By Aichinger E.

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Open P A subset n V(B). that = V(A) A point R. the u n i o n point R = V(AB) X A = X - V(A) is not A generic let A c B, t h e n V ( A n B) Zariski S and I} of i d e a l s of the R is a m a x i m a l irreducible V(P) ideals r a d A of A. ideals ideals So w e m a y only unit. 38. 1. Let A , B on prime = {P e X, P ~ A} ly on t h e r a d i c a l Topology. ring with Put X : S p e c R : { p r o p e r set V(A) SHEAVES of sets X for a topology is c l o s e d if a n d S c X is said t o be sets w h i c h set S are d i f f e r e n t is a P e X s u c h t h a t : S.

O is a r e s t r i c t e d 2. The extension PROOF. left The kernel pe of Jacobson ideals P of Qo(R), if pe = e A e then is r e s t r i c t e d . p e ( N Ae) ee The 1. ideal is a u - p e r f e c t PROOF. 1. and closed : Jacobson is the by the eorollary n'{Ae~ A e C'(~)}. radical to T h e o r e m if P = n {A, A 6 C'(o)} (N Ae) e = n A e e of Qo(R). intersection then : n A of the m a x i m a l 9 we get = (N Ae) c = e A and this conditions If A e C'(o) implies pe = P. : that g : (n A) e c N A e Hence, for an e l e m e n t A in C'(o) to is such that [A : R] e C(o) then A is of R.

B. Fix an index There exists a c t • K such that e t c t k • O K for all r + 1 4 k ~ n and ctCtl b r+t • : c b'zr+t • B n A' For suitable ~i • K1 = 1 for some r + 1 4 1 ~ n. ,r, = B 1. we have k=r+lEn ~(ctctk)Y k = i=lEn ~iYi, with ~(ctctl ) = 1, hence a contradiction. The following damental theorem characterizes for the application simple algebras THEOREM 56. unramified of pseudo-places pseudo-places; to the theory of central (section V). Let @ be an unramified pseudo-place of A/K then 1. A' is a free OK-mOdule of rank n : [A : K].

### 2-affine complete algebras need not be affine complete by Aichinger E.

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