All Questions
Tagged with rt.representation-theory homological-algebra
40 questions
15
votes
2
answers
863
views
What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
- 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
13
votes
1
answer
5k
views
What are tame and wild hereditary algebras?
What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
- 2,992
12
votes
1
answer
922
views
Does this algebra have finite global dimension ? (Human vs computer)
Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...
- 26.5k
11
votes
2
answers
558
views
Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
- 26.5k
10
votes
0
answers
1k
views
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
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
0
answers
228
views
Question on $n$-torsionless modules
Let $A$ be a finite dimensional algebra. Recall that a module $M$ is called $n$-torsionfree in case $Ext_A^i(D(A),\tau(M))=0$ for all $i=1,...,n$ when $\tau$ denotes the Auslander-Reiten translate. ...
- 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
3
votes
0
answers
205
views
Finitistic dimension via tilting modules
is the following true (all algebras and modules are assumed to be finite dimensional):
The finitistic dimension of an algebra is equal to the supremum of projective dimensions of tilting modules?
It ...
- 26.5k
28
votes
4
answers
3k
views
Yoga of six functors for group representations?
I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can ...
- 7,789
10
votes
1
answer
400
views
Derived equivalences of Dyck paths
Call two Dyck paths $D_1$ and $D_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top ...
- 26.5k
10
votes
1
answer
3k
views
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
8
votes
2
answers
1k
views
When is the exterior algebra a Hopf algebra?
I have several questions on the exterior algebra of a vector space:
Q1:When has the exterior algebra A (viewed just as an algebra, not considered as a graded algebra) of an $n$-dimensional vector ...
- 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
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
7
votes
0
answers
355
views
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
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
266
views
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
1
answer
361
views
How to check whether a module is an n-th syzygy
Given a finite dimensional algebra $A$, define $\Omega^{n}(mod-A)$ (modules here are always finite dimensional) to be the full subcategory of projective modules or modules $M$ such that $M \cong \...
- 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
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
5
votes
0
answers
209
views
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
88
views
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
1
answer
353
views
Existence of non-trivial reflexive modules
Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f_M:M \rightarrow M^{**}$ (where $M^{*}=Hom_R(M,R)$) is an isomorphism, where $f_M(m)=g$ with $g(h)=h(m)$, ...
- 26.5k
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
0
answers
76
views
Minimal rank of a permutation resolution of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
- 14.2k
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
3
votes
1
answer
240
views
Split monomorphisms 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. Further $X\subseteq M$ and for ...
- 696
3
votes
1
answer
339
views
If the Hom-space of finite length modules is generated by single elements, must the elements be conjugate?
Let $A$ be an Artin $k$-algebra for a commutative artinian ring $k$ (e.g. $A$ is a finite dimensional algebra over a field $k$). Let $X,Y$ be finite length left $A$-modules. If $\text{Hom}_A(X,Y)$ is ...
- 696
3
votes
2
answers
214
views
History of an open problem on partial tilting modules
The following is an open problem:
Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting ...
- 26.5k
3
votes
1
answer
187
views
Explicit proof that algebra is derived wild
Following the terminology of
Drozd, Yuriy A., Derived tame and derived wild algebras, Algebra Discrete Math. 2004, No. 1, 57-74 (2004). ZBL1067.16028.
let $A$ and $R$ be algebras over a field $k$. A ...
- 497
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
2
votes
1
answer
118
views
Example to periodic symmetric algebras
In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric.
Is there an easy counterexample for the following:
In case $A$ is additionally a periodic ...
- 26.5k
2
votes
1
answer
207
views
Question on Ext for finite dimensional algebras
Given a finite dimensional non-Gorenstein algebra $A$, do we have $Ext^i(D(A),A) \neq 0$ for infinitely many $i$? (We can assume A is local or commutative if that helps).
All I can show is that such ...
- 26.5k
2
votes
1
answer
389
views
Calabi-Yau algebra for finite dimensional algebras
I read the article "Defomrations of algebras in noncommutative geometry" by Schedler.
In Definition 3.7.9. he gives the definition of Calabi-Yau algebra of dimensi on d as algebras that are ...
- 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
1
vote
1
answer
127
views
Gorenstein projective modules of a certain triangular matrix algebra
Let $B$ be a finite dimensional selfinjective algebra over a field $k$ with a finite dimensional non-projective $B$-module $M$ and $$A=\pmatrix{k&M\\0&B}.$$
A module $N$ over an algebra $C$ ...
- 26.5k
1
vote
1
answer
110
views
Question on strong cotilting modules
In https://www.sciencedirect.com/science/article/pii/0001870891900378 section 6 a cotilting module T over an algebra A is said to be strong in case $\hat{add(T)}$ coincides with the subcategory of ...
- 26.5k
1
vote
1
answer
131
views
Finite add(N)-resolution
Let $A$ be a local selfinjective algebra with indecomposable module $M$.
Let $N=A \oplus M$.
When there is an indecomposable module $U$ not in $add(N)$, having finite $add(N)$-resolution for some ...
- 26.5k