All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
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
4
votes
0
answers
63
views
Algebras derived equivalent to a hereditary algebra
Let $A=KQ/I$ be a quiver algebra with relations in $I$ having only coefficients 1 or -1. This implies that $A=FQ/I$ is defined over any other field $F$ (possibly of even another characteristic).
...
- 26.5k
8
votes
0
answers
334
views
Dyck paths of Dynkin type
(The conjecture is a homological algebra question, but question 2 is a pure combinatorics question given that the conjecture is true)
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...
- 26.5k
7
votes
0
answers
142
views
When is an algebra derived indecomposable?
Call a finite dimensional (acyclic) quiver $K$-algebra A derived indecomposable in case $A$ is not derived equivalent to an algebra of the form $B \otimes_K C$.
For example when the number of simples ...
- 26.5k
4
votes
0
answers
241
views
Finding local algebra and relations lottery
This can be seen as an attempt for a mini Polymath project on homological properties of (local) finite dimensional algebras. You only need to know what a finite dimensional algebra is and have GAP to ...
- 26.5k
5
votes
1
answer
226
views
Frobenius algebras from symmetric polynomials
Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
- 26.5k
4
votes
0
answers
155
views
Commutative algebras associated to simple Lie algebras
In Section 2 of the article https://www.sciencedirect.com/science/article/pii/S0021869307000385, the authors study the center $Z=Z_Q$ of certain preprojective like algebras associated to the simply ...
- 26.5k
1
vote
0
answers
37
views
Coxeter period of representation-finite selfinjective algebras
Let $A$ be a representation-finite selfinjective (quiver) algebra, that we assume to be connected and non-semisimple. Define the Coxeter period $p_A$ of $A$ to be equal to the period of the Coxeter ...
- 26.5k
5
votes
0
answers
97
views
Periodics of Coxeter matrices for truncated Nakayama algebras
For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$).
So for example for $n=7$ and $r=3$ we ...
- 26.5k
6
votes
0
answers
328
views
When does a group and its pro-algebraic completion have equivalent categories of arbitrary representations?
In the following everything is over some field $k$.
Let $G$ be a discrete group. We write $G^{\text{alg}}$ for its pro-algebraic completion. The latter is a pro-affine pro-algebraic group which arises ...
- 1,635
5
votes
1
answer
829
views
Rigid monoidal and closed monoidal categories
I am trying to understand the relationship between rigid monoidal categories and closed monoidal
categories. First every rigid monoidal category is closed, with an adjoint to the functor $X \otimes -$ ...
- 1,144
4
votes
0
answers
55
views
Algebras with a simple preserving duality and finite global dimension
Algebras with a simple preserving duality (an anti-automorphism preserving pointwise a primitive full set of ortohogonal idempotents) and finite global dimension include important classes of algebras ...
- 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
11
votes
0
answers
202
views
Quiver and relations for blocks of category $\mathcal{O}$
In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ .
...
- 26.5k
10
votes
1
answer
3k
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An enumeration problem for Dyck paths from homological algebra
In their article "On n-Gorenstein rings and Auslander rings of low injective dimension" Fuller and Iwanaga gave a homological characterisation of 2-Gorenstein Nakayama algebras with 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
4
votes
0
answers
56
views
Which posets can occur from commutative Frobenius algebras?
Let $A$ be a commutative Frobenius algebra. We can assume that $A$ is local and $A=K[x_i]/(I)$ for some variables $x_1,...,x_n$ and an admissible ideal.
Then the non-zero monomials $u_i$(including 1) ...
- 26.5k
3
votes
1
answer
446
views
A set of objects classically generates the full subcategory of compact objects iff it generates the whole category
Sorry in advance if my question doesn't have the level of this community.
I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...
- 224
2
votes
0
answers
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
votes
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
5
votes
0
answers
132
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On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber
This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...
- 51
4
votes
1
answer
497
views
Semisimple Abelian categories with infinite sums
A semisimple category is an abelian category in which every object is a finite direct sum of simple objects.
A) Why does one impose the finiteness condition here?
B) If one condsiders infinite direct ...
- 1,144
2
votes
1
answer
127
views
Ext between a module and its higher Auslander-Reiten translate
Let $A$ be a representation-finite algebra and $M$ an indecomposable module with finite projective dimension $g >0$.
Question 1: Do we have $dim(Ext_A^g(M, \tau_g(M)))=1$? Here $\tau_g(M)=\tau ( \...
- 26.5k
3
votes
1
answer
98
views
Finding automorphisms and cyclic modules via QPA
Given a symmetric finite dimensional algebra $A$ over a finite field with enveloping algebra $A$.
Assume we know that $\Omega_{A^e}^i(A) \cong A_{f}$, where $f$ is some automorphism of the algebra $A$....
- 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
8
votes
2
answers
960
views
Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?
Let $k$ be a field and $X$ a topological space.
Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...
- 1,635
2
votes
1
answer
183
views
Almost split sequences coming from bimodules
Let $A$ be a finite dimensional algebra with enveloping algebra $A^e$.
Auslander and Reiten proved in "On a theorem of E. Green on the dual of the transpose" that
$Hom_A(Tr_{A^e}(A),M) \cong ...
- 26.5k
2
votes
0
answers
52
views
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
1
vote
0
answers
49
views
Invariance under derived equivalence of a Gorenstein projective bimodule
A module $M$ over an finite dimensional algebra $A$ is called Gorenstein projective in case there exists an exact complex $(P_i)$ of projective $A$-modules such that the complex stays exact after ...
- 26.5k
5
votes
0
answers
168
views
Higher analogue of the Auslander-Bridger transpose
Let $A$ be an Artin algebra and $M$ a module with $Ext^i(M,A)=0$ for $i=1,...,n-2$.
Then in case $P_{n-1} \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ is the beginning of a minimal ...
- 26.5k
6
votes
1
answer
203
views
Question on a subcategory being extension-closed
In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\...
- 26.5k
5
votes
2
answers
281
views
Isomorphism for Ext spaces for finite dimensional algebras
Let $A$ be an Artin algebra with enveloping algebra $A^e$.
Then we have $Hom_{A^e}(X,A^e) \cong Hom_A(D(A) \otimes_A X,A)$ for a bimodule $X$. (see for example in the article "A theorem of Green on ...
- 26.5k
6
votes
0
answers
182
views
On properties of an algebra as a bimodule
Let $A$ be a two-sided artinian ring.
Recall that a module $M$ is said to have dominant dimension at least $n$ in case the terms $I_i$ in the minimal injective coresolution of $M$ are projective for $...
- 26.5k
2
votes
1
answer
200
views
Characterisation of minimal projective resolutions via the Euler characteristic
Let $A$ be a finite dimensional $K$-algebra (where $K$ is a field) and $M$ a finitely generated $A$-module.
Let $\psi: 0 \rightarrow P_r \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$
...
- 26.5k
5
votes
1
answer
212
views
On tilting and cotilting modules
Let A be an Artin algebra and assume all modules are basic, then a classical result says that tilting modules $T$ are in bijection with complete cotorsion pairs $(T^{\perp}, \check{ add(T)})$ (with ...
- 26.5k
1
vote
0
answers
20
views
Finding minimal copresentations of projectives in stable endomorphism rings
Let $A$ be a finite dimensional algebra (you can assume it is selfinjective in case this helps) and $M$ an $A$-module without projective summands.
Let $B=\underline{End_A(M)}$, the stable endomorphism ...
- 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
1
vote
0
answers
92
views
Symmetric stable categories
Let $A$ and $B$ be Frobenius algebras that are stable equivalent.
In case $A$ is symmetric, is $B$ also symmetric? (no, see the comment of Jeremy Rickard) Does it hold in case $A$ and $B$ are ...
- 26.5k
2
votes
0
answers
85
views
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
views
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{...
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
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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
4
votes
0
answers
44
views
An analog of the representation dimension for algebras
The representation dimension of a finite dimensional algebra $A$ is defined as
$repdim(A)= \inf \{ gldim(B) | B=End_A(M)$ for a generator-cogenerator $M \}$.
It was shown by Iyama that it is always ...
- 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
6
votes
1
answer
256
views
On the global dimension of an endomorphism algebra
Let $G_n$ be the elementary abelian 2-group with $2^n$ elements and $R=R_n:=KG$ the group algebra over the field with 2 elements.
Let $M_n$ be the direct sum of all non-projective modules of the form ...
- 26.5k
5
votes
1
answer
145
views
Commutator of finite global dimension algebras
Let $A=KQ/I$ be a finite dimensional quiver algebra of finite global dimension.
Is it true that the dimension of $A/[A,A]$ is equal to the number of simples of $A$?
Here $[A,A]$ is the vector space ...
- 26.5k
4
votes
0
answers
67
views
Admissible relations for the quiver of the preprojective algebra
Let $K$ be a field of characteristic 0.
Let $Q_n$ be the quiver of the preprojective algebra of Dynkin type $A_n$.
So from each point $i$ to its neighbor $i+1$ there is an arrow $a_i$ and an arrow ...
- 26.5k
4
votes
1
answer
213
views
Tensor-indecomposable modules
Let $A$ be a finite dimensional algebra.
Call an $A$-bimodule $M$ tensor-indecomposable in case $M$ is not isomorphic to $X \otimes_K Y$ for a left $A$-module $X$ and a right $A$-module $Y$.
...
- 26.5k
5
votes
1
answer
225
views
Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
- 26.5k
2
votes
1
answer
250
views
Example of a projective bimodule with isomorphic left and right duals
What is an example of a non-free finitely generated $R$-bimodule $M$ satisfying
i) $M$ is projective as both a left and right $R$-module
ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual ...
