# zbMATH — the first resource for mathematics

Three-dimensional flops and noncommutative rings. (English) Zbl 1074.14013
This paper gives a new proof, based on noncommutative rings, of T. Bridgeland’s theorem [Invent. Math. 147, No. 3, 613–632 (2002; Zbl 1085.14017)] that says that two three dimensional smooth varieties $$Y$$, $$Y^+$$ related by a flopping transformation $$Y\to X\leftarrow Y^+$$ have equivalent bounded derived categories of coherent sheaves. There are extensions of this result for normal varieties with isolated smooth singularities, due to J.-C. Chen [J. Differ. Geom. 61, No. 2, 227–261 (2002; Zbl 1090.14003)] and for some non-Gorenstein singularities, due to Y. Kawamata [in: Algebraic geometry, de Gruyter, Berlin, 197–215 (2002; Zbl 1092.14023)]. The proof in this paper is really nice, and is based in the construction of vector bundles $$\mathcal P$$ and $$Y$$ and $$\mathcal Q^+$$ on $$Y^+$$ such that the direct image on $$X$$ of the endomorphism sheaves algebras of $$\mathcal P$$ and $$\mathcal Q^+$$ are isomorphic. If $$\mathcal A$$ is this noncommutative algebra, then the author proves that the bounded derived categories of coherent sheaves on $$Y$$ and on $$Y^+$$ are both equivalent to the bounded derived category of right $$\mathcal A$$-modules, being thus equivalent. The bundles $$\mathcal P$$ and $$\mathcal Q^+$$ are related to certain categories of perverse sheaves associated with the flop which appear also in the original Bridgeland proof. The author also describes the projective generators of these categories of perverse sheaves. Actually, some more general results are proven, among them, some higher dimensional generalizations (also considered by Chen).

##### MSC:
 14E05 Rational and birational maps 14E30 Minimal model program (Mori theory, extremal rays) 18E30 Derived categories, triangulated categories (MSC2010) 14A22 Noncommutative algebraic geometry
##### Keywords:
flops; flips; derived category
Full Text:
##### References:
 [1] M. Artin and J.-L. Verdier, Reflexive modules over rational double points , Math. Ann. 270 (1985), 79-82. · Zbl 0553.14001 · doi:10.1007/BF01455531 · eudml:182944 [2] M. Auslander and O. Goldman, Maximal orders , Trans. Amer. Math. Soc. 97 (1960), 1-24. · Zbl 0117.02506 · doi:10.2307/1993361 [3] 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. · Zbl 0703.14011 · doi:10.1070/IM1990v035n03ABEH000716 [4] 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. · Zbl 0996.18007 [5] —-, Semiorthogonal decomposition for algebraic varieties , · arxiv.org [6] A. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry , Moscow Math. J. 3 (2003), 1-36. · Zbl 1135.18302 [7] T. Bridgeland, Flops and derived categories , Invent. Math. 147 (2002), 613-632. · Zbl 1085.14017 · doi:10.1007/s002220100185 [8] T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories , J. Amer. Math. Soc. 14 (2001), 535-554. JSTOR: · Zbl 0966.14028 · doi:10.1090/S0894-0347-01-00368-X · links.jstor.org [9] E. Brieskorn, Die Auflösung der rationalen Singularitäten holomorpher Abbildungen , Math. Ann. 178 (1968), 255-270. · Zbl 0159.37703 · doi:10.1007/BF01352140 · eudml:161748 [10] K. A. Brown and C. R. Hajarnavis, Homologically homogeneous rings , Trans. Amer. Math. Soc. 281 (1984), 197-208. JSTOR: · Zbl 0531.16019 · doi:10.2307/1999529 · links.jstor.org [11] J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities , J. Differential Geom. 61 (2002), 227-261. · Zbl 1090.14003 [12] H. Clemens, J. Kollár, and S. Mori, Higher-Dimensional Complex Geometry , Astérisque 166 , Soc. Math. France, Montrouge, 1988. [13] R. M. Fossum, The Divisor Class Group of a Krull Domain , Ergeb. Math. Grenzgeb. (2) 74 , Springer, New York, 1973. · Zbl 0256.13001 [14] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I , Inst. Hautes Études Sci. Publ. Math. 11 (1961). [15] D. Happel, I. Reiten, and S. O. Smalø, Tilting in abelian categories and quasitilted algebras , Mem. Amer. Math. Soc. 120 (1996), no. 575. · Zbl 0849.16011 [16] 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. [17] —-, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. [18] M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras , Math. Ann. 316 (2000), 565-576. · Zbl 0997.14001 · doi:10.1007/s002080050344 [19] Y. Kawamata, “Francia’s flip and derived categories” in Algebraic Geometry: A Volume in Memory of Paolo Francia , de Gruyter, Berlin, 2002, 197-215. · Zbl 1092.14023 [20] D. S. Keeler, Ample filters of invertible sheaves , J. Algebra 259 (2003), 243-283. · Zbl 1082.14004 · doi:10.1016/S0021-8693(02)00557-4 [21] B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), 63-102. · Zbl 0799.18007 · numdam:ASENS_1994_4_27_1_63_0 · eudml:82359 [22] J. Kollár, Flops , Nagoya Math. J. 113 (1989), 15-36. [23] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties , Cambridge Tracts in Math. 134 , Cambridge Univ. Press, Cambridge, 1998. [24] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , J. Amer. Math. Soc. 9 (1996), 205-236. JSTOR: · Zbl 0864.14008 · doi:10.1090/S0894-0347-96-00174-9 · links.jstor.org [25] 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. · Zbl 0634.14003 [26] I. Reiner, Maximal Orders , London Math. Soc. Monogr. 5 , Academic Press, London, 1975. [27] M. Van den Bergh, Non-commutative crepant resolutions , to appear in the Proceedings of the Abel Bicentennial Conference, · Zbl 1082.14005 · arxiv.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.