Equivalence and finite determinacy of mappings
The aim of this article is to consider the following problem: Given two ideals \(I\) and \(J\) in \(R\) a power series ring over a field \(k\), or a ring of convergent power series, when is there an isomorphism \(R/I \simeq R/J\)? This article is the continuation of a previous article of the same authors where they give an answer in the \(j(I)\)-adic completion \(\widehat R\) of \(R\), where \(j(I)\) is the Jacobian ideal of \(I\). In the first part the authors give ``the extension to the analytic case'' of this previous result. Theorem: Let \(k\) be a field of characteristic zero with a non-trivial valuation, let \(I\) be a reduced and equidimensional ideal of the ring of convergent power series \(R = k \{ x_1 , \ldots , x_r \}\). Then there is an integer \(a\) depending only on \(I\), such that if \(J\) is any another reduced equidimensional ideal of the same height as \(I\), and if \(I + j(I) ^{ac} = J + j(I) ^{ac}\) for some \(c \geq 1\), then there exists an automorphism \(s\) of \(R\) such that \(s(I) = J\) and \(s \equiv\) id mod \(j(I) ^c\). This theorem is an extension of Hironaka's theorem for isolated singularities. To prove this result, the authors use their previous theorem for \(\widehat R\) and the approximation theorem of Artin for solutions of analytic equations. If \(F\) is an homomorphism, \(F \in \Hom(R^n,R)\), and \(H\) an ideal of \(R\), we say that \(F\) is finitely determined by \(H\) if there is a positive integer \(a\), such that \(F \cong G\bmod H\) implies that there exist an automorphism \(\phi\) of \(R\) and an \(R\)-isomorphism \(A\) of \(R^n\) such that \(\phi (F) = A G\). If \(I\) is the ideal of \(R\), image of \(F\), and if \(F\) is determined by an ideal \(H\) which depends only on \(I\), we get a criterion for determining when \(R/I\) would be isomorphic to \(R/J\). (The idea of finite determination was introduced by Mather for analytic and \(C^{\infty}\) mappings.) Then the authors give a condition for the existence of a non-trivial finite determination for an ideal \(I\) of \(R\), i.e. a condition to get the homomorphism \(F\) determined by an ideal \(H\) defining the non-smooth locus of \(R/I\). Let \(R = k x_1 , \ldots , x_r \) be a power series ring over a field \(k\) with maximal ideal \(m = ( x_1 , \ldots , x_r)\), let \(F = [ f_1 , \ldots , f_n ] \in R^n\), and \(W = [\partial f _i / \partial x_j]\) the Jacobian matrix, and \(I_t (W)\) the ideal of \(R\) generated by the \(t \times t\) minors of \(W\), for any integer \(t\). Theorem: Let \(J = ( f_1 , \ldots , f_n) + I_n (W)\), with \(J \not= R\). Let \(G = [ g_1 , \ldots , g_r ] \in R^n\), and suppose that \(F \equiv G\bmod J^s R^n\) with \(s \geq 3\), then there exists \(B \in \text{GL}_n (R)\) and a \(k\)-automorphism \(\phi\) of \(R\) such that: \[ \phi (F) = BG \in R^n, \quad \phi \equiv \text{id mod }J^{s-1}, \quad B \equiv \text{id mod } J^{s-1} R^n. \] There is also a converse result: Theorem: Suppose that \(k\) is field of characteristic zero and \(F \in m R^n\). If \(F\) is \(H\)-determined where \(H\) is a proper ideal of \(R\), then: \[ H \subset \text{Ann}_R \bigl (R^n/(\text{Image} (W)+ (f_1 ,\ldots,f_n) R^n)\bigr). \] From these two theorems we can deduce that we have a non-trivial finite determination for an ideal \(I\) of \(R\) if and only if \(I\) is a smooth complete intersection in a non-trivial open set. To get a general result of the form \(R/I \simeq R/J\) whenever some appropriate data of \(I\) is equivalent to the corresponding data of \(J\) modulo a power of an ideal \(H\), we consider the first part of a free resolution of \(I\). \smallskip Let \(R^l@> \Lambda>>R^n@>F>>R\) be an exact sequence with \(F = [ f_1 , \ldots, f_n ]\), \(I= (f_1, \ldots, f_n)\) and \(W= [\partial f _i / \partial x_j ]\). Let \(H = \sum _t I_{n-t} (\Lambda) I_t (W) + I\) be the ideal defining the non-smooth locus of \(R/I\). If \(R^l@>\Phi>> R^n@>G>>R\) is another exact sequence, we want to find a condition to get the existence of a \(k\)-automorphism \(\phi\) of \(R\) and \(A \in \text{GL}_n (R)\) such that \(GA = \phi (F)\). Theorem: Suppose that \(H=R\). There exists a positive integer \(a_0\), depending only on \(F\), such that if \(a > a_0\), \(F \equiv G\bmod m^a\) and \(\Lambda \equiv \Phi\bmod m\), then there exist \(\phi\) and \(A\) such that: \[ GA= \phi (F), \quad \phi \equiv \text{id mod } m^a, \quad A \equiv \text{id mod } m^{a - a_0}. \] Theorem: Suppose that \(H \not= R\). There exist positive integers \(a_0\), \(b_0\), \(c_0\) depending only on \(F\), such that if \(a \geq a_0\), \(F \equiv G \bmod H^a\) and \(\Lambda \equiv \Phi\bmod H^{c_0}\), then there exists \(\phi\) and \(A\) such that: \[ GA = \phi (F), \quad \phi \equiv \text{id mod } H^{a - b_0}, \quad A \equiv\text{id mod } H^{a - b_0}. \] If \(J\) is the ideal of \(R\) generated by \(G\), then we get a condition to fulfill \(R/I \simeq R/J\). In the last part of the article the authors generalize the previous result for an exact sequence \(F_2 @>{f_2}>> F_1 @>{f_1}>> F_0\) where \(F_2\), \(F_1\) and \(F_0\) are free \(R\)-modules. In a similar way they obtain condition for isomorphisms of homomorphisms \(A\) and \(B \in \Hom_R (F,G)\), with \(F\) and \(G\) free \(R\)-modules.
scientific article; zbMATH DE number 1013912