Noetherian ring

IX) If A is a local noetherian ring with maximal ideal the m-adic completion A is a flat A-algebra. 

Noetherian rings, and the decomposition theorem of ideals in these rings. 

Since k [Xi,..., Xm is a noetherian ring, the subsets of kn of the form V(A) axe exactly the closed algebraic sets. On the other hand, if X is a closed algebraic set, we define... 

Proposition 1. Let X be a scheme. If X can be covered by a finite number of open affine sets Ui = Spec (Ri) such that each Ri is a noetherian ring, then X is a noetherian scheme. 

Structure theorem of integrally closed noetherian rings. If R is a noethe-rian integral domain, then... 

Step I. Assume 3 =Spf A with A a J-adic noetherian ring. 

Lemma 3.1.7. hetd = Spf A,c/= Spf A with A, A adic noetherian rings and fid —a morphism. Equivalent conditions ... 

Prooft It suffices to prove this in case S= Spec A with A a noetherian ring and Y defined by an ideal J. Then 3= Spf A with A the J-adic completion of A. Introduce also S = Spec A and X = X A. By the remarks made in 3 1.5 aud by. 1.3 it suffices to prove that the inverse images s, of the on S, are regular and... 

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