An inductive analytic criterion for flatness (Q2846725)

The authors present a criterion for flatness of modules over \(K\)-analytic algebras, where \(K\) denotes either the field of real or complex numbers, of an inductive nature. The main result may be expressed as follows.NEWLINENEWLINELet \(R=K\{{\mathbf y}\}=K\{y_1, \dots,y_m\}\), \(S=K\{{\mathbf x}\}=K\{x_1, \dots, x_n\}\), \(A=K\{\mathbf y, \mathbf x\}= R\{\mathbf x\}\) (where \(y_1, \ldots, x_n\) are analytically independent variables), \({\mathcal M} =(x_1, \ldots, x_n)R\), \(J\) an ideal of \(R\), \(F\) a finite \(A\)-module. Via the inclusion \(R \subset A\), \(F\) is also an \(R\)-module. Then:NEWLINENEWLINE(a) There are an element \(g \in A\), a natural number \(q\), and a homomorphism of \(A\)-modules \(\psi: A^q \to F\) such that \(g(\mathbf 0,\mathbf x) \not= 0 \) (in \(S\)), \(gF \subseteq \mathrm{Im} (\psi)\) and \(\mathrm{Ker} (\psi) \subseteq {\mathcal M}A^q\);NEWLINENEWLINE(b) The \(R/J\)-module \(F {\otimes}'(R/J)\) is flat if and only if for all \(g, q\) and \(\psi\) as in (a) it is true that (i) \(\mathrm{Ker} (\psi) \subseteq JA^q\) and (ii) \(F/\mathrm{Im} (\psi) {\otimes}'(R/J)\) is a flat \(R/J\)-module. Here, \({\otimes}'\) denotes the analytic tensor product over \(R\).NEWLINENEWLINEIn (b), if \({g(\mathbf 0, \mathbf 0)} \not=0\) then \(F/\mathrm{Im} (\psi)=0\) and (ii) is trivial. If \({g(\mathbf 0, \mathbf 0)} = 0\) then by the Weierstrass Preparation Theorem (after a linear change of coordinates, if necessary) \(A/gA\) is a finite \(R\{x_1, \dots, x_{n-1} \}\)-module, and we may use induction on the number of variables \(x_j\).NEWLINENEWLINEThe proof involves, among other techniques, delicate work with matrices with coefficients in suitable rings.NEWLINENEWLINEMore geometric statements may be reduced to the mentioned algebraic result.NEWLINENEWLINEAs an application, the authors give new, more constructive proofs of several fundamental results of analytic geometry. Namely, the existence of a universal local flattener for an \({\mathcal O} _X\)-coherent module given a germ of analytic morphism \(\phi:X \to Y\), the ``openness of flatness Theorem'' and Frisch's ``generic flatness theorem''.