Dorin Popescu
Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

An extension $u:A\to A'$ of Noetherian rings is a filtered colimit of smooth algebras if and only if $u$ is a regular morphism. In non Noetherian rings the things are more difficult. The Zariski Uniformization Theorem says that a valuation ring containing a field $k$ of characteristic zero is a filtered union of its smooth $k$-subalgebras. More general, if $V\subset V'$ is an immediate extension of valuation rings of characteristic zero, that is it induces trivial extensions on value groups and residue fields, then $V'$ is a filtered union of its smooth $V$-subalgebras, When $V$ contains a field of characteristic $\not = 0$ and $V\subset V'$ is essentially finite then the above result does not hold by an example of Ostrowski. However, $V'$ is a filtered union of its complete intersection $V$-subalgebras.