Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T18:21:34.515Z Has data issue: false hasContentIssue false
  • English
  • Français

A Representation Theoretic Study of Non-Commutative Symmetric Algebras

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

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 ([email protected])
A. Nyman
Affiliation:
Western Washington University, Bellingham, WA, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Auslander, M., Platzeck, M. I. and Reiten, I., Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1–46.Google Scholar
2.Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, Volume 36 (Cambridge University Press, Cambridge, 1997).Google Scholar
3.Butler, M. C. R. and King, A. D., Minimal resolutions of algebras, J. Algebra 212 (1999), 323–362.Google Scholar
4.Chan, D. and Nyman, A., Species and noncommutative ℙ1's over non-algebraic bimodules, J. Algebra 460 (2016), 143–180.Google Scholar
5.Dlab, V., An introduction to diagrammatic methods in representation theory. Lecture notes written by Richard Dipper. Lecture Notes in Mathematics at the University of Essen, Volume 7 (Universität Essen, Fachbereich Mathematik, Essen, 1981).Google Scholar
6.Dlab, V. and Ringel, C., The preprojective algebra of a modulated graph, Representation Theory II (Proceedings of the 2nd International Conference, Carleton University, Ottawa, Ontario, 1979), Lecture Notes in Mathematics, Volume 832, pp. 216–231 (Springer-Verlag, 1980).Google Scholar
7.Kontsevich, M. and Rosenberg, A., Noncommutative smooth spaces, in The Gelfand Seminars 1996–1999, Gelfand Mathematical Seminars, pp. 85–108 (Birkhaüser Boston, Boston MA, 2000).Google Scholar
8.Minamoto, H., A noncommutative version of Beilinson's theorem, J. Algebra 320 (2008), 238–252.Google Scholar
9.Minamoto, H., Ampleness of two-sided tilting complexes, Int. Math. Res. Not. (2012), 67–101.Google Scholar
10.Nyman, A., An abstract characterization of noncommutative ℙ1-bundles, J. Noncommut. Geom., to appear.Google Scholar
11.Nyman, A., Serre duality for non-commutative ℙ1-bundles, Trans. Amer. Math. Soc. 357 (2005), 1349–1416.Google Scholar
12.Patrick, D., Noncommutative symmetric algebras of two-sided vector spaces, J. Algebra 233 (2000), 16–36.Google Scholar
13.Piontkovski, D., Coherent algebras and noncommutative projective lines, J. Algebra 319 (2008), 3280–3290.Google Scholar
14.Polishchuk, A., Non-commutative proj and coherent algebras, Math. Res. Lett. 12 (2005), 63–74.Google Scholar
15.Ringel, C., Representations of K-species and bimodules, J. Algebra 41 (1976), 269–302.Google Scholar
16.Van den Bergh, M., Non-commutative ℙ1-bundles over commutative schemes, Trans. Amer. Math. Soc. 364 (2012), 6279–6313.Google Scholar