All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
5
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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 ...
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4
votes
2
answers
337
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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{...
- 26.5k
5
votes
0
answers
88
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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 ...
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2
votes
0
answers
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 ...
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4
votes
0
answers
90
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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 ...
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5
votes
1
answer
151
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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 ...
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1
vote
0
answers
47
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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 ...
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4
votes
0
answers
81
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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....
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4
votes
0
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135
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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)...
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5
votes
0
answers
253
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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 ...
3
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0
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136
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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 ...
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1
vote
0
answers
117
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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 $\phi_A$...
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10
votes
1
answer
400
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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 ...
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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 ...
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4
votes
1
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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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2
votes
1
answer
206
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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\...
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5
votes
1
answer
353
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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 $g(h)=h(m)$, ...
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3
votes
1
answer
153
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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 $A$ ...
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3
votes
1
answer
211
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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 ...
- 55
1
vote
0
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77
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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
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380
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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 ...
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3
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0
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68
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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 ...
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2
votes
0
answers
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 ...
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3
votes
1
answer
252
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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
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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 ...
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3
votes
1
answer
258
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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 $M$ ...
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2
votes
1
answer
203
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$Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$
Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the ...
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8
votes
0
answers
173
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On constructible Hall algebra and instantons
I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
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4
votes
0
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210
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Conjecture on tilting modules for an Auslander algebra
On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism
classes of modules, occurring as the $i$-th summand of ...
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15
votes
2
answers
860
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What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
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1
vote
0
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77
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On $Ext_A^2(S,A)$ for algebras $A$
For Nakayama algebras $A$ with a linear quiver and at most 7 points (which are 196 algebras) the following was true:
$max \{ dim(Ext_A^2(S,A)) | S $ simple $\}= max \{ dim(Ext_A^2(X,A)) | X $ ...
- 26.5k
2
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0
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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 ...
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2
votes
0
answers
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
8
votes
1
answer
217
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Categorification of monotone maps via tilting modules?
It is well known that for the algebra of $n \times n$-upper triangular matrices over a field the number of tilting modules is equal to the Catalan number $C_n$. This is just the (hereditary) Nakayama ...
- 26.5k
8
votes
1
answer
291
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Eilenberg-Watts theorem for the derived category
Let $A$ and $B$ be $k$-algebras. And for convenience let's say $k$ is a field and both $A$ and $B$ are finite-dimensional.
A well known theorem independently discovered by Eilenberg and Watts states ...
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2
votes
1
answer
100
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In search of disconnected indecomposable self-injective finite-dimensional algebras
I wanted to know if it is possible to construct an indecomposable self-injective finite-dimensional algebra $\Lambda$ whose Auslander-Reiten quiver $\Gamma_\Lambda$ is not connected. I'd love to see ...
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2
votes
0
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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 ...
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2
votes
1
answer
118
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Example to periodic symmetric algebras
In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric.
Is there an easy counterexample for the following:
In case $A$ is additionally a periodic ...
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1
vote
0
answers
80
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When is a stable endomorphism ring selfinjective?
Let $A$ be a local symmetric finite dimensional algebra and $M$ an $A$-module with at least two non-isomorphic indecomposable non-projective summands.
In case $\Omega^1(M) \cong M$ in the stable ...
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3
votes
1
answer
106
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The kernel of the morphism from the Picard group to the stable Picard group of a self-injective algebra
Let $\Lambda$ be a finite-dimensional self-injective algebra (over an algebraically closed field, if necessary). Let $Pic(\Lambda)$ be the group of natural isomorphism classes of self-equivalences $...
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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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2
votes
1
answer
212
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Injective dimension of $A/AeA$
Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $e$ the idempotent such that $eA$ is the direct sum of all indecomposable projective-injective $A$-modules. Do we have $g=...
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2
votes
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 ...
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1
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0
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64
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Questions on holonomic modules
An Auslander-Gorenstein ring is a noetherian ring R that has finite left and right selfinjective dimension and such that $fd(I_i) \leq i$ for all $i \geq 0$ for an injective coresolution of the ...
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8
votes
3
answers
1k
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Intuition behind the canonical projective resolution of a quiver representation
Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective ...
- 1,161
3
votes
1
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237
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Finding all selforthogonal indecomposable modules
Given a finite dimensional algebra $A$ with finite global dimension such that there are only finitely many basic tilting modules. Then every selforthogonal indecomposable module $M$ (that is a module ...
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1
vote
0
answers
69
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Inequality for the global dimension of quiver algebras
Let $A$ be a finite dimensional algebra of finite global dimension given by connected quiver and relations.
Do we have $gldim(A) \geq \min \{ \text{injdim}(S)+\text{projdim}(S) | S$ simple $\} $ in ...
- 26.5k
3
votes
2
answers
214
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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 ...
- 26.5k
3
votes
0
answers
54
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Ext for simple modules in selfinjective algebras
Let $A$ be a finite dimensional symmetric algebra (given by a connected quiver and non-semisimple) and $S$ a simple $A$-module. (I am more interested in symmetric algebras, but selfinjective examples ...
- 26.5k
3
votes
0
answers
208
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A new characterisation of hereditary algebras?
Let $A$ be a quiver algebra with global dimension at most two (or more generally finite global dimension) and $A^e=A^{op} \otimes_K A$ be its enveloping algebra.
Guess:Is $A$ hereditary if and only ...
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