Abstract

We reformulate the theorem of Rees-Boger (19. 6) by use of the generalized multiplicity e(x,a,R) and give an application for complete intersections. Let (R,m) be a local ring and let p be a prime ideal of R. Recall that, by definition (10.10), s(p) − 1 is the dimension of the fibre of the morphism $$Bl(p,R) \to Spec(R)$$ at the closed point m of Spec(R) (this fibre being Proj (G(p,R)⊗RR/m) . Likewise, if q is any prime ideal of R containing p, then s(pRq) − 1 is the dimension of the fibre of the above morphism at the point q (by flat base change). Now s(pRq) ≦ s(p) by (10.11), and s(pRp) = dim Rp = ht(q) by Remark (10.11), a). This shows that ht(p) = s(p) if and only if the fibre dimension of \(Bl(p,R) \to Spec(R)\) is a constant function on V(p) ⊂Spec(R).

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