# Residues and Duality: Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963 /64 (Lecture Notes in Mathematics)

Language: English

Pages: 424

ISBN: 3540036032

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i~8 154 16o 16~ 176 18~ 195 200 21o page Chapter IV. 81. w w Dualizing Complexes and Local Duality 252 Introduction Example: duality for abelian groups Dualizing complexes Uniqueness of the dualizing complex Local cohomology on a prescheme Dualizing functors on a local noetherian ring Local duality Application to dualizing complexes Pointwise dualizing complexes and f~ Gorenstein preschemes Existence of dualizing complexes 252 254 257 266 272 Chapter VI. w 81. 82. 83. 84. 85. Residual

the triangulated categories But now, using [I.3.~] L and it is enough to define a morphism of functors M, we see that lO4 Rf~ R HOm'X (QF~ (where or Q M to QG') > R HOm'y(Rf~QF',= Rf~QG'= ) , denotes the localization functor from u-M-is) for F" 6 L and G" L to LQi s 6 M. We now make explicit the morphisms ~ between a functor and its derived functor (cf. definition of derived functor, and obtain the following diagram: [I.5]) f~ Qf..om'(F',G') (i) Q Hom'(f.F',f.G') ~

for G 6 D b (Y), and qc [VII 4.3]. c for F 6 D- (X) qc al-3 for and D + (Y) qc G 6 qDDc(Y) Here the exponent "b" denotes complexes which are bounded in both directions, i.e., finite. 10 (iv) Recently P. Deligne has shown (unpublished) for the category of noetherian preschemes, that one can construct I f" and Trf satisfying and G 6 D+(Qco(Y)), al,bl, and c, working with F 6 D(Qco(X)) the derived categories of the categories of quasi-coherent sheaves on X and Y, respectively.

map d: cP(~,F) as follows. components If of e 6 cP(qA, F)(V) du pj is a section, as above, then the are given by (~)i O , 9 9 9 , i p + 1 where > cP+I(~,F) ~( = -i )j pj ~i o ' ' ' ' ' i^ j' .." , I p + 1 is the appropriate restriction map on sections of F. Finally, we define an augmentation to the product of its restrictions by sending a section ~.~ ~ F(vnui )" Proposition 3.1. [G, of Xo II.5.2.1] Then the augmentation to the Cech complex r C'(%I,F) of Suppose that It

Proposition 6.2. Let X f > Y g > Z be two finite morphisms of locally noetherian preschemes (rasp. with finite Tor-dimension). Then there is a natural morphism (gf)~ > f~g~ f 167 of functors from Furthermore, (resp. D qc D+(Z) to D+(X) (resp. D(Z) to D(X)). this map is an isomorphism for all G" 6 D + (Z) qc (Z) 1. Proof. For G 6 Mod(Z) there is a natural isomorphism ~ (g'f) *H~ O"Z((g f )"x'O"X"G )-----> X YZ(g*O"Y, G ) ) , whence by [I.5.4] the morphism of functors