Applications of Local Cohomology to the Cohen-Macaulay-behaviour of Blowing Up Rings

Abstract

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 qν, where q is generated by a system of parameters.