Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T05:19:27.800Z Has data issue: false hasContentIssue false

NONNOETHERIAN HOMOTOPY DIMER ALGEBRAS AND NONCOMMUTATIVE CREPANT RESOLUTIONS

Published online by Cambridge University Press:  30 October 2017

CHARLIE BEIL*
Affiliation:
Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, NAWI Graz, Heinrichstrasse 36, A-8010 Graz, Austria e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Noetherian dimer algebras form a prominent class of examples of noncommutative crepant resolutions (NCCRs). However, dimer algebras that are noetherian are quite rare, and we consider the question: how close are nonnoetherian homotopy dimer algebras to being NCCRs? To address this question, we introduce a generalization of NCCRs to nonnoetherian tiled matrix rings. We show that if a noetherian dimer algebra is obtained from a nonnoetherian homotopy dimer algebra A by contracting each arrow whose head has indegree 1, then A is a noncommutative desingularization of its nonnoetherian centre. Furthermore, if any two arrows whose tails have indegree 1 are coprime, then A is a nonnoetherian NCCR.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Auslander, M. and Buchsbaum, D. A., Homological dimension in noetherian rings, Proc. Natl. Acad. Sci. USA 42 (1956).Google Scholar
2. Auslander, M. and , D. A., Homological dimension in local rings, Trans. Am. Math. Soc. 85 (1957), 390405.CrossRefGoogle Scholar
3. Baur, K., King, A. and Marsh, R., Dimer models and cluster categories of Grassmannians, Proc. Lond. Math. Soc. (2016).Google Scholar
4. Beil, C., Cyclic contractions of dimer algebras always exist, arXiv:1703.04450.Google Scholar
5. Beil, C., Morita equivalences and Azumaya loci from Higgsing dimer algebras, J. Algebra 453 (2016), 429455.Google Scholar
6. Beil, C., Nonnoetherian geometry, J. Algebra Appl. 15 (2016).Google Scholar
7. Beil, C., Homotopy dimer algebras and cyclic contractions, in preparation.Google Scholar
8. Beil, C., Noetherian criteria for dimer algebras, in preparation.Google Scholar
9. Beil, C., On the central geometry of nonnoetherian dimer algebras, in preparation.Google Scholar
10. Beil, C., The central nilradical of nonnoetherian dimer algebras, in preparation.Google Scholar
11. Beil, C., On the noncommutative geometry of square superpotential algebras, J. Algebra 371 (2012), 207249.Google Scholar
12. Benvenuti, S., Franco, S., Hanany, A., Martelli, D. and Sparks, J., An infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals, J. High Energy Phys. 6 (2005), 064.Google Scholar
13. Berenstein, D. and Douglas, M., Seiberg duality for quiver gauge theories arXiv:hep-th/0207027.Google Scholar
14. Bocklandt, R., Consistency conditions for dimer models, Glasgow Math. J. 54 (2012), 429447.Google Scholar
15. Broomhead, N., Dimer models and Calabi-Yau algebras, Memoirs AMS 215 (2012), 1011.Google Scholar
16. Brown, K. and Hajarnavis, C., Homologically homogeneous rings, Trans. Am. Math. Soc. 281 (1984), 197208.Google Scholar
17. Davison, B., Consistency conditions for Brane tilings, J. Algebra 338 (2011), 123.Google Scholar
18. Clark, P. L., Commutative Algebra. Available at: http://math.uga.edu/~pete/MATH8020.html, submitted.Google Scholar
19. Gulotta, D., Properly ordered dimers, R-charges, and an efficient inverse algorithm, J. High Energy Phys. 10 (2008).Google Scholar
20. Ishii, A. and Ueda, K., Dimer models and the special McKay correspondence, Geom. Topol. 19 (2015), 34053466.Google Scholar
21. Rotman, J., An introduction to homological algebra (Springer, 2009).Google Scholar
22. Serre, J.-P., Sur la dimension homologique des anneaux et des modules noethériens, in Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955 (Science Council of Japan, Tokyo, 1956), 175189 (in French).Google Scholar
23. Van den Bergh, M., Non-commutative crepant resolutions, in The legacy of Niels Henrik Abel (Springer, Berlin, 2004), 749770.Google Scholar