## Analytic Sets in Locally Convex Spaces by Pierre Mazet PDF

By Pierre Mazet

ISBN-10: 008087200X

ISBN-13: 9780080872001

ISBN-10: 0444868674

ISBN-13: 9780444868671

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Extra info for Analytic Sets in Locally Convex Spaces

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5. , A + B . Proposition A be a ring and i s n-Noetherian (resp. ,) t h e n s o is S-'A . A ; if A CHAPTER 3 36 Proof As regards he Noetherian property we hav . this in proposition 2 . 1 7 Suppose now that prime ideal in in I . A ring and let P be a n-closed n We denote by P' the inverse image of P is a A . M. to theorem 2 . 1 6 PI of finite type contained in gr I inf = { gr P' gr I 6 g r P that and , n + < . n 1 = ht P' such that and proposition 1 . 8 } A ; since h t P' = gr P' = gr I ht P there exists an ideal We therefore have i s n-closed in P' that lr ady established < grP and therefore ht P A .

Decreases strictly starting from Q, , whence one deduces that htP>,htQ b p . P (This statement does no more than translate the inequality between the homological codimension and the dimension (cf. [SR]). Corollary Let A be a n n - N o e t h e r i a n P we have grP 4 htP . ring. For e v e r y n - c l o s e d prime i d e a l CHAPTER 3 34 Proof ensures the existence of a regular sequence of Theorem 2 . 1 6 P elements of whose length is gr P . 2. g Ca 3 ~ 3 1 5 ~a2 Q. ring i f i t i s n - N o e t h e r i a n a n d e v e r y P v e r i f i z s ht P = gr P , u - .

We complete this chapter by giving the following important examp 1 e . 20. Proposition I n order t h a t a ring A be f a c t o r i a l i t i s n e c e s s a r y and s u f f i c i e n t t h a t i t be i n t e g r a l , I - N o e t h e r i a n i d e a l s o f h e i g h t 1 be p r i n c i p a l . and t h a t i t s p r i m e Proof T h e condition i b nEce6baty. we know that height 1 A Let A be a factorial ring. Then is integral and that its prime ideals o f are principal. principal ideals of A Furthermore we know that the set of is Noetherian for the- relationship of inclusion; it therefore suffices to prove that the I-closed ideals A are principal.