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 $\mathbb{ P}^1$.
Categories: math.AG math.RA