Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities

Iyama, O. and Wemyss, M. (2018) Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities. Journal für die Reine und Angewandte Mathematik (Crelles Journal), 2018(738), pp. 149-202. (doi: 10.1515/crelle-2015-0031)

Abstract

In this paper we define and study triangulated categories in which the Homspaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If C is such a category, we say that C is Calabi–Yau with dim C ≤ 1. We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories CM R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of CM R and both partial crepant resolutions and Q-factorial terminalizations of Spec R, and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cAn singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of CM R, we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [7, 10], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k D C we obtain many autoequivalences of the derived category of the Q-factorial terminalizations of Spec R.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Wemyss, Professor Michael
• Authors: Iyama, O., and Wemyss, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal für die Reine und Angewandte Mathematik (Crelles Journal)
• Publisher: De Gruyter
• ISSN: 0075-4102
• ISSN (Online): 0075-4102
• Published Online: 17 September 2015
• Copyright Holders: Copyright © 2015 De Gruyter
• First Published: First published in Journal für die Reine und Angewandte Mathematik (Crelles Journal) 2018(738): 149-202
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher