In [Ga] Gabrielov has given conditions under which the completion of the kernel of a morphism φ: A → B between analytic rings coincides with the kernel of the induced morphism φ̂: Â → B̂ between the completions. If B is a domain, a sufficient condition is that rk φ = dim(Â/ker φ̂), where rk φ is the rank of the jacobian matrix of φ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap83-1-4,
author = {F. Broglia and A. Elkhadiri and F. Pieroni},
title = {Formal relations between quasianalytic functions of some fixed class},
journal = {Annales Polonici Mathematici},
volume = {83},
year = {2004},
pages = {35-40},
zbl = {1062.32007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap83-1-4}
}
F. Broglia; A. Elkhadiri; F. Pieroni. Formal relations between quasianalytic functions of some fixed class. Annales Polonici Mathematici, Tome 83 (2004) pp. 35-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap83-1-4/