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

I. and if pr~(WX/y) differentials ~> Y is both Y' >Y = WX. /y , . and beliefs delicate, or a in the community is a base switch, This follows of subschemes entire and X' = X ~ Y', from the truth that are c o m p a t i b l e w i t h base extension. Lemma 1. four. enable X f> Y g> Z with g soft. Then F is in the neighborhood a whole g, for this reason F: X > X xz Y F. notice that PI: W F 1. 2 to v and WX/XXzY = f wy/Z. W = X ~ . > X is soft by means of base is an area whole of preschemes, be the graph morphism. intersection, We observe Proposition evidence. part permit be m o r p h i s m s intersection, and Y, X, and the extension from 142 = j/j2 ~ F. _I ~/X extension, . V for this reason WX/W = F WW/x. WW/x = P2~y/Z, * and f = P2F, yet back via base so as WX/w = f * W y ~ Z required. Definition 1. five. permit X f > Y --~-~Z in the neighborhood noetherian preschemes, be m o r p h i s m s and think that of f,g, and gf is every one both soft or a neighborhood entire intersection. Then we outline an isomorphism f,g: ~x/z ~ > f*~Y/z | ~x/Y" There are 4 instances to contemplate. a) f,g, and gf are all delicate. of the precise series of P r o p o s i t i o n b) J Then we take to be i~ f,g, and gf are all neighborhood entire intersections. is the correct of Y in Z, and okay is the appropriate of X If in X, then we've a precise series on X, o --+ f. (j/j2) We take ~ > (K/J)/(K/J) 2 --~ of this unique series, inverse i s o m o r p h i s m to be c) > K/K 2 then dualize, O. and take the ~. f is an area entire intersection, w i t h g and gf soft. We take ~ of the precise series of P r o p o s i t i o n WX/y , and take the inverse to get ~. 1. 2, tensor w i t h i~3 d) f and gf neighborhood entire intersections, Then by means of the lemma above, F and g soft ~ nine three- Y is a neighborhood P2 entire intersection, X XzY and we will be able to use b) above utilized to F and P2 to procure ~r,p2" ~x/Y ~> r*~x•174 ~~215 ~" ' Now via base extension, V ~xjx,zY = f ~/z" Proposition ~• Le~ X f > Y f,g,h,gf,hg,hgf whole intersection. >Z and taking the inverse, we receive g > Z --~-~W morphisms of in the neighborhood noetherian preschemes, of the morphisms gf = PlWx/z , and by means of the lemma, Transposing 1. 6. X Z be 3 and believe that every is both soft or an area Then the isomorphisms supply a commutative diagram ~h,g~f,hg = ~f,g~gf,h facts. *Remark. context " Use Lemma 1. three. The reader will discover later that the right kind for the inspiration of Gorenstein morphism, WX/Y simply studied is that of a and we are going to depart him to difficult at the 144 following symptoms. A morphism f: X > Y of in the community noetherian preschemes is termed Gorenstein whether it is in the community of I finite style, has finite Tor-dimension, in D+(X) sheaf to an invertible sheaf. and if f" (~y) is isomorphic Then we name that invertible WX/y, and end up that i f'(F') for all = | WX/y F" 6 D + (Y). quality controls delicate morphisms, and in the community entire intersection morphisms are Gorenstein, and in these circumstances the sheaf is the single we've already outlined.