# Questions tagged [koszul-algebras]

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

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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 ...
131 views

### 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 ...
157 views

### 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 ...
92 views

### 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 $i$. ...
364 views

### 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 ...
188 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 $P \subseteq k \oplus ... 1answer 415 views ### What bigrading is used in this spectral sequence? I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ... 1answer 94 views ### Description of Koszul dual of Sklyanin algebras It is well-known that Sklyanin algebras are Koszul, but, is it known an explicit description of the dual algebra Ext_A(k,k)? (I mean in terms of generators and relations) 1answer 360 views ### Non-commutative regular sequences and non-commutative Koszul complex I'm trying to understand the non-commutative Koszul complex, as can be found in Anick's nice paper "Non-Commutative Graded Algebras and Their Hilbert Series", J. of Algebra 78, (1982) and I'm stuck at ... 1answer 200 views ### Is Koszulity equivalent to the Lusztig character formula holding? Let$\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)$is ... 1answer 261 views ### Is the (super-)symmetric power of the exterior algebra free? Let$V$be a vector space over$k$of dimension$m$. (I'm only interested in the case$k=\mathbb{Q}$.) Let$R:=\Lambda^*V$be the exterior algebra. It carries the structure of a supercommutative ring: ... 1answer 175 views ### Koszul algebras deformations Do we know the maximal class of Koszul algebras for which any deformation is Koszul? 2answers 598 views ### Koszulness of the cohomology ring of moduli of stable genus zero curves Let$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 \...
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 ...
$\DeclareMathOperator\Cl{Cl}$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:...