By Mary Gray

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**Extra resources for A Radical Approach to Algebra (Addison-Wesley Series in Mathematics) **

**Example text**

PdAI I n . Indeed, - - ... - - - By H i l b e r t ' s s y z y g i e s theorem t h e r e i s a r e s o l u t i o n 0 where t h e Fn-l Fils A algebra over I. Fo are R-free. jective resolution of If i s an i d e a l I. I = IOA where I. Tensoring with A We c a n s e e t h a t 0 yields a pro- pdRIO = pdAI. ,t n ] ,A) TOT;(Z If characteristic A = p n , we = and s i m i l a r l y c o n c l u d e o for i > o . can s t i l l conclude A an 24 Homology o f L o c a l R i n g s S Tori(L,A) module. Tor:(iZ Indeed is a A / p Z ,A) = 0 Z!

Since I = I. I. Po Io, w i t h Let 0 f i n i t e l y gener- Pi has a f i n i t e projective resolution Q A by f i n i t e l y g e n e r a t e d A - p r o j e c t i v e m o d u l e s , we have t h a t pdAI = Sup { p d 1 I We may assume ZZ [ t l , . , , t n ] l o n g e r S - s e q u e n c e f o r any p r i m e g r a d e I. a). ,t n p,tl, As n-3 the pdSIO = n - 1 . - is a pdS(S/Io) and c o n s i d e r a minimal S - r e s o l u t i o n f o r 0-L-F (n > 2 (as ( [ E l o r ( 2 . 2 4 ) ) we c o n c l u d e pdAI < n - 1 S a t t h e i n v e r s e image o f t h e pdSIO f n - 1 A.

0 . p . assumption i s t h a t (x"). then (xn)/(xn)' = A/(xn), that is (x2") The : xn = We c l a i m t h a t x i s n o t a z e r o d i v i s o r : i n f a c t , i f r x = 0 n r x = 0 and so r E (x") f o r a l l n . By t h e i n t e r s e c t i o n theorem r = 0. dim A > 1 : Assume t h e s t a t e m e n t t r u e f o r r i n g s o f l o w e r d i m e n s i o n . ,xd be a s . 0 . p . Pass t o A ' = A/(xl); it i s c l e a r t h a t A ' i n h e r i t s t h e independence p r o p e r t y f o r i d e a l s g e n e r a t e d by s .