By Stanislaw Saks

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Then we have p ( k + a + 1 ) c I but i=l p ( k + a ) <* I. Proof. Let n = pA /IA denote the maximal ideal in A /IA . Then as f , t 1, f 2 t 2, ... f f^ , t d-1 i s a regular sequence in G(pA ) (cf. 3) (3)), the graded ring G(n) is isomorphic to G(pA )/(fjt 1, f 2 t k 2 , ... , f d l t k d - l ) G ( p A p ) . 6)], so that p k + a + 1 A c I A p but P k + a A p

Proof. Because [H 1 M (R)l kn = [ H ^ ( R ( k ) ) l n for all i, n e Z (here N denotes the unique graded maximal ideal of lr , cf. 1)]), v( k ) it suffices to show that R ' is a Cohen-Macaulay ring for some integer k > 1. 3) that the field A/m is infinite. Let k > 1 be an integer such that [ p ( k ) ] n = p ( k n ) for all n e Z. f 2 . - « f d-i of l s o t h a t 1 <=Ja. where J « (fj, f2 f d-pA- Then for each x e m \ p we have the equality length A (A/(x, f v ... 11) (1), whence the sequence fjt f^it k , f2t k , x is G-regular (cf.

As f,, f2, ... , f ^ j is anA-regular sequence, Claim 2 guarantees that G( I) is a Cohen-Macaulay ring (cf. [VV]). 10)], a s T d _ 1 = J T d " 2 by Claim 1. Because ( x ) n l n = x l n (recall that x is not contained in p), R(I)/xR(I) is naturally isomorphic to R(I), so that fr ' = R(I) is a Cohen-Macaulay ring. 4). 1)] also), where they explored the Cohen-Macaulay and Gorenstein properties of Rces 44 SHIRO GOTO algebras R(I) associated to ideals I in a Noetherian local ring in connection with the corresponding ring-theoretic properties of the associated graded rings G(I).