Shin Ikeda

Remarks on lifting of Cohen-Macaulay property

Let ( R, m ) be a local noetherian ring and I a proper ideal in R . Let ( I ) be the Rees-ring ⊕ n≥0 I n with respect to I . In this note we describe conditions for I and R in order that the Cohen-Macaulay property (C-M for short) of R/I can be lifted to R and ( I ), see Propositions 1.2, 1.3. and 1.4.

On the Gorenstein Property of Rees Algebras.

Let (A,M) be a noetherian local ring. For certain equimultiple ideals I in A we try to relate the Gorenstein property of the Rees algebra⊕n and of A itself. In particular we n≥0 treat the case of equimultiple prime ideals of height two and the case I=M. The results underscore a natural conjecture, s. Thm. 2.6. and 3.2.

Various Notions of Equimultiple and Permissible Ideals

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

Equimultiplicity and Cohen-Macaulay Property of Blowing Up Rings

Bl(p,R) \to Spec(R)

Generalized Cohen-Macaulay Rings and Blowing Up

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).

Asymptotic Sequences and Quasi-Unmixed Rings

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 Our presentation uses also ideas of the unpublished thesis of E.C. Dade [5].

Local Cohomology and Duality of Graded Rings

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.

Review of Multiplicity Theory

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.