All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
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 ...
- 1,244
11
votes
0
answers
818
views
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 ...
- 63.9k
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
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
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 ...
CommunityBot
- 1
8
votes
3
answers
1k
views
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 ...
- 63.9k
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
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 ...
- 16.9k
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$.
...
- 12.3k
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 ...
- 1,042
3
votes
1
answer
244
views
Left module which cannot be made into a bimodule?
Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a ...
- 12.3k
3
votes
2
answers
1k
views
Dual of a projective module
Let $R$ be a noncommutative ring with unit, let $P$ be a projective left $R$-module, and denote $^{\vee}\!P := \,_R\mathrm{Hom}(P,R)$. One often sees it written that projectivity implies an ...
- 12.3k
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
32
votes
3
answers
4k
views
Replacing triangulated categories with something better
Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at ...
- 9,229
1
vote
0
answers
30
views
Right approximations for special modules in Frobenius algebras
Let $A$ be a commutative Frobenius algebra (we can assume $A$ is also local) given by quiver and relations.
Let $M_i=A/p_iA$ be a module where $p_i$ is a path in Q.
Let $N:=A \oplus \bigoplus\limits_{...
- 26.5k
14
votes
1
answer
1k
views
Factorization and vertex algebra cohomology
A chiral algebra on a smooth curve $X$, in the sense of Beilinson-Drinfeld, is a right $D_{X}$-module with a chiral bracket, which is a map $\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta_{*...
- 3,242
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
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 ...
CommunityBot
- 1
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)$ ...
CommunityBot
- 1
4
votes
0
answers
210
views
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 ...
- 10.5k
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
6
votes
1
answer
339
views
Monoidal categories from the projective modules of a ring
Let $R$ be a not necessarily commutative ring, and denote by $_R\mathrm{lp}_R$ the category of $R$-bimodules, which are finitely generated projective as left modules, with morphism $R$-bimodule maps, ...
- 55.9k
2
votes
1
answer
98
views
A weaker version of strongly graded algebras
Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that
$$...
- 39.3k
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^...
- 38k
6
votes
1
answer
205
views
When is the Jacobson radical reflexive?
Let algebras be Artin algebras.
It is well known that a an algebra has global dimension at most one if and only if the Jacobson radical is projective. As reflexive is a natural generalisation of ...
CommunityBot
- 1
6
votes
1
answer
220
views
Derived invariant for Gorenstein algebras?
Let $A$ be a finite dimensional algebra with simple modules $S_i$ and projective indecomposable modules $P_l$ and global dimension $g< \infty$ (and $n$ is the number of simple modules).
I noted ...
- 35.2k
5
votes
1
answer
268
views
Global dimension 2 Nakayama algebras , 321-avoiding permutations and Fibonacci combinatorics
A $n$-LNakayama algebra (for $n \geq 2$) is a list $[c_0,c_1,...,c_{n-1}]$ with $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i -1 \leq c_{i+1}$ for all $i$.
They are in bijection with Dyck paths, ...
- 24.2k
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
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
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 ...
CommunityBot
- 1
6
votes
1
answer
368
views
Indecomposable objects in bounded derived category of $\mathbb C[x]/x^2$-mod
We know for any principal ideal domain, objects in the bounded derived category are all formal hence we can classify those objects with finitely generated cohomology using structure theorem for ...
- 35.2k
2
votes
0
answers
104
views
Tate cohomology for group algebras
Let $A=kG$ be a group algebra with a finite group $G$ and a field $k$.
Let $T^i(M,N)= \underline{Hom_A}(\Omega^i(M),N)$ be the $i$-th Tate cohomology group. Note $T^i(M,N)= Ext_A^i(M,N)$ in case $i \...
- 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 $End_A(M)$ ...
CommunityBot
- 1
2
votes
1
answer
307
views
Gaps in the projective dimensions of simple modules
Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $n$ simple modules.
Let $d_1<d_2<...<d_r$ be the sequence of projective dimension of simple $A$-modules in ...
- 1,690
1
vote
1
answer
132
views
Representation-finite implies planar for quiver algebras?
Let $A=KQ/I$ be a finite dimensional quiver algebra with an admissible ideal $I$.
Is it true that in case $A$ is representation-finite, $Q$ has to be planar?
In case it is true a possible approach ...
- 35.2k
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 $...
CommunityBot
- 1
1
vote
1
answer
190
views
Ext in Nakayama algebras
Let $A$ be a Nakayama algebra, that is an Artin algebra such that any indecomposable module has a unique composition series. The easiest examples of such algebras are $K[x]/(x^n)$.
Question 1: In ...
CommunityBot
- 1
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
8
votes
1
answer
193
views
Maximal numbers of summands in middle terms of short exact sequences
Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split ...
CommunityBot
- 1
14
votes
2
answers
514
views
Classification of shod Dyck paths
A sequence $[c_0,c_1,...,c_{n-1}]$ with $n \geq 2$ is called a Dyck path in case $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i-1 \leq c_{i+1}$ for each $i$.
For example the Dyck paths for $n=4$ ...
- 7,875
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 ...
- 44.7k
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
2
votes
1
answer
200
views
Projective dimensions of simple modules in acyclic quiver algebras
Given a quiver algebra $A=KQ/I$ with acyclic $Q$. Then $A$ has finite global dimension, lets say $g$.
Question: Is there for any $0 \leq i \leq g$ a simple module with projective dimension equal to ...
- 35.2k
4
votes
1
answer
346
views
Verma module and vanishing of extension groups
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$.
Let $W$ be the associated Weyl group and let $\Phi$ be its root system.
We write $\Phi^+...
- 1,875