I'm trying to understand an argument in Hartshorne's proof that if the local rings of dimension 1 in $X$ are regular, then so are the local ring of dimension 1 in $X \times \mathbb{A}^1$. My problem is in the "first type" points:
We can reduce to the affine case $X = \operatorname{Spec}\left(R\right)$, and in that case first type points are of the form $x := \mathfrak{p}\left[t\right] \subset R \otimes \mathbb{Z}\left[t\right] \simeq R\left[t\right]$ where $y := \mathfrak{p} \subset R$ is a point of dimension 1 in $X$. In that case, the claim is that the local ring $\mathscr{O}_x$ is isomorphic to $\mathscr{O}_y[t]_{\mathfrak{m}_y}$, and then since $\mathscr{O}_y$ is a DVR it is "obvious" that so is $\mathscr{O}_x$.
First, I guess $\mathscr{O}_y[t]_{\mathfrak{m}_y}$ means $\mathscr{O}_y[t]_{\mathfrak{m}_y\left[t\right]}$, am I right? Then why is it obvious that $\mathscr{O}_x$ is a DVR?
Best Answer
Yes, you're right that the $\mathfrak{m}_y$ in $\mathcal{O}_y[t]_{\mathfrak{m}_y}$ means $\mathfrak{m}_y[t]$ (or equivalently, $(\mathfrak{m}_y)$, the ideal generated by $\mathfrak{m}_y$ inside $\mathcal{O}_y[t]$) - we invert everything not in the prime ideal generated by $\mathfrak{m}_y\subset \mathcal{O}_y[t]$ (this ideal is prime because the quotient is $\kappa(y)[t]$, a domain). The discrete valuation on $\mathcal{O}_y[t]_{\mathfrak{m}_y[t]}$ is given by defining the valuation to be the valuation on $\mathcal{O}_y$ for any element of $\mathcal{O}_y[t]$ which in that subring and setting $v(t)=0$, then extending by the properties of valuations.