The point variety of quantum polynomial rings

Abstract

We show that the reduced point variety of a quantum polynomial algebra is the union of specific linear subspaces in $\mathbb{P}^n$, we describe its irreducible components and give a combinatorial description of the possible configurations in small dimensions.Comment: 10 pages, an extended version of arxiv.org/abs/1506.0651

Combinatorial classification of $(\pm 1)$-skew projective spaces

“…Thanks to the following result due to Vitoria [26] and independently Belmans, De Laet and Le Bruyn [4], we can compute the point scheme of S α explicitly. ).…”

On Knörrer Periodicity for Quadric Hypersurfaces in Skew Projective Spaces

“…The following is the classification of the point schemes of graded skew polynomial algebras in 4 variables. Proposition 2.4 ([9, Corollary 5.1], [2,Section 4.2]). Let S = k x 1 , x 2 , x 3 , x 4 /(x i x j − α ij x j x i ) be a graded skew polynomial algebra in 4 variables.…”

“…. , x n ] be a graded polynomial algebra generated in degree 1, and f = x 2 1 + x 2 2 + · · · + x 2 n . Let CM Z (S/(f )) denote the stable category of graded maximal Cohen-Macaulay module over S/(f ).…”

“…. , x n /(x i x j − α ij x j x i ) 1≤i,j≤n where α ii = 1 for every 1 ≤ i ≤ n, α ij α ji = 1 for every 1 ≤ i, j ≤ n, and deg x i = 1 for every 1 ≤ i ≤ n. (2) We say that S is a graded (±1)-skew polynomial algebra if S = k x 1 , . .…”

“…. , x n /(x i x j − ε ij x j x i ) be a graded (±1)-skew polynomial algebra so that f = x 2 1 + x 2 2 + · · · + x 2 n ∈ S is a homogeneous regular central element. Let A be the graded quotient algebra S/(f ).…”