Laura Ghezzi

This is joint work with S.D. Cutkosky.

Let $k$ be a field of characteristic zero, $K$ an algebraic function field over $k$, and $V$ a $k$-valuation ring of $K$. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings $R_i$ with quotient field $K$ which are dominated by $V$, and such that the direct limit $\cup R_i=V.$

We investigate the ring $T=\cup \hat R_i$. The ring $T$ is Henselian and thus can be considered to be a "completion" of the valuation ring $V$. We compare this completion with other notions of completion of a valuation ring. We give an example showing that $T$ is in general not a valuation ring.