ginzburg and Soergel, published in 1996.
Let $A=T_{A_0}A_1/\langle R\rangle$ be a ...
- 21
2
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0
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138
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Construction of a certain long exact sequence
Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field.
Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...
- 26.5k
2
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0
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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
2
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0
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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 ...
- 26.5k
2
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0
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70
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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
2
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0
answers
91
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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
2
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0
answers
135
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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
2
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0
answers
116
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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
2
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0
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77
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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
2
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0
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56
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On periods of symmetric algebras
Let $A$ be a symmetric finite dimensional algebra over a field of characteristic two (or even over the field with two elements) such that every simple $A$-module has the same period equal to $n$.
...
- 26.5k
2
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0
answers
56
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Invertible bimodule for hereditary algebras
Let $A=kQ$ be a path algebra over a field $k$ for a finite acyclic quiver with enveloping algebra $A^e$.
Question: When is it true that $\tau_{A^e}(A) \cong A$ as a left and as a right $A$-modules? (...
- 26.5k
2
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0
answers
52
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Bimodule isomorphism for representation-finite blocks of the Schur algebra
Let $A$ be a representation-finite block of a schur algebra with $n \geq 2$ simple modules. Then the global and dominant dimension of $A$ are equal to $g=2n-2$. You can find quiver and relations for ...
- 26.5k
2
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0
answers
85
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Algebras from a basis of a Frobenius algebra
Let $A$ be a commutative Frobenius algebra over a field $K$ (we can assume that $A$ is local).
We can assume $A=K[x_1,...,x_r]/I$ for an ideal $I$ with $J^n \subseteq I \subseteq J^2$ where $J=<x_i&...
- 26.5k
2
votes
0
answers
110
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Generalising injective modules
Free modules over a ring generalise to projective modules over a ring, which generalise to flat modules, which generalise to torsion free modules:
$$
\textrm{free} \to
\textrm{projective}
\to
\textrm{...
2
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0
answers
104
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Tate cohomology for group algebras
Let $A=kG$ be a group algebra with a finite group $G$ and a field $k$.
Let $T^i(M,N)= \underline{Hom_A}(\Omega^i(M),N)$ be the $i$-th Tate cohomology group. Note $T^i(M,N)= Ext_A^i(M,N)$ in case $i \...
- 26.5k
2
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0
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51
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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 ...
- 26.5k
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 ...
- 26.5k
2
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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 ...
- 26.5k
2
votes
0
answers
69
views
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
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0
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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 $i>0$.
...
- 26.5k
2
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0
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60
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$\Omega$-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
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0
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65
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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
2
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0
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69
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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 ...
- 26.5k
2
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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
2
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0
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83
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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
2
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0
answers
164
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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 ...
- 26.5k
2
votes
0
answers
55
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$Ext_{A^e}^i(D(A),A)$ 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
2
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0
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43
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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 ...
- 26.5k
2
votes
0
answers
201
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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 ...
- 26.5k
2
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0
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74
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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
2
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207
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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
2
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66
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Injective dimension of $A/AfA$
Let $A$ be an algebra of finite global dimension and $Af$ the direct sum of all indecomposable projective-injective left $A$-modules.
Using right modules, left $g$ denote the injective dimension of $A/...
- 26.5k
2
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53
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Periodic algebras from periodic simple modules
As continuation of the previous thread Example to periodic symmetric algebras , I have the following question:
Is there a counterexample to the following:
Let A be a symmetric algebra and W the ...
- 26.5k
2
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0
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84
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Super global dimension
Let $R$ be a ring of finite global dimension. Define the small super global dimension as $sgl(R):= \sup \{ pd(X)+id(X) | X \in mod-R$ and indecomposable $\}$.
Here $id(X)$ stands for the injective ...
- 26.5k
2
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0
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75
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Double dual of the simple module in local algebras
Let $A$ be a local Artin algebra that is not selfinjective with simple module $S$.
Questions:
Can $S^{**}$ be indecomposable?
$S^{**}$ be somehow generally be described (for example as a ...
- 26.5k
2
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0
answers
103
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Representation dimension of Auslander algebras
Is the representation dimension of Auslander algebras known? Is there an example of such algebras with representation dimension larger than 4?
- 26.5k
2
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121
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Ext of a Schur algebra
Let $A=A_n$ be the representation-finite block of a Schur algebra with $n$ simple modules for $n \geq 2$. Quiver and relations of $A$ can be found in 6.1. of https://arxiv.org/pdf/1607.05965.pdf . Let ...
- 26.5k
2
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0
answers
61
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Question on outer Ext-products
For group algebras $A=KG$ over a field $K$ with finite group $G$ there exists an outer product on Ext:
$Ext_A^i(M,N) \otimes_K Ext_A^j(M',N') \rightarrow Ext_A^{i+j}(M \otimes_K M',N \otimes_K N')$.
...
- 26.5k
2
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0
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63
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QF-3 monoid algebras
A finite dimensional algebra $A$ is called QF-3 in case the injective envelope of the regular module is projective. For example all Frobenius algebras are QF-3.
Given a monoid algebra $kG$ of a finite ...
- 26.5k
2
votes
0
answers
135
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Ext over a certain commutative algebra
Let $A$ be the algebra $K[x,y]/(x^2,y^2,xy,yx)$. Then $A$ is a 3-dimensional commutative algebra and $Ext^i(M,A) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$. Namely: $...
- 26.5k
2
votes
0
answers
33
views
Bounds on global and dominant dimension of certain algebras
Algebras are always finite dimensional over a field $K$.
Let $X_n$ be the set of representation-finite algebras over $K$ with $n$ simple modules.
Define $g(X_n):= sup \{ gldim(A) | A \in X_n $ and $A$...
- 26.5k
2
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0
answers
48
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Special modules over symmetric algebras
Let $A$ be a symmetric connected finite dimensional algebra over a field $k$.
Call a tuple of two modules $(X,M)$ (having no projective direct summands) cute in case $Ext^l(X,M) \neq 0$ for some $l \...
- 26.5k
2
votes
0
answers
81
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Characterisation of Frobenius algebras via sequences
Given a commutative Frobenius algebra, finite dimensional over a field $k$.
We assume that the algebra is connected and in fact given by quiver and relations. Let $S$ be the unique simple modules of ...
- 26.5k
2
votes
0
answers
191
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Global dimension of quiver algebras
Given a finite quiver $Q$, let $Y(Q)$ be the set of (isomorphism classes of) algebra $kQ/I$ (with admissible ideal $I$) that have finite global dimension.
For example in case $Q$ is acyclic, all ...
- 26.5k