Finite determinacy and approximation of flat maps

TL;DR

This paper investigates the finite determinacy and approximation of flat analytic maps between germs of real or complex analytic spaces, establishing conditions under which flatness is finitely determined and how such maps can be closely approximated.

Contribution

It proves flatness is finitely determined for maps from Cohen-Macaulay germs and shows flat maps can be approximated by algebraic or Nash maps while preserving key invariants.

Findings

Flatness is finitely determined for maps from Cohen-Macaulay germs.

Flat maps from complete intersection and Cohen-Macaulay germs can be approximated while preserving the Hilbert-Samuel function.

Preservation of the Hilbert-Samuel function implies preservation of Whitney's tangent cone in the complex case.

Abstract

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real or complex analytic spaces. It is shown that the flatness of analytic maps from germs of real or complex analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection and Cohen-Macaulay analytic germs can be arbitrarily closely approximated by algebraic and Nash maps respectively in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone.