On Bayer's deformation and the associativity formula (Q1180165)

Let \(k\) be a noetherian commutative ring, \(I\) an ideal in a multivariate polynomial ring over \(k\) and \(M\) the monomial ideal associated to \(I\) with respect to a given term ordering. For a ground field \(k\) \textit{D. Bayer} constructed a deformation having \(I\) and \(M\) as fibres [``The division algorithm and the Hilbert scheme'', Thesis (Harvard Univ. 1982)]. He showed that this deformation is flat. This implies Macaulay's theorem: if \(I\) is homogeneous, then the Hilbert functions of \(I\) and \(M\) are the same. The author studies this deformation for a noetherian ground ring \(k\). First the gives necessary and sufficient conditions for the flatness of Bayer's deformation. Then he proves the following generalization of Macaulay's theorem: if \(I\) is homogeneous and \(f\) an additive function on the category of finitely generated \(k\)-modules, then \(f(I_ n)=f(M_ n)\), for all nonnegative integers \(n\) (here \(I_ n\) is the \(n\)-th homogeneous component of \(I\)).