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

Abstract

The noncommutative projective scheme $\mathsf{Pro}{\mathsf{j}}_{\mathsf{nc}}S$ of a $(\pm 1)$-skew polynomial algebra $S$ in $n$ variables is considered to be a $(\pm 1)$-skew projective space of dimension $n-1$. In this paper, using combinatorial methods, we give a classification theorem for $(\pm 1)$-skew projective spaces. Specifically, among other equivalences, we prove that $(\pm 1)$-skew projective spaces $\mathsf{Pro}{\mathsf{j}}_{\mathsf{nc}}S$ and $\mathsf{Pro}{\mathsf{j}}_{\mathsf{nc}}{S}^{\prime }$ are isomorphic if and only if certain graphs associated to $S$ and $S^{\prime }$ are switching (or mutation) equivalent. We also discuss invariants of $(\pm 1)$-skew projective spaces from a combinatorial point of view.

14 pages, v2: minor modifications, v3: removed an example

References in corpus (1)

• Noncommutative Knörrer's periodicity theorem and noncommutative quadric hypersurfaces