The Rising Sea

Update

December 29, 2009 on 10:35 am | In Uncategorized | Comments Off

Unfortunately this website was compromised in the recent wave of attacks on Wordpress blogs, but hopefully everything is now OK. I’d like to thank Graham Leuschke for sounding the alarm.

In other news, I’ve been thinking lately about duality in Orlov’s singularity category (or, for the commutative algebraists, the stable category of Cohen-Macaulay modules) and the product of these musings is now available on the arXiv. The paper originated in an attempt to understand the geometric version of a paper of Henning Krause and Jue Le which uses compactly generated triangulated categories to study Auslander-Reiten translation. This made me curious about trace maps for the Serre functor in singularity categories, and it turns out that using the machinery of duality in compactly generated triangulated categories it was possible to recover the rather beautiful formula of the string theorists Anton Kapustin and Yi Li in the case of isolated hypersurface singularities.

For reasons of length the connection to the work of Krause and Le is almost completely suppressed in the paper, although this point of view is what led me to the main arguments. I still hope to write a sequel treating the global case using homotopy categories of sheaves, as for example developed in my thesis and joint work with Shokrollah Salarian, in which the “compactly generated” point of view is made explicit.