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Finite determinacy of matrices and ideals in arbitrary characteristic
| Content Provider | Semantic Scholar |
|---|---|
| Author | Greuel, Gert-Martin Phạm, Thúy Hương |
| Copyright Year | 2017 |
| Abstract | Let M be the ring of m x n matrices A with entries in R=K[[x1,.,xs]], the ring of formal power series over an arbitrary field K. We call A finitely determined if any matrix B, with entries of A-B in ^k for some k, is left-right equivalent to A, i.e. B is contained in the G-orbit of A, where G is the group of automorphisms of R combined with the multiplication of invertible matrices from the left and from the right. Finite determinacy is an important property, which implies that A is left-right equivalent to a matrix with polynomial entries. It has been intensively studied for one power series over the complex and real numbers in connection with the classification of singularities. In positive characteristic the problem is more subtle since the tangent image may differ from the tangent space to the orbit, as was shown in one of our previous papers. There we show that finite codimension of the tangent image is sufficient for finite determinacy. The question whether it is also necessary remains open for matrices of arbitrary size in positive characteristic. In this paper we answer this question positively for 1-column matrices. For this we prove that the Fitting ideals of a finitely determined matrix have maximal height. 1-column matrices are of particular interest, since left-right equivalence of matrices corresponds to contact equivalence of the ideals in R generated by their entries. Here two ideals I and J are contact equivalent if the K-algebras R/I and R/J are isomorphic. Our main result on matrices implies that a positive dimensional ideal is finitely contact-determined if and only if it is an isolated complete intersection singularity. In addition we give explicitly computable and semicontinuous determinacy bounds. We discuss also several open problems which are of independent interest. |
| File Format | PDF HTM / HTML |
| DOI | 10.1016/j.jalgebra.2019.04.013 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1708.02442v2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
