All Questions
50 questions
16
votes
3
answers
1k
views
Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 26.5k
1
vote
0
answers
124
views
Computing the induced homomorphisms of derived functors using acyclic resolutions
Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
- 38.6k
1
vote
0
answers
106
views
Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
8
votes
1
answer
356
views
Homological conjectures for finite dimensional commutative algebras
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
- 26.5k
3
votes
1
answer
173
views
$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
2
votes
0
answers
138
views
Construction of a certain long exact sequence
Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field.
Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...
- 26.5k
2
votes
0
answers
84
views
Representation finite Hopf algebras up to stable equivalence
It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra.
Question: Is it true that every representation-finite Hopf algebra is stable ...
- 26.5k
5
votes
1
answer
207
views
Finite lattices that are Koszul
Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$.
It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
- 26.5k
2
votes
0
answers
70
views
Rigid modules for hereditary algebras
Let $A=KQ$ be a path algebra of a connected quiver. (K algebraically closed if it helps)
Question: Is there an explicit classification of all indecomposable $A$-modules $M$ that are rigid, that is $...
- 26.5k
3
votes
1
answer
177
views
Quiver and relations for ADE singularities in dimension one
Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...
- 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
6
votes
1
answer
244
views
What is a Serre-smooth algebra?
Let $A$ be an $R$-algebra.
In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction.
But no formal ...
- 26.5k
2
votes
0
answers
91
views
When does a stable endomorphism ring have injective dimension at most one?
tLet $A$ be a Frobenius algebra (we can assume that $A$ is given by quiver and relations) and let $M$ be a basic $A$-module without projective direct summands (we can assume we know the decomposition ...
- 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
10
votes
1
answer
307
views
Rings where all indecomposable projective modules are finitely generated
Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.
Question 1: Is there a nice equivalent ...
- 26.5k
5
votes
0
answers
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
6
votes
1
answer
186
views
Endomorphism ring of trivial source modules for abelian p-groups
Bernhard Böhmler (who is also on MO) and myself had the following idea:
Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
- 26.5k
3
votes
0
answers
48
views
Questions on piecewise hereditary algebras
Let $A$ be a finite dimensional quiver algebra over a field $k$ that is quasi-tilted and representation-finite (this implies that $A$ is a tilted algebra). Assume that the Coxeter polynomial of $A$ is ...
- 26.5k
10
votes
1
answer
396
views
Generalising the union-closed sets conjecture from lattice to a larger class of posets
(edit: I decided to simplify the question and only pose it for bounded posets first)
The Union-closed sets conjecture is equivalent for lattices P to:
There exists a join-irreducible element $a$ with ...
- 26.5k
5
votes
0
answers
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
132
views
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
6
votes
1
answer
203
views
Question on a subcategory being extension-closed
In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\...
- 26.5k
5
votes
2
answers
281
views
Isomorphism for Ext spaces for finite dimensional algebras
Let $A$ be an Artin algebra with enveloping algebra $A^e$.
Then we have $Hom_{A^e}(X,A^e) \cong Hom_A(D(A) \otimes_A X,A)$ for a bimodule $X$. (see for example in the article "A theorem of Green on ...
- 26.5k
2
votes
1
answer
200
views
Characterisation of minimal projective resolutions via the Euler characteristic
Let $A$ be a finite dimensional $K$-algebra (where $K$ is a field) and $M$ a finitely generated $A$-module.
Let $\psi: 0 \rightarrow P_r \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$
...
- 26.5k
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
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
5
votes
1
answer
225
views
Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
- 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
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
4
votes
2
answers
770
views
Finitistic dimension conjecture for quadratic algebras
The finitistic dimension of a finite dimensional algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. The finitistic dimension conjecture says ...
- 26.5k
5
votes
0
answers
140
views
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/...
- 26.5k
5
votes
0
answers
91
views
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
votes
0
answers
125
views
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
6
votes
2
answers
268
views
Derived invariance of the Cartan determinant
The Cartan matrix $C$ of a finite quiver algebra $A$ with points $e_i$ is defined as the matrix having entries $c_{i,j}=\dim(e_i A e_j)$. The Cartan determinant is defined as the determinant of the ...
- 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
1
vote
0
answers
77
views
n-Gorenstein algebras and tilting modules
Let $\Lambda$ be an artin algebra over a commutative artinian ring $R$. $\Lambda$ is said to be $n$-Gorenstein, for some natural number $n$, provided it have finite self-injective dimension at most $n$...
- 775
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
1
vote
1
answer
174
views
Reference for a result of Auslander about the global dimension
One of Auslanders famous theorems is that he proved that the global dimension of a semiprimary ring is equal to the maximum of the projective dimensions of the simple modules of the ring. This result ...
- 26.5k
2
votes
0
answers
61
views
Question on outer Ext-products
For group algebras $A=KG$ over a field $K$ with finite group $G$ there exists an outer product on Ext:
$Ext_A^i(M,N) \otimes_K Ext_A^j(M',N') \rightarrow Ext_A^{i+j}(M \otimes_K M',N \otimes_K N')$.
...
- 26.5k
1
vote
0
answers
63
views
Reference request for formula on global dimension
Given a finite dimensional algebra $A$ over an algebraically closed field $K$.
Let $A^e=A^{op} \otimes_K A$ be the enveloping algebra of $A$.
Who noted first that the global dimension of $A$ is equal ...
- 26.5k
15
votes
1
answer
960
views
Who conjectured the Cartan determinant conjecture
The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this ...
- 26.5k
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 ...
- 507
5
votes
0
answers
303
views
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
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
0
votes
1
answer
337
views
Homological dimensions of tensor products of algebras
Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A ...
- 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
6
votes
0
answers
209
views
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
10
votes
1
answer
842
views
Is there a "correct" general setting for the principle: "tensoring any object with a projective object yields another projective"?
Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$. (See Exercise 2 on p. ...
- 52.9k
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