# Localization of the coordinate ring $K[V]$, $K$ not necessarily algebraically closed.

In his first chapter of "The Arithmetic of Elliptic Curves", Silverman develops some necessary background in algebraic geometry. He works over ground fields $$K$$, which are perfect but not necessarily algebraically closed. For example, we get the coordinate ring of $$V/K$$: $$K[V] = \frac{K[X_1, \dots, X_n]}{I(V/K)}.$$

I am then wondering why Silverman defines the maximal ideal $$M_P = \{f \in \bar{K}[V] : f(P) = 0\}$$ and the corresponding local ring $$\bar{K}[V]_{M_P}$$ only for algebraically closed fields?

It seems to me that (for $$P$$ a $$K$$-rational point) we could define a maximal ideal $$m_P = \{f \in K[V] : f(P) = 0\}$$ and a corresponding local ring $$K[V]_{m_P}$$ — although I am not sure if this is still a DVR?

Yes, all of this still works for non-algebraically-closed fields. One doesn't even need $$P$$ to be $$K$$-rational: it's perfectly fine to think about the local ring of $$\operatorname{Spec} \Bbb R[x]$$ at the point $$(x^2+1)$$, for instance.
As for whether the local ring is a DVR or not, this is only the case when $$V$$ is of dimension one - in general, the local ring of a point has dimension equal to the codimension of the closure of the point in the variety. So for $$K[V]_{I(P)}$$ to have dimension one, $$\overline{\{P\}}$$ should be a codimension one subvariety of $$V$$, which if $$P$$ is a closed point can only happen when $$V$$ is of dimension one.
• at the maximal ideal $(x^2+1) = \{f \in K[V] : f(i) = 0\}$ (in elliptic curves we want a group law on the points so it is weird to call a maximal ideal a point) Aug 7 '19 at 18:17