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A Representation Theoretic Study of Non-Commutative Symmetric Algebras

Part of: Algebraic geometry: Foundations Rings and algebras arising under various constructions Homological methods

Published online by Cambridge University Press:  14 February 2019

D. Chan  and
A. Nyman
D. Chan
Affiliation:
University of New South Wales, Sydney, NSW, Australia (danielc@unsw.edu.au)
A. Nyman
Affiliation:
Western Washington University, Bellingham, WA, USA (adam.nyman@wwu.edu)
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Abstract

We study Van den Bergh's non-commutative symmetric algebra 𝕊nc(M) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show that 𝕊nc(M) is coherent, and its proj category ℙnc(M) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ℙnc(M) is hereditary and there is a structure theorem for sheaves on ℙnc(M) analogous to that for ℙ1.

Keywords

non-commutative algebraic geometrynon-commutative curvenon-commutative symmetric

MSC classification

Primary: 14A22: Noncommutative algebraic geometry 16S38: Rings arising from non-commutative algebraic geometry
Secondary: 16E35: Derived categories
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 875 - 887
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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