Singular derived categories of -factorial terminalizations and maximal modification algebras

Abstract Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D sg ( X ) and its idempotent completion D sg ( X ) ¯ , we give necessary and sufficient categorical conditions for X to be Q -factorial and complete locally Q -factorial respectively. We then relate this information to maximal modification algebras (= MMAs), introduced in [20] , by showing that if an algebra Λ is derived equivalent to X as above, then X is Q -factorial if and only if Λ is an MMA. Thus all rings derived equivalent to Q -factorial terminalizations in dimension three are MMAs. As an application, we extend some of the algebraic results in [6] and [14] using geometric arguments.

[1]  Xiao-Wu Chen.  Unifying two results of Orlov on singularity categories , 2010 .

[2]  Osamu Iyama,et al.  Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities , 2013, 1304.5259.

[3]  Hui-wen Lin.  On crepant resolution of some hypersurface singularities and a criterion for UFD , 2002 .

[4]  R. Hartshorne.  Residues And Duality , 1966 .

[5]  M. Bergh,et al.  On two examples by Iyama and Yoshino , 2008, Compositio Mathematica.

[6]  Ragnar-Olaf Buchweitz,et al.  Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Rings , 1986 .

[7]  H. Dao.  Decent intersection and Tor-rigidity for modules over local hypersurfaces , 2006, math/0611568.

[8]  D. Orlov,et al.  Formal completions and idempotent completions of triangulated categories of singularities , 2009, 0901.1859.

[9]  H. Dao.  Remarks on non-commutative crepant resolutions of complete intersections , 2009, 0911.4541.

[10]  Victor Ginzburg.  Calabi-Yau algebras , 2006 .

[11]  Michel Van den Bergh.  Three-dimensional flops and noncommutative rings , 2002 .

[12]  Jiun C. Chen.  Flops and Equivalences of Derived Categories for Threefolds with Only Terminal Gorenstein Singularities , 2002 .

[13]  M. Raynaud,et al.  Critères de platitude et de projectivité , 1971 .

[14]  Qing Liu,et al.  Algebraic Geometry and Arithmetic Curves , 2002 .

[15]  R. Rouquier,et al.  Picard Groups for Derived Module Categories , 2003 .

[16]  A. Neeman.  Chapter 1. Definition and elementary properties of triangulated categories , 2001 .

[17]  Miles Reid,et al.  Mukai implies McKay: the McKay correspondence as an equivalence of derived categories , 1999, math/9908027.

[18]  Dmitri Orlov.  Triangulated categories of singularities and D-branes in Landau-Ginzburg models , 2003 .

[19]  Miles Reid,et al.  Young person''s guide to canonical singularities , 1985 .

[20]  Amnon Neeman,et al.  The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , 1996 .

[21]  H. Dao.  Decency and rigidity over hypersurfaces , 2006 .

[22]  J. Kollár,et al.  Birational Geometry of Algebraic Varieties: Index , 1998 .

[23]  C. U. Jensen.  On the vanishing of lim←(i) , 1970 .

[24]  I. Reiten,et al.  Cluster tilting for one-dimensional hypersurface singularities , 2007, 0704.1249.

[25]  M. Reid.  Minimal Models of Canonical 3-Folds , 1983 .

[26]  Osamu Iyama,et al.  Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras , 2006 .

[27]  H. Dao,et al.  Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free , 2010, Compositio Mathematica.

[28]  M. Wemyss.  The $${{\rm GL}(2,\mathbb{C})}$$ McKay correspondence , 2011 .

[29]  P. Roberts.  Multiplicities and Chern Classes in Local Algebra , 1998 .

[30]  Tom Bridgeland.  Flops and derived categories , 2000 .

[31]  R. Thomason,et al.  Higher Algebraic K-Theory of Schemes and of Derived Categories , 1990 .

[32]  A. Duncan.  COHEN‐MACAULAY MODULES OVER COHEN‐MACAULAY RINGS , 1992 .

[33]  P. Jothilingam.  A note on grade , 1975 .

[34]  Ø. Solberg.  Hypersurface Singularities of Finite Cohen‐Macaulay Type , 1989 .

[35]  H. Foxby.  Isomorphims between complexes with applications to the homological theory of modules. , 1977 .

[36]  Raphael Rouquier,et al.  Dimensions of triangulated categories , 2003, math/0310134.

[37]  Y. Kawamata.  Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces , 1988 .

[38]  Osamu Iyama,et al.  Maximal modifications and Auslander–Reiten duality for non-isolated singularities , 2010, 1007.1296.

[39]  Maurice Auslander,et al.  Isolated singularities and existence of almost split sequences , 1986 .

[40]  Den Bergh,et al.  Non-commutative Crepant Resolutions , 2002 .

[41]  Hailong Dao,et al.  Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings , 2010, 1005.5359.

[42]  Yu Ye,et al.  Algebras of Derived Dimension Zero , 2006 .

[43]  Jeremy Rickard,et al.  Morita Theory for Derived Categories , 1989 .

[44]  Bernhard Keller,et al.  On differential graded categories , 2006, math/0601185.

[45]  T. Wassmer.  6 , 1900, EXILE.

[46]  Martin Kalck,et al.  Singularity category of a non-commutative resolution of singularities , 2011, 1103.3936.

[47]  A. Ooishi.  Matlis duality and the width of a module , 1976 .