# Questions tagged [koszul-algebras]

Questions about Koszul algebras as defined by Priddy (1970) and generalizations.

Questions about Koszul algebras as defined by Priddy (1970) and generalizations.

20
questions

9
votes

1
answer

241
views

It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain ...

3
votes

2
answers

305
views

What is an example of a Frobenius algebra that is not Koszul? Are there reasonable requirements for a Frobenius to be Koszul?

6
votes

1
answer

103
views

Let $A,B$ be a pair of quasi-hereditary algebras and assume that $A$ and $B$ are both standard Koszul. Further assume that the graded decomposition matrices of $A$ and $B$ coincide (that is, the ...

4
votes

0
answers

168
views

Let $A$ be a finite dimensional local Frobenius algebra that is Koszul.
Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...

7
votes

0
answers

164
views

The short question is the following: If a positively graded Lie algebra $\mathfrak g$ over a field $F$ is Koszul, is the center of $\mathfrak g$ concentrated in degree $1$?
Let us be more precise. A ...

4
votes

1
answer

174
views

Let $A = k[x_1 , \dots , x_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x_1 , \dots , x_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset ...

1
vote

0
answers

86
views

Let $A$ be a finite dimensional Koszul algebra of finite global dimension.
Question: When is $A$ derived equivalent to its Koszul dual algebra?
I wonder whether there is an exact condition to ...

4
votes

0
answers

106
views

The Beilinson algebra $A=A_n$ is a finite dimensional quiver algebra that is derived equivalent to the category of coherent sheaves of $\mathcal{P}^n$. See for example https://link.springer.com/...

1
vote

1
answer

96
views

Let $A$ be a non-negatively graded algebra such that $A_0 = k$. We say that $A$ is Koszul if $k$ has a projective resolution by projective modules such that the i-th piece is generated in degree $i$. ...

2
votes

1
answer

473
views

I am not able to find any literature which studies complete intersection subvarieties of a projective variety, all good references consider CI in projective space.
My guess is, a subvariety X of ...

5
votes

0
answers

197
views

A Quadratic-Linear-Constant (QLC) algebra $U$ is an algebra which can be written as $T(V)/P$ where
$T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ is the tensor algebra
and $P \subseteq k \oplus ...

3
votes

1
answer

434
views

I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ...

2
votes

1
answer

96
views

It is well-known that Sklyanin algebras are Koszul, but, is it known an explicit description of the dual algebra Ext_A(k,k)? (I mean in terms of generators and relations)

6
votes

1
answer

414
views

I'm trying to understand the non-commutative Koszul complex, as can be found in Anick's nice paper "Non-Commutative Graded Algebras and Their Hilbert Series", J. of Algebra 78, (1982) and I'm stuck at ...

3
votes

1
answer

209
views

Let $\pi$ denote a saturated set of weights. Let $S_q(\pi)$ denote the associated generalised $q$-Schur algebra. I was wondering if the following claim is true:
Claim: The algebra $S_q(\pi)$ is ...

6
votes

1
answer

278
views

Let $V$ be a vector space over $k$ of dimension $m$. (I'm only interested in the case $k=\mathbb{Q}$.) Let $R:=\Lambda^*V$ be the exterior algebra. It carries the structure of a supercommutative ring: ...

2
votes

1
answer

177
views

Do we know the maximal class of Koszul algebras for which any deformation is Koszul?

14
votes

2
answers

637
views

Let $n \geq 3$. The ring $H^\bullet(\overline{M}_{0,n},\mathbf Q)$ was determined by Sean Keel. It is generated by the cohomology classes of boundary divisors $D_{A,B}$ corresponding to partitions $A \...

21
votes

2
answers

2k
views

Koszul duality
Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can ...

14
votes

2
answers

1k
views

$\DeclareMathOperator\Cl{Cl}$Update Feb 3 '12: now with a question 2 which is much more elementary (and should be well-known!).
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:...