Maximal Closed Subsets

Zieschang, P.-H. On maximal closed subsets in association schemes, J. Combin. Theory Ser. A 80, 151-157 (1997)... 

In Section 2.4, we apply some of the previously obtained results on closed subsets in order to derive a sufficient condition for a closed subset to be maximal. 

From Lemma 2.5.2(iv) and Lemma 2.3.6(i) we obtain that a thin scheme possesses a uniquely determined maximal normal closed subset of odd valency. We shall denote this closed subset by 0(5). 

Throwing away unneeded sets, we can now write any closed 5 as Xi u u Xm with the X, closed irreducible and no Xt contained in any other. Let Y X be irreducible. An easy induction shows that an irreducible space is not a finite union of proper closed subsets hence Y = (Y n X ) implies Y = Y n X for some j. Thus the Xt are the maximal irreducible subsets, and are therefore uniquely determined. ... 

Corollary 1. Let X be a variety and Z a maximal closed irreducible subset, smaller than X itself. Then dim Z = dim X — 1. 

Each closed subset V (p)) is a copy of z, but they have all been pasted together here. The whole set-up is called a surface for 2 reasons 1) all maximal chains of irreducible proper closed subsets have length 2, just as in A. 2) If O is the local ring at a closed point x, then O has Krull dimension 2. In fact, if x = [(p, /)], then its maximal ideal is generated by p and /, and there is no single element g O such that m — / g). 

Let v be a closed point of V and let u be the image of v in U. Let us take affine open subsets Spec(A) resp. Spec(B) in U resp. V containing u resp. v such that Spec(B) maps into Spec(A). Let p C A (resp. q C B) be the maximal ideal corresponding to u (resp. v). 

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