Local Cohomology by Hartshorne R.

By Hartshorne R.

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Be of i d e a l of an I A, and let with support in Proof. H o m A ( 9 , I). i s in~ective. Let Let I In = H~ V( ~w ). 7, (I) In Ab) I~ be the largest submodule is injective. be the exact functor = lirn T(A/~ n) 55 We n o w c o m e t o t h e s t u d y o f d u a l i z i n g f u n c t o r s . Proposition A (i. e . 9. For all 444 A/~ 9 be an ideal of finite colength in is an Artin ring), and let Then the following conditions are equivalent: M a ~f , T(M) is an A-module of finite type, and the natural morphism M TT(M) -~ (defined via the isomorphism (ii) where T ~ .

I P on the closed subset V(p) Since a constant sheaf on an irreducible Noetherian space is flasque, the contention follows. N o w w e c o m e to the m a i n t h e o r e m of this section, relating the local c o h o m o l o g y groups on a p r e s c h e m e Let and let X F be a p r e s c h e m e , be a q u a s i - c o h e r e n t be a q u a s i - c o h e r e n t let ~n = for each let ~X / n Y be a c l o s e d s u b s p a c e o f s h e a f of ~X-mOdules. s h e a f of i d e a l s d e f i n i n g ~n.

For s o m e coherent sheaf and for all integers prescheme, be a coherent Then the following conditions Supp G C Y, and for all integers Supp G = Y, be a locally Noetherian i < n. G with 44 (ii) depthyF > n. (iibis) depth F for all > n x s Y. 8. 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.

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