All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
8
votes
1
answer
217
views
Categorification of monotone maps via tilting modules?
It is well known that for the algebra of $n \times n$-upper triangular matrices over a field the number of tilting modules is equal to the Catalan number $C_n$. This is just the (hereditary) Nakayama ...
- 26.5k
8
votes
0
answers
334
views
Dyck paths of Dynkin type
(The conjecture is a homological algebra question, but question 2 is a pure combinatorics question given that the conjecture is true)
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...
- 26.5k
8
votes
0
answers
140
views
$n$-fold tensor products of $D(A)$ for finite dimensional algebras
Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected).
Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
- 26.5k
8
votes
0
answers
173
views
On constructible Hall algebra and instantons
I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
- 661
8
votes
0
answers
399
views
Reference/ elementary proof of a result about projective dimension in group rings
Hello- I've had to use a result that sounds like it should be well-known, but I couldn't find any references, and my proof is rather unsatisfactory. I was hoping someone here could help! The problem ...
- 298
7
votes
3
answers
3k
views
Beilinson-Bernstein and Koszul duality
For geometric representation theorists down here.
Consider the Beilinson-Bernstein theorem:
Functor of global sections establishes
the correspondence between twisted
D-modules with fixed ...
- 15.1k
7
votes
2
answers
2k
views
Whitehead lemmas in Lie algebra cohomology for non-algebraically closed fields
I read in Weibel's homological algebra that Whitehead's first and second lemmas are true for any characteristic 0 field. I mean the following:
Whitehead Lemma(s): Let g be a semisimple Lie algebra ...
- 349
7
votes
1
answer
370
views
Gorenstein symmetric conjecture for arbitrary rings
The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension ...
- 26.5k
7
votes
1
answer
287
views
Semi-projective complexes of modules over a finite group
Let $G$ be a finite group and $k$ a field of characteristic $p$ dividing $|G|$. A perfect complex of $kG$-modules is by definition a finite complex of finitely generated projective ($=$ injective $=$ ...
- 16.2k
7
votes
1
answer
462
views
On a problem for determinants associated to Cartan matrices of certain algebras
This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
- 26.5k
7
votes
1
answer
495
views
Serre functor on the category $Perf(A)$, $A$ - k-algebra
Consider a finite-dimensional $k$-algebra $A$ of finite global dimension. Then it is known that the Serre functor on $D^b(mod-A)$ exists and is given by the Nakayama functor. The proof goes something ...
- 1,071
7
votes
1
answer
180
views
Complexity of rational $\mathrm{GL}_{n(r)}$-modules
Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
- 768
7
votes
0
answers
275
views
Split epimorphism 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. Suppose further we have an ...
- 696
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
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
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
433
views
What is the endomorphism cooperad?
In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...
- 2,074
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
3
answers
368
views
Is there a finite dimensional algebra with left finitistic dimension different from its right finitistic dimension?
Let $\Lambda$ be finite dimensional algebra over a field $k$. The (left) finitistic dimension of a finite dimensional algebra is defined as
$$\operatorname{findim}(\Lambda)=\sup\{\operatorname{pd}M | ...
- 497
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
6
votes
2
answers
2k
views
Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan
Where can I find a comprehensive treatment of this important result at the level of a very advanced undergraduate/beginning graduate student? What works develop the relevant material in a cohesive and ...
- 1,539
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
6
votes
2
answers
269
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
6
votes
2
answers
332
views
Ext in symmetric algebras and group algebras
Let $A$ be a selfinjective algebra and for an indecomposable module $M$ define $\psi_M:= \inf \{ i \geq 1 | Ext_A^i(M,M) \neq 0 \}$.
Questions:
In case $A$ is symmetric, do we have $\psi_M \leq max \...
- 26.5k
6
votes
1
answer
775
views
Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra
Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$.
Tilting module theory play an important role in the ...
- 159
6
votes
1
answer
197
views
Known posets of tilting modules for finite dimensional algebras
Question: For which classes of finite dimensional algebras $A$ is the poset of tilting $A$-modules known?
Here two famous examples:
-For the path algebra of a linear oriented quiver of Dynkin type $...
- 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
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
6
votes
2
answers
768
views
Auslander-Reiten theory for Gorenstein algebras
In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
- 26.5k
6
votes
2
answers
229
views
Bounding the projective dimension of modules by the number of points and arrows
Let $A=KQ/I$ be a finite dimensional connected quiver algebra with admissible ideal $I$ with n points and m arrows and $M$ an $A$-module.
Question: Is the projective dimension of $M$ bounded by $n+m$ ...
- 26.5k
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 ...
- 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
6
votes
1
answer
420
views
Calculating the Ext-algebra with a computer
Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension.
Let $B$ be the Ext algebra of $M$, that is $B:=\...
- 26.5k
6
votes
1
answer
505
views
Global dimension of quiver algebra
Given a representation-finite (finite dimensional over a field) quiver algebra of finite global dimension. Is $eAe$ isomorphic to the field for at least one primitive idempotent $e$?
This is true for ...
- 26.5k
6
votes
1
answer
448
views
Hochschild cohomology of certain local algebras
Let $K$ be a field and $A$ the algebra $K\langle x_1,...,x_n\rangle/J^m$ for $n \geq 1$ and $m \geq 2$, where $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring in $n$ variables over ...
- 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
1
answer
310
views
A formula for the projective dimension of finite dimensional algebras
Let $A$ be a finite dimensional ring-indecomposable $K$-algebra that is not selfinjective for $K$ a field and let $I(A)$ denote the injective envelope of the regular module $A_A$. Define the stable $A$...
- 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
6
votes
1
answer
689
views
Tensor product of bimodules
Im not sure whether this question is appropriate for MO, but I do not have much experience with bimodules (or I forgot many things).
Let $A$ be a finite dimensional (connected) algebra over a field $...
- 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
6
votes
1
answer
894
views
Generators of the derived category
For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}_R$ is generated by the indecomposable projective ...
- 456
6
votes
1
answer
281
views
An identity for Ext for rings
Let $A$ be a two-sided noetherian ring (which we should assume to be Gorenstein first so that everything is well defined, otherwise it is only well defined up to a conjecture, which states that every ...
- 26.5k
6
votes
1
answer
204
views
On the injective dimension of $A$ as a bimodule
(Note: I modified the question and instead of looking at simple modules in the question, I look at all indecomposable modules.)
Let $A$ be a finite dimensional algebra over a field $K$ given by an ...
- 26.5k
6
votes
0
answers
178
views
Ext for commutative Gorenstein algebras
Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$.
Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...
- 26.5k
6
votes
0
answers
328
views
When does a group and its pro-algebraic completion have equivalent categories of arbitrary representations?
In the following everything is over some field $k$.
Let $G$ be a discrete group. We write $G^{\text{alg}}$ for its pro-algebraic completion. The latter is a pro-affine pro-algebraic group which arises ...
- 1,635
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
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
6
votes
0
answers
293
views
Representation-finiteness vs. $\tau$-tilting-finiteness
Setting: Throughout, $\Lambda$ is a finite dimensional associative algebra, $\operatorname{mod} \Lambda$ is the category of all finitely generated left $\Lambda$-modules, and all subcategories are ...
- 493
6
votes
0
answers
266
views
Tachikawa conjecture for commutative algebras
Let $A$ be a selfinjective algebra. A famous conjecture by Tachikawa says that $Ext^{i}(M,M) \neq 0$ for some $i>0$ and any nonprojective module $M$ (all algebras and modules are finite dimensional ...
- 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