I encountered the phrase "normalized valuation" similar to the following:
Let $A_i$ be the valuation ring $k[x_1,…,x_n]_{\langle x_i\rangle}$ and $v_i$ be the normalized valuation defined by $A_i$.
I didn't know this term before, and a short internet search did not help me.
What I know: we can define a map $k[x_1,…,x_n]\smallsetminus\{0\}\to\mathbb{Z}$ by sending $f=gx_i^{n_f}$ with $x_i\nmid g$ to $n_f\in\mathbb{Z}$. Then extend this to $v:Q(k[x_1,…,x_n])^*=k(x_1,…,x_n)^*\to\mathbb{Z}$ via $\frac{f}{g}\mapsto n_f-n_g$, and this is a discrete valuation on $k(x_1,…,x_n)$ with $k[x_1,…,x_n]_v=k[x_1,…,x_n]_{\langle x_i\rangle}$ as its discrete valuation ring. Is the map $v$ already the $v_i$ mentioned above? Is it "normalized", and what does this mean?
Also, the definition of the map $v$ relies on $k[x_1,…,x_n]$ being a UFD. So by the same argument, $R_{\langle p\rangle}$ is a DVR if $R$ is a UFD and $p\in R$ prime. So I guess this does no longer hold for rings of the form $R_\mathfrak{p}$ where $\mathfrak{p}\subset R$ is prime in general? What about $k[x_1,…,x_n]_{\langle x_1,x_2\rangle}$?
Thank you!
Best Answer
A normalized discrete valuation $v$ of a field $K$ means that $v$ is a discrete valuation of $K$ such that $v(K^*) = \mathbb{Z}$. In general, $v(K^*)$ can be any discrete subgroup of $\mathbb{R}$.