Constructible Invariants

A local numerical invariant is a map ω which assigns to a local ring R a natural number ω(R). It induces on any scheme X a partition given by the sets consisting of all points x of X for which ω(OX,x) takes a fixed value. Criteria are given for this partition to be constructible, in case X is a scheme of finite type over a field. It follows that if the partition is constructible, then it is finite, so that the invariant takes only finitely many different values onX . Examples of local numerical invariants to which these results apply, are the regularity defect, the Cohen-Macaulay defect, the Gorenstein defect, the complete intersection defect, the Betti numbers and the (twisted) Bass numbers. As an application, we obtain that for an affine scheme X of finite type over a field K, there is a number δ(X), such that for any n and any closed immersion X ↪→ AK , we can realize X as the scheme-theoretic intersection of δ(X) + n hypersurfaces. Moreover, this bound δ(X) is uniform in families.