1
vote
0answers
99 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} ...
7
votes
1answer
148 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 ...
0
votes
0answers
242 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 ...
3
votes
0answers
227 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", ...
2
votes
0answers
114 views

Criteria for a finite-dimensional $k$-Algebra to be basic and elementary

I have the following question: Suppose, I have a finite dimensional $k$-Algebra $A$ over an arbitrary field $k$ and a finite dimensional module $M$ that is a generator-cogenerator of mod-$A$. I'm ...
1
vote
0answers
128 views

Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...
4
votes
1answer
397 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 ...
11
votes
1answer
268 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) $ ...
1
vote
0answers
122 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 ...
14
votes
1answer
667 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" ...
4
votes
1answer
174 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 ...
2
votes
1answer
155 views

Proving indecomposability of special modules

Hi, I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf On page 6 there are the definitions (in a diagrammatical way) of some $A_n$ modules, whereupon ...
2
votes
2answers
239 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 ...
0
votes
1answer
177 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 ...
7
votes
0answers
283 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 ...
4
votes
0answers
139 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 ...
4
votes
2answers
556 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 ...
5
votes
1answer
385 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 ...
6
votes
1answer
367 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 ...
2
votes
3answers
277 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 ...
7
votes
1answer
606 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 ...
3
votes
2answers
405 views

An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
3
votes
2answers
503 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 ...
10
votes
1answer
482 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. ...
1
vote
0answers
95 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?
7
votes
1answer
589 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 ...
4
votes
0answers
363 views

Strange boundary-like map on tensor algebra: what is its kernel?

Let $k$ be a commutative ring and $L$ a $k$-module. The tensor algebra $\otimes L$ is $\mathbb{Z}$-graded and $\mathbb{Z}_2$-graded (an element of $L^{\otimes n}$ has degree $n$ and ...
5
votes
2answers
910 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 ...
7
votes
4answers
1k 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 ...
6
votes
2answers
867 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 ...
10
votes
0answers
730 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 ...
5
votes
1answer
1k 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. ...
7
votes
3answers
2k 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 ...