All Questions
Tagged with rt.representation-theory homological-algebra
255 questions with no upvoted or accepted answers
14
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0
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891
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Local proof of Grothendieck-Riemann-Roch theorem
There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of ...
- 7,377
12
votes
0
answers
402
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Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
- 1,161
12
votes
0
answers
516
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Does $\mathrm{Ext}^1(M,M) \neq 0$ imply $\mathrm{Ext}^2(M,M) \neq 0$?
$\DeclareMathOperator{\Ext}{\operatorname{Ext}}$The first question is about group algebras:
Question 1: Let $A=kG$ be a group algebra (with $G$ finite) and let $M$ be an indecomposable $A$-module. ...
- 26.5k
11
votes
0
answers
202
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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
11
votes
0
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818
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How to compute Ext-groups for categories without enough injectives/projectives?
I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
- 507
10
votes
0
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236
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Is being derived equivalent independent of the field?
Let $Q_1, Q_2$ be (connected) acyclic quivers and $I_1, I_2$ admissible ideals (in which the relations have only coefficients 1 or -1).
Let $K$ and $F$ be two fields.
Question 1: Is $KQ_1/I_1$ ...
- 26.5k
10
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0
answers
1k
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Complexes of representations with complementary central charges
This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
- 15.6k
9
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0
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366
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A characterisation of symmetric algebras using Hochschild (co)homology
A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
- 26.5k
9
votes
0
answers
123
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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
8
votes
0
answers
334
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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 $[c_1,c_2,...
- 26.5k
8
votes
0
answers
140
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$n$-fold tensor products of $D(A)$ for finite dimensional algebras
Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected).
Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
- 26.5k
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-...
- 661
8
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0
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399
views
Reference/ elementary proof of a result about projective dimension in group rings
Hello- I've had to use a result that sounds like it should be well-known, but I couldn't find any references, and my proof is rather unsatisfactory. I was hoping someone here could help! The problem ...
- 298
7
votes
0
answers
275
views
Split epimorphism of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...
- 696
7
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355
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A homological algebra approach to the Union-closed sets conjecture
I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
- 26.5k
7
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0
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142
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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
7
votes
0
answers
296
views
Secret exact sequence in path algebras of Dynkin type
Given a connected finite dimensional path algebra $A=KQ$ of Dynkin type with enveloping algebra $A^e= A^{op} \otimes_K A$.
I can prove that there is a canonical exact sequence connecting the ...
- 26.5k
7
votes
0
answers
433
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What is the endomorphism cooperad?
In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...
- 2,074
7
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266
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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_{...
- 26.5k
6
votes
0
answers
178
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Ext for commutative Gorenstein algebras
Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$.
Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...
- 26.5k
6
votes
0
answers
328
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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
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
6
votes
0
answers
94
views
Injective dimension of the radical series of an algebra
Let $A$ be a (connected) finite dimensional algebra with Jacobson radical $J$.
Question: Is the sequence $injdim(J^i)$ for $i=1,2,...,$ monotone decreasing?
(one can ask the same question for $...
- 26.5k
6
votes
0
answers
293
views
Representation-finiteness vs. $\tau$-tilting-finiteness
Setting: Throughout, $\Lambda$ is a finite dimensional associative algebra, $\operatorname{mod} \Lambda$ is the category of all finitely generated left $\Lambda$-modules, and all subcategories are ...
- 493
6
votes
0
answers
266
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Tachikawa conjecture for commutative algebras
Let $A$ be a selfinjective algebra. A famous conjecture by Tachikawa says that $Ext^{i}(M,M) \neq 0$ for some $i>0$ and any nonprojective module $M$ (all algebras and modules are finite dimensional ...
- 26.5k
6
votes
0
answers
209
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Classification of representation-finite algebras up to stable equivalence of Morita type
Assume $K$ is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
- 26.5k
6
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0
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506
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How can you see the minimal relations on a quiver from its bimodule resolution?
Suppose that you are given an algebra $KQ/I$, coming from a quiver Q, of finite global dimension. Suppose also that you know its minimal bimodule resolution over its enveloping algebra. Can you get a ...
- 61
5
votes
0
answers
213
views
Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
- 26.5k
5
votes
0
answers
190
views
On the not so clear relationship between torsion theories and localization for a newcomer
Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so ...
- 83
5
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0
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83
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It there an algebra of the form $B_T$ with global dimension 3?
Let $A$ be the (symmetric Frobenius) algebra $A=K[x]/(x^3) \otimes_K K[x]/(x^3)$ over a field $K$, which is isomorphic to the group algebra of $C_3 \times C_3$, with $C_3$ cyclic of order 3, when $K$ ...
- 26.5k
5
votes
0
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142
views
A practical way to check whether a module is periodic
A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
- 26.5k
5
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0
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116
views
An intelligent ant living on a symmetric quiver algebra - Does it have a way to find out whether it lives on a trivial extension?
For a given algebra $B$ over a field $K$ the trivial extension $T(B)$ of $B$ is defined as follows:
The underlying vectorspace is $T(B)=B \oplus D(B)$ where $D(B)=Hom_K(B,K)$ and the multiplication is ...
- 26.5k
5
votes
0
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76
views
Reference on two numbers associated to a module of finite homological dimension
Let $A$ be a finite dimensional algebra over a field $K$ with a module $M$ which has finite projective dimension and finite injective dimension.
Let $n \geq 1$.
Let $(P_i)$ be a minimal projective ...
- 26.5k
5
votes
0
answers
97
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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
5
votes
0
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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
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
5
votes
0
answers
113
views
On algebras where all indecomposables have no selfextensions
Let $A$ be a finite dimensional algebra (we can assume it is a connected quiver algebra).
Call $A$ extfree in case for every indecomposable $A$-module $M$ we have $\operatorname{Ext}_A^i(M,M)=0$ for ...
- 26.5k
5
votes
0
answers
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 ...
- 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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5
votes
0
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91
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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
5
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0
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125
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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 ...
- 26.5k
5
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0
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105
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Derived invariant acyclic algebras
Call a connected quiver algebra $A=KQ/I$ (finite quiver Q and admissible ideal I) derived-invariant in case every quiver algebra derived equivalent to $A$ is even isomorphic to $A$.
For example local ...
- 26.5k
5
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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 ...
- 26.5k
5
votes
0
answers
253
views
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 ...
5
votes
0
answers
380
views
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 ...
- 1,382
5
votes
0
answers
92
views
Criteria for being representation-infinite for subcategories of quiver algebras
Let $A$ be a quiver algebra over a field $K$ (maybe we need algebraically closed?).
Then the following is two statements are well known:
In case $A$ is representation-infinite, every Auslander-Reiten ...
- 26.5k
5
votes
0
answers
120
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Ext$^1(D(A),A)$ for hereditary algebras
Let $A$ be a hereditary (non-semisimple) finite-dimensional algebra over a field $K$. Let $M:={\rm Ext}^{1}_A(D(A),A)$ ($ \cong D(\tau((D(A)))$ as left modules) and let $A^e$ be the enveloping algebra ...
- 26.5k
5
votes
0
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303
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Recovering an A-infinity structure on an Ext-algebra from a quiver presentation
Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules.
According to [Keller: A-infinity algebras in ...
- 2,450
5
votes
0
answers
209
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Ext^1 for a local finite dimensional selfinjective algebra
Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here:
http://arxiv.org/pdf/1609.00588.pdf.
There it is ...
- 26.5k
5
votes
0
answers
259
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divided power structure on Hocschild cohomology?
Does Hochschild cohomology of a cocommutative Hopf algebra over a field of positive characteristic have a natural divided power structure?
If not, perhaps a certain natural extra structure on the ...
- 1,526