Finite determinacy of matrices and ideals

TL;DR

This paper characterizes when ideals in power series rings are finitely determined by contact equivalence, linking this property to isolated complete intersection singularities, and develops bounds and methods applicable in arbitrary characteristic.

Contribution

It provides a new characterization of finite determinacy for ideals in power series rings using Fitting ideals and matrix equivalences, extending known results to positive characteristic.

Findings

Ideals with maximal height Fitting ideals are finitely contact determined.

The characterization applies to arbitrary characteristic, including positive characteristic.

Provides computable bounds for determinacy.

Abstract

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x1,...,xs]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim(R/I) > 0 and K is an infinite field (of arbitrary characteristic). Here two ideals I and J are contact equivalent if the local K-algebras R/I and R/J are isomorphic. If I is minimally generated by a1,...,am, we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b1,...,bm with ai - bi in <x1,...,xs>^k for some integer k. We give also computable and semicontinuous determinacy bounds. The above result is proved by considering left-right equivalence on the ring M of m x n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in M have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture.