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By Robin Hartshorne

Shipped from united kingdom, please enable 10 to 21 company days for arrival. Lecture Notes in arithmetic forty-one. 106pp. good shape ex. lib.

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Extra info for Local Cohomology: A seminar given by A. Grothendieck Harvard University Fall, 1961

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Where Let A X- i ~ n, ~ be a Gorenstein r i n g of d i m e n s i o n E x t n (M, A) and its associated Spec A, Proof 9 First and Y = {~t } = 0. is a n exact s e q u e n c e in ~'k 9 Indeed, i. M e M = k. ,,f~ --~ 0 by induction on the length of for all M s if ExtI(M,,,A) -~ E x t I ( M , A ) for a n y n H e , (A) - H y ( X , is the closed point. ) 4. 13. M any (We r e c a l l Any M, , so is the middle one. one shows that = 0 and all i ~n, s i n c e by h y p o t h e s i s it i s t r u e O'x), 65 It f o l l o w s n o w t h a t t h e f u n c t o r M is exact for ~ M s Extn(M,A) ~~ On the other hand, hypothesis, so we have a dualizing functor.

Let X be a locally Noetherian p r e s c h e m e , be a closed subset, and let n be an integer9 i (i) H y ( F ) (ii) = 0 depthyF> Proof. (i) for all G and for all by i n d u c t i o n o n n > 0. So s u p p o s e Therefore, G ~ X, n ; We proceed in the category Y, be a coherent sheaf on n is satisfied for > n - 1. F T h e n the following conditions are equivalent: (i) => (ii). depthyF support in let for all i< there is nothing to prove. 7, of c o h e r e n t n - E x t n" 1 ~X(G, F) i. y Then, n. Ext,, sheaves (G, F) = 0 X on X with In other words, the functor 45 from ~ Y of i d e a l s of to sheaves Y, on ~rn Let ~y be the sheaf m _- m, are in the category ~ r n ' --" m' > m is left-exact.

Is an isomorphism, ~ since the functor Hom is. exact, and let chosen above, M. 5. Proof. n then T On t h e o t h e r h a n d , s u p p o s e ~f is left exact, T be a m o d u l e a n n i h i l a t e d by is left 4M n. Then the morphism n I T(M) --"H o r n A ( M,T(A / ~ is an i s o m o r p h i s m , by P r o p o s i t i o n 4. Z. 6. ~ )) Taking the limit as n I is also an isomorphism. The categories Sex( ~ fo , Ab) and a r e m a d e e q u i v a l e n t by t h e f u n c t o r s T ~ lirn T(A/,~M n) n for T zSex( ~fo Ab) 53 and I ~ Proof.

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