All Questions
Tagged with rt.representation-theory homological-algebra
481 questions
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
6
votes
0
answers
506
views
How can you see the minimal relations on a quiver from its bimodule resolution?
Suppose that you are given an algebra $KQ/I$, coming from a quiver Q, of finite global dimension. Suppose also that you know its minimal bimodule resolution over its enveloping algebra. Can you get a ...
- 61
1
vote
0
answers
120
views
Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$
Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...
- 257
2
votes
0
answers
293
views
Definition of 'Koszul Ring' (in BGS)
In the paper 'Koszul Duality Patterns in Representation Theory' by Beilinson et. al, they give the definition of a Koszul Ring:
A Koszul ring is a positively graded ring $A = \bigoplus_{j \geq 0} A_j$...
- 922
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 ...
- 52.9k
1
vote
0
answers
336
views
generators for derived category
Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...
CommunityBot
- 1
3
votes
0
answers
324
views
2 - Calabi Yau algebras and bimodule coherence
Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero.
In H. Minamoto "Ampleness of two-sided tilting complexes", ...
5
votes
1
answer
573
views
"as close to being semisimple as it can possibly be."
I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.
In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality Patterns......
- 12.1k
12
votes
1
answer
509
views
When is Ext*(M,N) finitely generated as a Ext*(M,M) module?
Let A be a finite dimensional algebra over a field k and M,N a finitely generated A-module.
Im searching for examples where the module $ Ext^{o} (M,N) $ is a finitely generated $ Ext^{o}(M,M) $ -...
- 3,686
3
votes
0
answers
318
views
When does Ext^2 vanish in a category of group representations.
Let $G$ be a linear algebraic group over field $k$ of characteristic zero. It is well known that the category of finite dimensional $k$--linear representations of $G$ is abelian, and that it is ...
- 4,015
18
votes
1
answer
1k
views
Homology in the $A_\infty$ World
This question is turning out to be a little long so let me start off with the headline. Given a differential graded algebra $A$, we can recover $A$ from its homology $HA$ if we know "the" $A_\infty$-...
- 13.2k
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
4
votes
1
answer
286
views
Why is the representation dimension of an Artin algebra never equal to 1?
Hi,
in 1971 M.Auslander showed that the representation dimension of $A$ is $\neq 1$ for every Artin algebra $A$.
Does anybody have a reference paper or book proving this? Is the proof easy and / or ...
- 6,292
1
vote
2
answers
291
views
Question about an exact sequence
Hello,
I would be glad, if someone could answer a question concerning the following:
http://www.math.uni-bonn.de/people/schroer/preprints/repdim.pdf
On page 5 they show (3)=>(1). The last step is ...
- 15.6k
0
votes
1
answer
223
views
Representation dimension of a special algebra
Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
I've come across a piece of information, which I don't understand, and wanted to ask, if I ...
- 6,292
4
votes
0
answers
157
views
Endomorphismrings of maximal submodules.
The question I am interested in answering is the following:
Suppose that for a pair of $d$-dimensional modules $M$ and $N$ over a $k$-algebra ($k$ a field) $R$ we have that $\dim_k \rm{Hom}_R(X,M)\...
- 236
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 ...
- 52.9k
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 ...
- 44.7k
8
votes
1
answer
800
views
How to recognize a finite dimensional algebra is Koszul or quadratic?
I have a family of finite dimensional algebras that are directed quasihereditary. I think they might be Koszul algebras and I am wondering what approaches there are to check Koszulness or even ...
- 38.6k
8
votes
1
answer
575
views
Generalizing Representation Theory of Finite Groups to Module Theory
My question is essentially this: which parts of the representation theory of finite groups are really just applications of module theory, and which are not? Here is an example of each case. Induction ...
- 38.6k
2
votes
3
answers
299
views
Can the 'linkages' between equivalent extensions of modules of an algebraic group be taken to have bounded length?
I have a conjecture concerning how "tightly" two equivalent $n$-fold extensions of modules over an algebraic group over a field might
be "linked". I suspect that the
question has been already ...
- 3,504
5
votes
2
answers
1k
views
Describing the kernel of the exponential map as a homology group
I am reading Deligne: Hodge III, and am puzzled by a certain statement in section 10. If anyone could give a reference or a hint for how to prove this, I would be grateful. Maybe it is obvious and I ...
- 23k
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
- 1,335
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. ...
CommunityBot
- 1
2
votes
0
answers
107
views
In which non-gorenstein algebras, are all maximal ideals Gorenstein injective modules?
In which non-gorenstein algebras, are all maximal ideals Gorenstein injective modules?
or Are the Gorenstein injective dimensions of all maximal ideals finite?
- 21
9
votes
1
answer
926
views
Koszul duality and modules over the Chevalley complex
Let $g$ be a Lie algebra over $\mathbb{C}$. Then the equivalence between the derived category of modules over $U(g)$ and the coderived category of co-modules over it's Chevalley complex $C_*(g)$ in ...
- 15.6k
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 ...
- 10.7k
8
votes
2
answers
2k
views
Question about Ext
I heard that $Ext(M,N)$ is naturally isomorphic to $Ext(M^*\otimes N,1)$ where 1 is the trivial representation and $M,N$ some representations of a group $G$.
Can anyone explain why?
Is there an ...
- 3,246
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
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. ...
- 47.3k
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 ...
- 24.1k