# Questions tagged [koszul-algebras]

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

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### Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul

In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...

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### Koszul algebras among finite dimensional commutative algebras

Given a local commutative artinian algebra $A$ of the form $K[x_1,...,x_n]/I$ with quadratic ideal $I$ and $K$ a field.
Question 1: Is there a computer algebra system that can check whether such an ...

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### Filtrations and Koszul algebras

I was looking at this question and asked my self the following:
Let $A$ be graded algebra, which is also an $\mathbb{N}_0$-filtered algebra. If its associated
graded algebra $\mathrm{gr}(A)$ is ...

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### When is a group algebra Koszul?

Let $KG$ be a group algebra of a finite group $G$ such that the characteristic of $K$ divides the group order.
Question: When is a block of a group algebra (or even the whole group aglebra) a Koszul ...

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### Quotients of Koszul algebras

Let $A$ be a noncommutative Koszul algebra (see here for a definition of Koszul) and let $c \in A$ be a central element. Will the quotient of $A$ by the ideal generated by $c$ again be Koszul. If not ...

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### Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request

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 ...

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### What is an example of a Frobenius algebra that is not Koszul?

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

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### Are standard Koszul algebras with the same Kazhdan-Lusztig polynomials Morita equivalent?

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 ...

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### Quillen–Suslin theorem in a more general context

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 ...

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### On the center of Koszul Lie algebras

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 ...

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### In a commutative Koszul algebra, does every ideal generated by a subset of variables have linear resolution?

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 ...

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### When is a Koszul algebra derived equivalent to its dual

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 ...

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### Perfect modules for the Beilinson algebra

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/...

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### Augmented algebras over semisimple ring

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 $i$. ...

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### Complete intersection subvariety of projective variety

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 ...

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### Koszul duality between QLS algebras and cdg algebras

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 ...

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### What bigrading is used in this spectral sequence?

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 ...

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### Description of Koszul dual of Sklyanin algebras

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)

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### Non-commutative regular sequences and non-commutative Koszul complex

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 ...

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### Is Koszulity equivalent to the Lusztig character formula holding?

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 ...

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### Is the (super-)symmetric power of the exterior algebra free?

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: ...

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### Koszul algebras deformations

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

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### Koszulness of the cohomology ring of moduli of stable genus zero curves

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 \...

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### Koszul duality between Weyl and Clifford algebras?

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 ...

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### Clifford PBW theorem for quadratic form

$\DeclareMathOperator\Cl{Cl}$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:...