Formal relations between quasianalytic functions of some fixed class
Volume 83 / 2004
Abstract
In [Ga] Gabrielov has given conditions under which the completion of the kernel of a morphism $\varphi : A \to B$ between analytic rings coincides with the kernel of the induced morphism $\widehat {\varphi } : \widehat {A} \to \widehat {B}$ between the completions. If $B$ is a domain, a sufficient condition is that $\mathop {\rm rk}\nolimits \varphi = \mathop {\rm dim}\nolimits \! {(\widehat {A}/\! \mathop {\rm ker}\nolimits \widehat {\varphi })} $, where $\mathop {\rm rk}\nolimits \varphi $ is the rank of the jacobian matrix of $\varphi $ considered as a matrix over the quotient field of $B$. We prove that the above property holds in a fixed quasianalytic Denjoy–Carleman class if and only if the class coincides with the ring of germs of analytic functions.
