# Tag Info

Accepted

### Does every map from a noetherian ring to a valuation ring factor through a DVR?

The answer is no. I give two examples, which are standard non-dvr points on the Riemann-Zariski space of the plane. (1) Let $R=k[x,y]$ and let $V\subseteq k(x,y)$ be the subring consisting of rational ...
• 14.4k
Accepted

### Extension of 2-adic valuation to the real numbers

No. The important thing to know is that, if $K \subseteq L$ is a field extension and $v: K \to \mathbb{R}$ is a valuation, then $v$ can be extended to $L$. So I can answer all of your questions by ...
• 142k
Accepted

### Is every field the residue field of a discretely valued field of characteristic 0?

Yes, by Hasse-Schmidt ("Die Struktur diskret bewerteter Koerper", Crelle's Journal, 1934) for any field $k$ of characteristic $p$ there exists a strict Cohen ring $A$, which is a Noetherian, ...
• 7,714
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### Classifying Space of "Valuation Ringed Spaces over a Topos"

Since the axioms describing what a valuation ring can be put as what's called geometric sequents [*], by the fundamental theorem on classifying toposes, there is a topos $T_{val}$ with precisely the ...
• 2,919
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• 61

### Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

EDIT I may have overlooked the assumption that $L$ is contained in $\hat{K}$. In general, if $v$ is a valuation of $K$ and $L/K$ is finite separable then $L \otimes_K \hat{K}$ is reduced and thus ...
• 19.9k
Accepted

### Torsors over complete local fields

I am rewriting my comment as an answer. That is false in characteristic $p$ for torsors for the finite, flat group scheme $\mu_p=\text{Spec}\ \mathbb{Z}[t]/\langle t^p -1 \rangle$ with the usual ...