Equimultiplicity and Blowing Up

In this chapter we collect all the basic facts about multiplicities, Hilbert functions and reductions of ideals. At the same time we will introduce the notations to be used throughout the book.

This chapter contains some general facts about graded rings that arise in connection with blowing up. We compute the dimensions of these rings and for certain cases we construct special systems of parameters. We also relate the multiplicities and Hilbert functions of the original ring to those of the various graded rings derived from it. Then we recall the theory of standard bases, and finally we show how to translate some well known results on flatness to the graded case. Our presentation uses also ideas of the unpublished thesis of E.C. Dade [5].

In this chapter we give various characterizations of quasi-unmixed local rings. Most of the results are contained in papers by Ratliff, although with different proofs. Recently it has been recognized that a very useful tool for these characterizations are asymptotic sequences, which are somewhat analogous to regular sequences for the characterization of local Cohen-Macaulay rings. The theory of asymptotic sequences has been developed by Ratliff [10] and independently by Katz [8]. Our treatment follows closely the treatment by Katz [8]. We start by giving some auxiliary results.

We reformulate the theorem of Rees-Böger (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 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(pR_q) − 1 is the dimension of the fibre of the above morphism at the point q (by flat base change). Now s(pR_q) ≦ s(p) by (10.11), and s(pR_p) = dim R_p = 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).

The problem of describing the behaviour of a given variety X under blowing up a closed subvariety Y ⊂ X should be phrased as follows: How does the blowing up morphism X′ → X depend on properties of Y ? Classically Y was chosen to be non-singular and equimultiple. For the non-hypersurface case equimultiplicity was refined to the notion of normal flatness by Hironaka, still assuming Y non-singular. But there are reasons to admit singular centers Y too. For example, in his theory of quasi-ordinary singularities, Zariski used generic projections of a surface to a plane, and blowing up a point in this plane induces the blowing up of a singular center in the original surface. In Chapter IV we gave three different algebraic generalizations of the classical equimultiplicity together with a numerical description of each condition. In Chapter VI we will indicate that the new conditions are useful in the study of the numerical behaviour of singularities under blowing up singular centers. In this Chapter V we want to show that these conditions are also of some use to investigate Cohen-Macaulay properties under blowing up, which are essential for the local and global study of algebraic varieties. Finally in Chapter IX we shall describe a general criterion of the Cohen-Macaulayness of Rees algebras in terms of local cohomology.

In this chapter we mainly study the behaviour of (generalized) Hilbert functions and (generalized) multiplicities of local rings R under blowing up an ideal I ⊂ R such that R/I need not be regular. After some preliminaries in Section 28 we have to present in Section 29 a result of Singh on Hilbert functions under quadratic transformations. Using this result one can prove in Section 30 the semicontinuity of Hilbert functions by desingularizing curves. Finally for inequalities of Hilbert functions under blowing up other centers one has to apply this semicontinuity. The last Section 32 is related to equisingularity theory via flat families. As before (R,m,k) is again a noetherian local ring and I a proper ideal of R.

In this chapter we give a summary of the theory of local cohomology and duality over graded rings, see [4],[6] and [13*]. To make the text as self-contained as possible we begin in § 33 with elementary properties of the category of graded modules over a graded ring A = _{n ε} ^⊗ _Z A_n. One should remark that most results in this chapter hold for any noetherian ring or any noetherian local ring R by regarding R as a graded ring with the trivial grading R₀ = R and R_n = 0 for n ≠ 0. On the other hand our theory of graded rings can be extended to any Z ⁿ-graded rings as Goto and Watanabe have done in [17].

In this chapter we investigate the properties of local rings (A,m,k) such that λ_A (H _m ⁱ (A)) < ∞ for i < dim A. Rings of this type appear in algebraic geometry frequently. For example, if X⊆P _K ⁿ is an irreducible, non-singular projective variety over a field k, then the local ring at the vertex of the affine cone over X satisfies this property (cf. Hartshorne [1]; see also the remark at the end of § 35 in Chapter VII). The purpose of this chapter is to present the results on “generalized Cohen-Macaulay rings” in a unified manner. We develop the theory according to S. Goto [7] and N.V. Trung [17]. Throughout the next two chapters (A,m,k) denotes a noetherian local ring with dim A = d. The reason for this deviation from our principle to denote local rings by R and graded rings by A is the fact that we want to use R for “Rees rings” in the sequel.

Here the results of Chapter V are partially extended and rephrased in a different context. One motivation for this chapter is the fact, that the Cohen-Macaulay (CM) properties of an algebraic variety X and its blowing up X′ with center Y ⊂ X are totally unrelated, unless we have suitable properties for Y and e.g. the local cohomology modules of the affine vertex over X have finite length in all orders ≦ dim X. Hence, replacing again X by a local ring (A,m) and Y by an ideal I ⊂ A we want to relate the CM-property of the Rees ring B(I,A) = _{n ≧} ^⊗ ₀Iⁿ to the CM-properties of A, B(m,A) and G(I,A) ⊗ A/m under suitable cohomological conditions. Our first aim is to give a general criterion of the Cohen-Macaulayness of Rees algebras in terms of local cohomology, see main Theorem (44.1). Then we ask this question for Rees rings of equimultiple ideals I, in particular of m-primary ideals and of ideals q and q^ν, where q is generated by a system of parameters.