Almeida Bastos Carvalho, Paula Alexandra de
(1997)
On the prime spectra of some Noetherian rings.
PhD thesis, University of Glasgow.
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Abstract
This thesis is devoted to the description of the graph of links of some skewpolynomial rings and skewLaurent rings; and the characterization of some crossed products which are Azumaya. The characterization of the Azumaya locus and relation with the singular locus is also studied for some crossed products.
In chapter 2 we describe the links between prime ideals in skewLaurent rings of the form S = R[1, 11, ..., n, 1n; 1, ..., n] and in skewpolynomial rings of the form T = R[1,..., n; 1, ..., n] with basis ring R, commutative and Noetherian, where 1,..., n are pairwise commuting automorphisms of R. In order to do so, we start by studying the strong second layer condition.
Theorem The ring S is ARseparated.
Corollary The ring S satisfies the strong second layer condition.
Corollary The ring T satisfies the strong second layer condition.
We show that, in determining the clique of a prime of S or T, there is no loss of generality in assuming that R is semilocal and that the primes contract in the basis ring R to an ideal of the form N = Mg where M is a maximal ideal of R and the indicated intersection is finite. We can then describe the links in S and T.
Proposition Let P and Q be prime ideals of S, with P R = N. Suppose that P Q. Then Q R = N and one of the following holds:
1. 0 NS = P = Q;
2. NS P = Q;
3. 0 NS P Q and there exist a prime ideal P# of S#2 lying over 2/MS2 and i {1,...,u} such that i (P#) lies over 2/MS2 where K# is the algebraic closure of K = R/M, S2 is a skewLaurent ring, S2 S, S#2 = K# K S2, 2 and 2 are minimal primes over P S2 and Q S2, respectively, such that 2 R = M = 2 R, and i are the automorphisms determined by the action of S#2 in K# K M/M2.
Conversely, if one of case 1,2 or 3 holds, then P Q.
The description of cliques in T will, in some cases, depend on the description of cliques in S given before.
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