All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
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 ...
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 ...
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
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_{*...
user108998
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
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
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, ...
- 993
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
$$...
4
votes
1
answer
375
views
Invertible bimodules and projectivity
Let $A$ be a noncommutative algebra over a field, say $\mathbb{C}$ or $\mathbb{R}$, and let $L$ be a bimodule over $A$. If $L$ is invertible, that is, if the dual right $A$-module $L^*$ satisfies
$$
L^...
4
votes
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
10
votes
0
answers
236
views
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
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 ...
- 26.5k
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, ...
- 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
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
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 ...
- 6,622
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
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 ...
- 26.5k
12
votes
0
answers
402
views
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
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 $End_A(M)$ ...
- 26.5k
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 ...
- 26.5k
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 ...
- 26.5k
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 ...
- 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 ...
- 6,292
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
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
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 $n=4$ ...
- 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 ...
- 26.5k
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
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 ...
- 26.5k
5
votes
0
answers
114
views
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
2
votes
1
answer
97
views
When is $N^{*} \otimes_K M$ projective for a local Hopf algebra?
Given a finite dimensional local Hopf algebra $A$ over a field $K$ and two finite dimensional indecomposable modules $N$ and $M$. Is it known when the module $N^{*} \otimes_K M$ is projective? Can ...
- 26.5k
12
votes
0
answers
516
views
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
4
votes
0
answers
69
views
Auslander-Solberg algebras from non-rigid modules
Let $A$ be a Nakayama algebra and $M$ be the direct sum of all indecomposable $A$-modules $N$ with $Ext_A^1(N,N) \neq 0$.
The following is suggested by computer experiments with QPA:
Question: Is ...
- 26.5k
2
votes
1
answer
169
views
Combinatorial problem on periodic dyck paths from homological algebra
edit: I added conjecture 2 that looks much more accessible.
Here is the elementary combinatorial translation of the problem (read below for the homological background):
Let $n \geq 2$.
A Nakayama ...
- 26.5k
2
votes
0
answers
51
views
Selfinjective algebras with loops
Given a selfinjective finite dimensional algebra $A$ with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$.
Question:
Is A derived equivalent to an algebra with a loop in the quiver in ...
- 26.5k
2
votes
0
answers
53
views
Strong no loop conjecture for uniserial modules
Let $A$ be a an Artin algebra. The strong no loop conjecture states that a simple $A$-module with $Ext_A^1(S,S) \neq 0$ has infinite projective dimension.
This conjecture was recently proved for ...
- 26.5k
2
votes
0
answers
45
views
On monomial and $\Omega^d$-finite algebras
Let $Q$ be a finite quiver and $I$ a monomial admissible ideal of the path algebra $KQ$ for a field $K$. Then an algebra $A=KQ/I$ is called a monomial algebra.
It is well known that monomial algebras ...
- 26.5k
1
vote
0
answers
55
views
$\mathrm{Ext}^1$-ordering on ${}^IW^{\Sigma_\mu}$
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
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
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
2
votes
0
answers
69
views
Stable m-Calabi Yau property for Frobenius categories
Let $C$ be a Frobenius category. The stable category $\underline{C}$ is called $m$-Calabi Yau in case it is Hom-finite and there is a functorial duality
$D \underline{Hom}(X,Y)=\underline{Hom}(Y,\...
- 26.5k
2
votes
0
answers
73
views
Equivalence from a tilting module
Let $A$ be a finite dimensional algebra. For a subcategory $C$ of $mod-A$ let $\overline{C}$ be the objects $X \in mod-A$ such that there exists an exact sequence $0 \rightarrow C_n \rightarrow ... \...
- 26.5k
4
votes
0
answers
82
views
On strongly simply connected quiver algebras
Let $A$ be a representation-finite quiver algebra. In this case $A$ is simply connected if and only if its first Hochschild cohomology vanishes by a result of Buchweitz and Liu. $A$ is called strongly ...
- 26.5k
2
votes
0
answers
56
views
Characterisation of representation-directed algebras
A representation-directed algebra $A$ over a field $K$ has the property that for every indecomposable module $M$ we have $\mathrm{End}_A(M) \cong K$ and $\mathrm{Ext}_A^i(M,M)=0$ for all $i>0$.
...
- 26.5k
16
votes
2
answers
694
views
How complicated can a finite double complex over a field be?
A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is ...
- 63.9k