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 $\mathbb{P}^1$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.05868
- arXiv:
- arXiv:1710.05868
- Bibcode:
- 2017arXiv171005868C
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Rings and Algebras;
- 14A22;
- 16S38
- E-Print:
- 12 pages. Minor improvements for clarity. Hypothesis added to Theorem 5.2. Main results unchanged