Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and residue field $K = A/\mathfrak{m}$ and suppose that $A$ is also a $k$-algebra. Under what conditions can I say that $K/k$ is a finite field extension?
More specifically, I have a non-singular curve $X$ over a field $k$. For every $P$ the local ring $\mathscr{O}_{X, P}$ is a DVR and a $k$-algebra and I would like to show that the residue field $k_P = \mathscr{O}_{X, P}/\mathfrak{m}_{X, P}$ is a finite extension of $k$.
This question comes from trying to understand Definition 3.2 in Isabel Longbottom's note "Riemann-Roch for nonsingular complete curves".
Thank you for your time. Any answers or comments will be much appreciated.
Talking about DVRs immediately is the wrong way to solve this problem. The right way is to use the fact that you're dealing with a scheme of finite type over $k$:
Suppose $C$ is our curve and let $U\subset C$ be an affine open subset containing the closed point $p$. Then $k[U]$ is a finitely generated $k$-algebra and $\kappa(p)=k[U]/\mathfrak{m}_p$, so the residue field $\kappa(p)$ is a field which is finitely generated as a $k$-algebra. By Zariski's lemma, $\kappa(p)$ is finite over $k$.
The DVR $k(x)[y]_{(y)}$ is an example of a DVR over $k$ which has a residue field which is not finite over $k$ - in general, the local ring of a codimension one subvariety in a normal variety of dimension at least two will furnish many examples of DVRs over $k$ which have a residue field which is not a finite extension of $k$.