Rigid Dualizing Complexes on Schemes

In this paper we present a new approach to Grothendieck duality on schemes. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. We obtain most of the important features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to finite type schemes over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic geometry.

[1]  P. Sastry.  Base change and Grothendieck duality for Cohen–Macaulay maps , 2000, Compositio Mathematica.

[2]  S. Iyengar,et al.  Dualizing Differential Graded Modules and Gorenstein Differential Graded Algebras , 2003 .

[3]  F. S. D. Salas.  Residue: A Geometric Construction , 2002 .

[4]  James J. Zhang,et al.  RESIDUE COMPLEXES OVER NONCOMMUTATIVE RINGS , 2001, math/0103075.

[5]  B. Conrad.  Grothendieck Duality and Base Change , 2001 .

[6]  A. Kapustin,et al.  Noncommutative Instantons and Twistor Transform , 2000, hep-th/0002193.

[7]  V. Ginzburg,et al.  Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism , 2000, math/0011114.

[8]  T. Bridgeland.  Flops and derived categories , 2000, math/0009053.

[9]  P. Sastry,et al.  Pseudofunctorial behavior of cousin complexes on formal schemes , 2003, math/0310032.

[10]  James J. Zhang,et al.  Rings with Auslander Dualizing Complexes , 1998, math/9804005.

[11]  M. Bergh.  Existence Theorems for Dualizing Complexes over Non-commutative Graded and Filtered Rings , 1997 .

[12]  V. Hinich.  Homological algebra of homotopy algebras , 1997, q-alg/9702015.

[13]  Amnon Yekutieli.  RESIDUES AND DIFFERENTIAL OPERATORS ON SCHEMES , 1996, alg-geom/9602011.

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

[15]  Bernhard Keller,et al.  Deriving DG categories , 1994 .

[16]  Amnon Yekutieli.  Dualizing complexes over noncommutative graded algebras , 1992 .

[17]  P. Sastry,et al.  An explicit construction of the Grothendieck residue complex , 1992 .

[18]  E. Kunz,et al.  Regular differential forms and duality for projective morphisms. , 1990 .

[19]  Masaki Kashiwara,et al.  Sheaves on Manifolds , 1990 .

[20]  N. Spaltenstein.  Resolutions of unbounded complexes , 1988 .

[21]  J. Lipman.  Dualizing sheaves, differentials and residues on algebraic varieties , 1984 .

[22]  S. Kleiman.  Relative duality for quasi-coherent sheaves , 1980 .

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