By Pierre Mazet

ISBN-10: 008087200X

ISBN-13: 9780080872001

ISBN-10: 0444868674

ISBN-13: 9780444868671

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**Extra resources for Analytic Sets in Locally Convex Spaces **

**Sample text**

Then if a E Pn but is minimal among those prime . Proposition 3 . 1 3 1 + ht P n - l = n . guarantees Since the inequality i s evident, we obtain the desired result. Remark The results of the propositions 3 . 1 3 for Noetherian rings. M. and 3 . 1 5 are classical I n order to prove them in the case of rings one could u s e the Noetherian case if it were possible to use localization in a prime ideal of finite height. would be so if one could prove that the ideal proposition 3 . 1 3 (resp. height.

Sxp) = we clearly have Axl p E , IN and x1 P symmetric p-linear x , V in in 0 a , and that if E s p(a+x) V a (I is is a balanced open neigha + V c U, then for all which verifies 1 = Zc! ,x, E E, Ax ~ ( a ) . For a O . . , x). , 1 - Xp(x,x, x), P! as long as the series o n the right hand side of the equation is convergent. It suffices therefore to establish this conver- gence in order to prove the Let x (since V ; there exists E V z = G-analyticity of A E ] 1, + 'P such that m [ is open); w e deduce, for each .

Let A be an without n-Noetherian ring; in o r d e r that an and sufficient n-dense). A-module be n-torsion (resp. an n-torsion module) 5t is necessary that the elements of Ass M be n-oZosed (resp. In particular an ideal I is n-cZosed Iresp. n-dense) if and only if the elements of Ass A/I Indeed, this condition signifies (resp. Ass M/'In(M) (resp. Tn(M) = M) 0 = . ) are. Ass Tn(M) and therefore = Tn(M) fl = {O} Corollary 2 . Let A be an verify gr 1 n-Noetherian ring. < n I n o r d e r that an ideal it is necessary and sufficient contained in a prime ideal P which verifies The condition is evidently sufficient.

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

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