18 - Applications to reductions of ideals

Summary

Graded local cohomology theory has played a substantial rôle in the study of Rees rings and associated graded rings of proper ideals in local rings. We do not have enough space in this book to include all we would like about the applications of local cohomology to this area, and so we have decided to select a small portion of the theory which gives some idea of the flavour. The part we have chosen to present in this chapter concerns links between the theory of reductions of ideals in local rings and the concept of Castelnuovo regularity, discussed in Chapter 16. The highlight will be a theorem of L. T. Hoa which asserts that, if b is a proper ideal in a local ring having infinite residue field, then there exist t₀ ∈ ℕ and c ∈ ℕ₀ such that, for all t > t₀ and every minimal reduction a of b^t, the reduction number r_a(b^t) of b^t with respect to a is equal to c. This statement of Hoa's Theorem is satisfyingly simple, and makes no mention of local cohomology, and yet Hoa's proof, which we present towards the end of this chapter, makes significant use of graded local cohomology.

Throughout this chapter, all graded rings and modules are to be understood to be ℤ-graded, and all polynomial rings R[X₁, …, X_t] (and R[T]) over R are to be understood to be (positively) ℤ-graded so that each indeterminate has degree 1 and deg a = 0 for all a ∈ R \ {0}.