All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
3
votes
1
answer
251
views
Some places I don't know of the paper "On the stable module category of a self-injective algebra"
Recently I am reading the paper "On the stable module category of a self-injective algebra", the link is here: http://www.ams.org/journals/tran/2000-352-05/S0002-9947-00-02232-7/S0002-9947-00-02232-7....
- 775
3
votes
1
answer
187
views
Explicit proof that algebra is derived wild
Following the terminology of
Drozd, Yuriy A., Derived tame and derived wild algebras, Algebra Discrete Math. 2004, No. 1, 57-74 (2004). ZBL1067.16028.
let $A$ and $R$ be algebras over a field $k$. A ...
- 497
3
votes
1
answer
325
views
Question on $Ext^1$
Given a finite dimensional algebra $A$ with two indecomposable modules $M$ and $N$. Define $H(M,N)$ as the largest number of indecomposable summands of a module $X$ such that there exists a non-split ...
- 26.5k
3
votes
1
answer
149
views
Bounds for the number of edges in an Alperin diagram
If $A$ is an algebra over a field $k$ and $M$ is a finite-dimensional $A$-module, then Alperin showed in a paper [Diagrams for modules, JPAA, 1980] how to associate a diagram to $M$ with the vertices ...
- 631
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 $A$ ...
- 26.5k
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 ...
- 55
3
votes
1
answer
348
views
Finitistic dimension equal to cofinistic dimension for QF-3?
Given a finite dimensional algebra A such that the regular module embeds into a projective-injective module (such algebras are called QF-3 algebras and generalise Frobenius algebras).
Define the ...
- 26.5k
3
votes
1
answer
70
views
2-periodic modules over p-group algebras
Given the group algebra of a p-group over a field of characteristic p. Can the 2-periodic indecomposable modules $M$ ($M$ with $\Omega^{2}(M)=M$) be classified? I am not experienced much with modular ...
- 26.5k
3
votes
0
answers
107
views
Dimension of hom spaces between indecomposable modules
Undergraduate-Level Background
Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A$...
- 5,230
3
votes
0
answers
83
views
Connection between certain finite groups and Frobenius algebras
This question is maybe not so precise yet, but contains some ideas that maybe just search for the right definition.
Let $G=F/U$ be a finite group with presentation $F/U$ (here the presentation really ...
- 26.5k
3
votes
0
answers
104
views
A depth version of a conjecture of Yamagata
Let $A$ be a finite-dimensional $K$-algebra.
Recall that the grade of an $A$-module $M$ is defined as the smallest $i$ such that $\operatorname{Ext}_A^i(M,A) \neq 0$ and the depth of $A$ is defined ...
- 26.5k
3
votes
0
answers
73
views
Turning a Frobenius algebra into a symmetric algebra via tensor products
Let $A$ be a finite dimensional Frobenius algebra over a field $K$, which means that $A \cong D(A)$ as right $A$-modules. Being symmetric means that $A \cong D(A)$ as $A$-bimodules. Here $D(-)=Hom_K(-,...
- 26.5k
3
votes
0
answers
85
views
Exterior powers of the Cartan matrix and Dyck paths
(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
- 26.5k
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
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
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
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
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
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
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
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
3
votes
0
answers
78
views
Quiver algebras of Dynkin type
Let $kQ$ be one of the Dynin path algebras of type $A_n , D_n $ or $E_i$ for $i=6,7,8$.
Question 1: How many (up to isomorphism) quiver algebras are there that are derived equivalent to $kQ$?
...
- 26.5k
3
votes
0
answers
129
views
Algebras with symmetric Cartan matrix
Let $A$ be a finite dimensional algebra with Cartan matrix $C_A$.$C_A$ being a symmetric matrix is equivalent to the Coxeter matrix being minus the identity matrix in case $A$ has finite global ...
- 26.5k
3
votes
0
answers
82
views
Derived equivalence for two modules
Let $A=K[x]/(x^n)$ and $M_1$ and $M_2$ two basic generator of mod-A and let $B_i=End_A(M_i)$.
$B_1$ and $B_2$ are derived equivalent in case $M_1 \cong \Omega^1(M_2)$ in the stable category.
Question:...
- 26.5k
3
votes
0
answers
61
views
On grades of torsion modules in noetherian rings
Let $A$ be a (not necessarily commutative) two-sided noetherian ring with minimal injective coresolution $(I_i)$ of the regular module $A$ as a right module.
Say that $A$ has dominant dimension $n$ in ...
- 26.5k
3
votes
0
answers
180
views
On a formula for the Auslander-Reiten translate
For an Artin algebra $A$ and an indecomposable non-projective module $M$ we should have that $\tau(M) \cong \nu \Omega^2(M)$ iff $Ext_A^i(M,A)=0$ for $i=1,2$. ($\nu$ being the Nakayama functor)
...
- 26.5k
3
votes
0
answers
56
views
Weakly symmetric rings and derived equivalences
A ring $R$ with Jacobson radical $J$ is called Frobenius in case $R/J \cong soc(R)$ as left and right $R$-modules and weakly symmetric in case we even have $R/J \cong soc(R)$ as $R$-bimodules.
...
- 26.5k
3
votes
0
answers
54
views
Classes of algebras where derived equivalence preserves the global dimension
Question: Are there known classes $X$ of finite dimensional algebras in the literature that have the property that in case $A, B \in X$ are derived equivalent, they share the same global dimension?
...
- 26.5k
3
votes
0
answers
54
views
Properties of sequences associated to Nakayama algebras
Assume Nakayama algebras are connected and given by quiver and relations. Note that such Nakayama algebras have global dimension at most $2n-2$ in case it is finite and the algebra has $n$ simples.
...
- 26.5k
3
votes
0
answers
45
views
Magnitude of ADR algebras
Let $A$ be a connected quiver algebra with $n$ simple modules and Jacobson radical $J$ and Loewy length $n+1$ (that is $J^{n+1}=0$ and $n$ is minimal with this property).
The ADR-algebra $B_A$ of $A$ ...
- 26.5k
3
votes
0
answers
176
views
Quiver algebras with finite global dimension
Given a fixed connected quiver $Q$.
Are there only finitely many quiver algebra $KQ/I$ (I an admissible ideal) up to isomorphism which have finite representation type and finite global dimension?
- 26.5k
3
votes
0
answers
92
views
On NCR for finite dimensional algebras
Let $A$ be a finite dimensional algebra.
A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ ...
- 26.5k
3
votes
0
answers
53
views
Inequality for the magnitude of quiver algebras
A conjecture on the global dimension of quiver algebras of finite global dimension states that the global dimension is bounded by the vector space dimension of the algebra.
The magnitude of a finite ...
- 26.5k
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
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
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 ...
- 26.5k
3
votes
0
answers
134
views
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=...
- 26.5k
3
votes
0
answers
156
views
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) ...
- 61
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
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
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
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
3
votes
0
answers
54
views
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
views
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 ...
- 26.5k
3
votes
0
answers
175
views
Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
- 26.5k
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
3
votes
0
answers
169
views
Characterisation of reflexive modules for general rings
A module $M$ over a general ring A is called reflexive in case the canonical evaluation map $e_M: M \rightarrow M^{**}$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. Is $M$ reflexive iff $M \cong M^{*...
- 26.5k
3
votes
0
answers
81
views
Number of generalised tilting modules
This question is a modification of Number of tilting modules and was suggested by a comment of Dag Madsen. I wondered about this question too some time ago, but did not do much with it since my ...
- 26.5k