  We prove that the algorithm for desingularization of algebraic varieties in
characteristic zero of the first two authors is functorial with respect to
regular morphisms. For this purpose, we show that, in characteristic zero, a
regular morphism with connected affine source can be factored into a smooth
morphism, a ground-field extension and a generic-fibre embedding. Every variety
of characteristic zero admits a regular morphism to a Q-variety. The
desingularization algorithm is therefore Q-universal or absolute in the sense
that it is induced from its restriction to varieties over Q. As a consequence,
for example, the algorithm extends functorially to localizations and
Henselizations of varieties.
