Abstract
Building upon ideas of Eisenbud, Buchweitz, Positselski, and others, we introduce the notion of a factorization category. We then develop some essential tools for working with factorization categories, including constructions of resolutions of factorizations from resolutions of their components and derived functors. Using these resolutions, we lift fully-faithfulness and equivalence statements from derived categories of Abelian categories to derived categories of factorizations. Some immediate geometric consequences include a realization of the derived category of a projective hypersurface as matrix factorizations over a noncommutative algebra and recover of a theorem of Baranovsky and Pecharich.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 195-249 |
| Number of pages | 55 |
| Journal | Advances in Mathematics |
| Volume | 295 |
| DOIs | |
| State | Published - Jun 4 2016 |
| Externally published | Yes |
Bibliographical note
Funding Information:The authors would like to thank Leonid Positselski for his insightful ideas. They would also like to thank the referee for his very careful reading (which corrected a nontrivial amount of errors), valuable insight and perspective. Funding for this research was provided through the following grants: NSF DMS 0636606 RTG , NSF DMS 0838210 RTG , NSF DMS 0854977 FRG , NSF DMS 0600800 , NSF DMS 0652633 FRG , NSF DMS 0854977 , NSF DMS 0901330 , NSF DMS 660778 , NSF DMS 1501813 , NSF DMS 66923E , FWF P 24572 N25 , FWF P20778 , Simons Collaboration grant, and by an ERC grant.
Publisher Copyright:
© 2016.
Keywords
- Derived categories
- Homological algebra
- Matrix factorizations