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Extra resources for An introduction to differentiable manifolds and riemannian geometry
6. 17. Finally, P Then suppose height P = 2. cannot be principal Remarks: P e A (I), and assume that For large n, Pp since is a Krull domain. R? (a) The class of domains is not minimal over is a prime divisor of in which Thus A (I) q, (i) ~ I. which therefore (iii). always equals A (I) will be studied in a later chapter. 7 was first proved by P. Eakin 21]. 8. Let multiplicity If e. We use Eakin's arguments (R,M) I be a 2-dimensional is an ideal and in our next result. local Cohen-Macaulay domain with M e A (I), then M e Ass(R/I n ) M ~ Ass(R/In).
The Rees ring of Proof: sequences over This was done by Katz in [Kz2]. is an asymptotic We begin with a nice result of Rees, PROPOSITION asymptotic (I,Y). ~ of However, since Let ~(I,Y) with respect to and write ~: R[Y] + R onto ~(l,x). ~(I,Y) via I and for the Y + x. This We claim that we will defer the proof of the claim briefly. x e M, (t'I,M,Y)~(I,Y) we see that onto ideal ker ~ ~ rad[(t'l,M,Y)~(l,Y)]. (t-l,M)~(l,x), ~(I,Y)/$(t'I,M,Y)~(I,Y) carries the maximal R we see that onto ~(l,x)/J(t-l,M)~(i,x).
If one of the following three conditions (i) Since generated by three elements. with if and only if However since if it the argument used above shows that containing ca e R. Let Let However, IR=aR. 6. p=P° is not principal [$2, Theorem i], a e P Then for some valuation overrings of R. is normal if necessary, such that R N [ca(NV)] = R N R ~(I) = i showing that c e R so that be a local domain with integral closure Then By going to P (X,Y,Z,W) there is an prime (R,M) be an ideal of TR this would give has two generators.
An introduction to differentiable manifolds and riemannian geometry by Author Unknown