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 ↪ → AnK, we can realize X as the scheme-theoretic intersection of δ(X) + n hypersurfaces. Moreover, this bound δ(X) is uniform in families