Duke Mathematical Journal

Three-dimensional flops and noncommutative rings

Michel Van den Bergh

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Abstract

For $Y,Y^+$ three-dimensional smooth varieties related by a flop, Bondal and Orlov conjectured that the derived categories $D^b({\rm coh}(Y))$ and $D^b({\rm coh}(Y^+))$ are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using noncommutative rings. The new proof also covers some mild singular and higher-dimensional situations (including those occuring in the recent paper by Chen [11]).

Article information

Source
Duke Math. J. Volume 122, Number 3 (2004), 423-455.

Dates
First available in Project Euclid: 22 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1082665284

Digital Object Identifier
doi:10.1215/S0012-7094-04-12231-6

Mathematical Reviews number (MathSciNet)
MR2057015

Zentralblatt MATH identifier
02133073

Subjects
Primary: 18E30: Derived categories, triangulated categories 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14A22: Noncommutative algebraic geometry [See also 16S38]

Citation

Van den Bergh, Michel. Three-dimensional flops and noncommutative rings. Duke Mathematical Journal 122 (2004), no. 3, 423--455. doi:10.1215/S0012-7094-04-12231-6. http://projecteuclid.org/euclid.dmj/1082665284.


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