All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
5
votes
0
answers
246
views
Partial formality of A-infinity structure implies formality
Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that $\operatorname{Ext}...
- 2,450
4
votes
4
answers
596
views
Homology of solvable (nilpotent) Lie algebras
Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for ...
- 332
4
votes
1
answer
375
views
Invertible bimodules and projectivity
Let $A$ be a noncommutative algebra over a field, say $\mathbb{C}$ or $\mathbb{R}$, and let $L$ be a bimodule over $A$. If $L$ is invertible, that is, if the dual right $A$-module $L^*$ satisfies
$$
L^...
4
votes
2
answers
453
views
Global dimension of the tensor algebra
Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$.
Question 1: Is there a simple proof that $...
- 26.5k
4
votes
2
answers
299
views
Relation of the first Hochschild cohomology and the outer automorphism group
Let $R$ be a ring.
Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?
(It is not true, by the two answers. Is it ...
- 26.5k
4
votes
2
answers
337
views
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
4
votes
1
answer
101
views
Extension of scalars for bounded chain complexes of $kG$-modules
I'm wondering if a generalization regarding a statement from Curtis-Reiner holds. The original statement is as follows:
(30.33) Theorem: Let $R$ and $S$ be complete discrete valuation rings, with $S$ ...
- 175
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
4
votes
1
answer
423
views
Alternating sum of symmetric and exterior powers vanishes
Let $V$ be a vector space with some extra structure (maybe to be general, an object of an abelian tensor category?), where we can form tensor products and exterior powers $\Lambda^i V$ and symmetric ...
- 3,149
4
votes
1
answer
198
views
Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?
All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...
- 696
4
votes
1
answer
302
views
Hattori-Stallings trace
Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
- 985
4
votes
1
answer
107
views
For a finite dimensional $k$-Algebra $A$ does infinite representation type imply $(\text{rad}_A^{\omega})^2 \neq 0$?
Let $k$ be a field, $A$ a finite dimensional $k$-Algebra, $\text{mod}\,A$ the category of finite dimensional left $A$-modules and $\text{rad}_A$ the collection of radical morphisms in $\text{mod}\,A$. ...
- 696
4
votes
1
answer
463
views
Global dimension of a graded algebra
Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...
- 26.5k
4
votes
1
answer
655
views
Are all algebras Igusa-Todorov?
A finite dimensional algebra A is called (n-)Igusa-Todorov in case there exists a module V such that for any module M there is an exact sequence:
$0 \rightarrow V_2 \rightarrow V_1 \rightarrow \Omega^{...
- 26.5k
4
votes
1
answer
286
views
Why is the representation dimension of an Artin algebra never equal to 1?
Hi,
in 1971 M.Auslander showed that the representation dimension of $A$ is $\neq 1$ for every Artin algebra $A$.
Does anybody have a reference paper or book proving this? Is the proof easy and / or ...
- 1,755
4
votes
1
answer
184
views
FI-homology of a spectral sequence of rational FI-modules
Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
- 275
4
votes
1
answer
158
views
Can we always choose 2 nonisomorphic simple modules to satisfy the following nonvanishing extension conditions?
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$Let A be a finite dimensional algebra over some field k and S a nonprojective simple left A-module. Suppose the projective dimension $\pd_A(S)$...
- 775
4
votes
1
answer
615
views
Characterisation of reflexive modules
Let $A$ be a semiperfect noetherian ring.
A module $M$ is called reflexive in case the canonical map $f_M: M^{**} \cong M$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. This is equivalent to say that ...
- 26.5k
4
votes
1
answer
273
views
Homological characterisation of standardly stratified algebras using Ext
Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...
- 26.5k
4
votes
1
answer
233
views
Derived equivalences and Tachikawa conjecture
The first Tachikawa conjecture states that for a finite dimensional algebra $A$, $Ext_A^i(D(A),A)=0$ for all $i \geq 1$ implies that $A$ is selfinjective.
Question: In case $A$ has the property that $...
- 26.5k
4
votes
1
answer
149
views
Testing whether a module generates $K_0(\mbox{mod-}A)$
Given a representation-finite (connected) quiver algebra $A$ and a module $M$.
Is there a good way to test whether the set $\{ [N] \mid N \in \mathrm{add}(M) \}$ generates $K_0(\mbox{mod-}A)$?
Can ...
- 26.5k
4
votes
1
answer
303
views
Algebras derived equivalent to quasi-hereditary algebras
Let an algebra always be finite dimensional over a field and connected.
It is well known that a quasi-hereditary algebra with $n$ simple modules has global dimension at most $2n-2$.
Questions:
1. ...
- 26.5k
4
votes
1
answer
179
views
Reference for isomorphism $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$
Let $A$ be a finite dimensional algebra and $X,Y$ indecomposable modules and $n \geq 2$.
We know that $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$.
Now such an isomorphism should be given by ...
- 26.5k
4
votes
1
answer
233
views
Right approximation in certain subcategories
Let $A$ be an Artin algebra and $C$ a subcategory of mod-$A$ that contains all projective modules and is closed under finite direct sums (but not necessarily under direct summands).
Let $T:=add(C)$.
...
- 26.5k
4
votes
1
answer
685
views
Quadratic algebras and Koszul algebras
Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$.
In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...
- 26.5k
4
votes
1
answer
159
views
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. ...
- 26.5k
4
votes
1
answer
189
views
Property of non-Gorenstein algebras
In the article http://www.sciencedirect.com/science/article/pii/S0021869301991306?via%3Dihub (see also the MO thread Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\...
- 26.5k
4
votes
1
answer
290
views
An inequality for Ext
$\DeclareMathOperator\Ext{Ext}$Consider the following statement for a $K$-algebra $A$:
$\dim(\Ext^1 (M, N )) \ge \min( \dim(\Ext^1(M, M )), \dim(\Ext^1 (N, N )) ),$
for all finite dimensional ...
- 26.5k
4
votes
1
answer
104
views
Which Auslander algebras satisfy $Ext_B^1(D(B),B)=0$?
Let $B$ be the Auslander algebra of a representation-finite algebra $A$.
Question: When do we have $Ext_B^1(D(B),B)=0$? Can this be expressed in terms of nice properties of $A$?
This is for example ...
- 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
4
votes
1
answer
122
views
Postprojective components of quiver algebras
Let $A=kQ/I$ be a quiver algebra with acyclic quiver $Q$.
An indecomposable module $M$ is called postprojective in case $M \cong \tau^{-1}(P)$ for an indecomposble projective module $P$. A component ...
- 26.5k
4
votes
1
answer
241
views
Questions in the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences"
I am reading the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences", the link is here:https://pub.uni-bielefeld.de/publication/1780235.
In the paper, $A$ ...
- 775
4
votes
0
answers
75
views
Gluing the outer functors between two recollements
Assume there are two recollements of triangulated categories and the functors $f_1$ and $f_3$ below.
\begin{align*}
\begin{array}{rcccc}
{\mathbf{T}_1} & \underset{\underset{i_R}\leftarrow}{\...
- 41
4
votes
0
answers
114
views
Classification of 2-periodic triangulated categories
Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$.
Question 1: Is there a ...
- 26.5k
4
votes
0
answers
227
views
Orbits of group representation over $\mathbb{F}_2$
Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
- 161
4
votes
0
answers
196
views
Quillen–Suslin theorem in a more general context
Let $A$ be a finite dimensional local Frobenius algebra that is Koszul.
Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...
- 26.5k
4
votes
0
answers
231
views
How far is the sphere spectrum to be tilting/silting
In the last years the following definition of a tilting/silting object in an arbitrary triangulated category with coproducts emerged:
Let $\mathcal D$ be a triangulated category with coproducts and ...
- 666
4
votes
0
answers
62
views
Deforming relations for a symmetric Frobenius algebra to obtain an algebra of finite global dimension
By doing computer experiments with the GAP-package QPA I found a (connected non-semisimple) symmetric quiver algebra $A=KQ/I$ with admissible relations $I$ containing relations of the form $w_1 - w_2$ ...
- 26.5k
4
votes
0
answers
83
views
Number of K-generators of an algebra and type $D_n$-parking functions
Let $A$ be a representation-finite quiver algebra.
When $A$ has $n$ simple modules a basic module $M$ with $n$ indecomposable summands $M_i$ is called a K-generator when the $M_i$ generate $K_0(A)$, ...
- 26.5k
4
votes
0
answers
121
views
Perfect modules for the Beilinson algebra
The Beilinson algebra $A=A_n$ is a finite dimensional quiver algebra that is derived equivalent to the category of coherent sheaves of $\mathcal{P}^n$. See for example https://link.springer.com/...
- 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
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
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
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
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
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
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
0
answers
73
views
Frobenius dimensions of Nakayama algebras
The Frobenius dimension $F(A)$ of an Artin algebra $A$ is given by the dimension of $Hom(D(A),A)$ (see the answer in On nearly Frobenius algebras ).
Question 1: Is it true that $F(A) \geq gldim(A)$ ...
- 26.5k
4
votes
0
answers
192
views
Extended double 2-cocycle conditions: Mathematical structure behind?
Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly.
The ordinary group 2-cocycle condition:
Let us remind the usual so-called homogeneous group 2-cocycle $...
- 10.5k
4
votes
0
answers
61
views
Characterisation of algebras with Euler trivial modules
Let $A$ be an algebra of finite global dimension.
The Euler form on $A$ for an indecomposable module $M$ is defined as $\psi(M)=\sum\limits_{k=0}^{\infty}{(-1)^k dim( \operatorname{Ext}_A^k(M,M)) }$.
...
- 26.5k