Algebraic Geometry

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Definition. relative size Proposition a) n9 allow f: X > Y Then we outline be a gentle morphism of WX/Y = A n ~ / y9 observe via i. i that wvl Y ~ / is a in the neighborhood unfastened sheaf of rank one. 141 b) enable an entire f: X intersection via n equations), we outline >Y be a closed immersion which of c o d i m e n s i o n and permit J in the community be the sheaf of beliefs WX/Y = (An(j/J2)) v , the place j/j2 is in the community n (i. e. , freed from rank n on X, v denotes in order that is in the neighborhood outlined of X. twin. Then be aware that WX/Y is a in the community unfastened sheaf of rank one on Y. feedback. immersion, f: X > Y is gentle and a closed so the 2 coincide. If f: X intersection, then be aware if then it really is in the neighborhood an isomorphism, definitions 2. i. and if pr~(WX/y) differentials ~> Y is both Y' >Y = WX. /y , . and beliefs tender, 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. permit X f> Y g> Z with g delicate. 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 gentle by means of base is a neighborhood 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 hence 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. allow X f > Y --~-~Z in the community noetherian preschemes, be m o r p h i s m s and think that of f,g, and gf is each one both tender or an area whole intersection. Then we outline an isomorphism f,g: ~x/z ~ > f*~Y/z | ~x/Y" There are 4 instances to think about. a) f,g, and gf are all gentle. 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 whole intersections. is the fitting of Y in Z, and ok is the perfect of X If in X, then now we have a precise series on X, o --+ f. (j/j2) We take ~ > (K/J)/(K/J) 2 --~ of this specific 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 whole 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 through the lemma above, F and g tender ~ nine three- Y is an area P2 whole intersection, X XzY and we will be able to use b) above utilized to F and P2 to acquire ~r,p2" ~x/Y ~> r*~x•174 ~~215 ~" ' Now by way of base extension, V ~xjx,zY = f ~/z" Proposition ~• Le~ X f > Y f,g,h,gf,hg,hgf entire intersection. >Z and taking the inverse, we receive g > Z --~-~W morphisms of in the community noetherian preschemes, of the morphisms gf = PlWx/z , and via the lemma, Transposing 1. 6. X Z be 3 and believe that every is both gentle or an area Then the isomorphisms supply a commutative diagram ~h,g~f,hg = ~f,g~gf,h evidence.

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