## Duke Mathematical Journal

### Three-dimensional flops and noncommutative rings

Michel Van den Bergh

#### 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

https://projecteuclid.org/euclid.dmj/1082665284

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

Mathematical Reviews number (MathSciNet)
MR2057015

Zentralblatt MATH identifier
1074.14013

#### Citation

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

#### References

• M. Artin and J.-L. Verdier, Reflexive modules over rational double points, Math. Ann. 270 (1985), 79–82.
• M. Auslander and O. Goldman, Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24.
• A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53, no. 6 (1989), 1183–1205., 1337; English translation in Math. USSR-Izv. 35, no. 3 (1990), 519–541.
• A. Bondal and D. Orlov, “Derived categories of coherent sheaves” in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed., Beijing, 2002, 47–56.
• ––––, Semiorthogonal decomposition for algebraic varieties, preprint.
• A. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Moscow Math. J. 3 (2003), 1–36.
• T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613–632.
• T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), 535–554.
• E. Brieskorn, Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann. 178 (1968), 255–270.
• K. A. Brown and C. R. Hajarnavis, Homologically homogeneous rings, Trans. Amer. Math. Soc. 281 (1984), 197–208.
• J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Differential Geom. 61 (2002), 227–261.
• H. Clemens, J. Kollár, and S. Mori, Higher-Dimensional Complex Geometry, Astérisque 166, Soc. Math. France, Montrouge, 1988.
• R. M. Fossum, The Divisor Class Group of a Krull Domain, Ergeb. Math. Grenzgeb. (2) 74, Springer, New York, 1973.
• A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I, Inst. Hautes Études Sci. Publ. Math. 11 (1961).
• D. Happel, I. Reiten, and S. O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575.
• R. Hartshorne, Residues and Duality: Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963/64, appendix by P. Deligne, Lecture Notes in Math. 20, Springer, Berlin, 1966.
• ––––, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
• M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), 565–576.
• Y. Kawamata, “Francia's flip and derived categories” in Algebraic Geometry: A Volume in Memory of Paolo Francia, de Gruyter, Berlin, 2002, 197–215.
• D. S. Keeler, Ample filters of invertible sheaves, J. Algebra 259 (2003), 243–283.
• B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), 63–102.
• J. Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36.
• J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
• A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.
• M. Reid, “Young person's guide to canonical singularities” in Algebraic Geometry (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Part 1, Amer. Math. Soc., Providence, 1987, 345–414.
• I. Reiner, Maximal Orders, London Math. Soc. Monogr. 5, Academic Press, London, 1975.
• M. Van den Bergh, Non-commutative crepant resolutions, to appear in the Proceedings of the Abel Bicentennial Conference, preprint.