A representation theoretic study of noncommutative symmetric algebras

Abstract

We study Van den Bergh's noncommutative symmetric algebra ${\mathbb{S}}^{nc}(M)$ (over division rings) via Minamoto's theory of Fano algebras. In particular, we show ${\mathbb{S}}^{nc}(M)$ is coherent, and its proj category ${\mathbb{P}}^{nc}(M)$ is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of \cite{minamoto}, which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ${\mathbb{P}}^{nc}(M)$ is hereditary and there is a structure theorem for sheaves on ${\mathbb{P}}^{nc}(M)$ analogous to that for ${\mathbb{P}}^1$.

12 pages. Minor improvements for clarity. Hypothesis added to Theorem 5.2. Main results unchanged