A representation theoretic study of noncommutative symmetric algebras

TL;DR

This paper explores noncommutative symmetric algebras using representation theory, establishing coherence, derived equivalences, and structural properties that generalize classical results like Beilinson's theorem.

Contribution

It demonstrates that Van den Bergh's noncommutative symmetric algebra is coherent and its proj category is derived equivalent to bimodule species, extending known derived equivalences.

Findings

Proves coherence of noncommutative symmetric algebra

Shows derived equivalence to bimodule species

Establishes hereditary property and sheaf structure theorem

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 $\mathbb{P}^1$.