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Divisor Theory in Module Categories by Leopoldo Nachbin and W.V. Vasconcelos (Eds.)

By Leopoldo Nachbin and W.V. Vasconcelos (Eds.)

Vasconcelos W.V. Divisor conception in module different types (1974)(ISBN 0444107371)

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1 4 ) w e can c o m p l e t e t h e p r o o f o f t h e n r o p o s i t i o n a s l ( ( O : p-) / x ( O : p ) ) = l ( A / ( p , x ) ) . uO(p;W. 17) 33 C o r o l l a r y . The t y p e o f a Macaulay r i n g d o e s n o t i n c r e a s e s under l o c a l i z a t i o n . 4 & i n j e c t i v e dimensions. The e m p h a s i s i n t h i s s e c t i o n i s on h o m o l o g i c a l d i m e n s i o n s o f f i n i t e l y g e n e r a t e d modules o v e r a l o c a l N o e t h e r i a n r i n g A . m_ w i l l d e n o t e t h e maximal i d e a l o f A and k - i t s residue f i e l d .

F o r o t h e r w i s e I and J would be c o n t a i n e d i n t h e same minimal nrime E o f A : Y would t h e n be a n o n z e r o , n o n f a i t h f u 1 , i n j e c t i v e P ! Let p t h e n be a e;rade module o v e r t h e l o c a l A r t i n i a n r i n g A ?!. J + J ' ; it i s e a s i l y seen t h a t one prime i d e a l c o n t a i n i n g h a s d e p t h o n e . J n_ A # 0 - t h u s implying t h a t I which i s n o t t r i v i a l , c o n s i s t s o f z e r o d i v i s o r s . But t h i s indeed t h e case ,J' # A as P E i s c l e a r l y impossible.

X;E) If Proof. ) = E/IE, If q # n p i c k an which i s n o t a z e r o d i v i s o r o f t h e e x a c t sequence 0 + E. are zero d i v i s o r s f o r s a t i s f y i n g o u r assumption. ) in I We c a l l t h i s number t h e I - d e p t h o f (0 : I ) # 0 and s o a l l e l e m e n t s o f I E E. We u s e d e c r e a s i n g i n d u c t i o n on q . element be t h e c o r r e s p o n d - Then a l l maximal r e g u l a r E - s e q u e n c e s i n t h e same l e n g t h n - q . a E E/aE + 0 ; by t h e f u n c t o r i a l i t y o f t h e Koszul complex w e g e t E.

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