Let $K$ be a field and $L$ be its field extension, $K\subset L.$ Let $V \subset L^n$ and let $\mathcal{I}(V)\subset K[x_{1},...,x_{n}]$ be its vanishing ideal. Let $R=K[x_{1},..,x_{n}]/\mathcal{I}(V)$ and $\mathfrak{m}$ be a maximal ideal in $R.$
Consider the local ring $R_{\mathfrak{m}},$ then I think $$L=R_{\mathfrak{m}}/\mathfrak{m}.$$ Thus the cotantangent space $\mathfrak{m}/\mathfrak{m}^2$ is an $L$-vector space.
i am kind of confused whether the residue field should be $L$ or $K.$
The residue field is neither $K$ nor $L$ in general!
For example, take $K=\mathbb Q\subset L=\mathbb C, n=1, V=\{i,-i\}\subset \mathbb C^1$, so that $\mathcal I(V)=(X^2+1)\subset \mathbb Q[X]$ .
Then $R=\mathbb Q[X]/(X^2+1)=\mathbb Q(i)$ and necessarily $\mathfrak m=(0)\subset R$, so that the residue field $\kappa(\mathfrak m)=\mathbb Q(i)$ is different from both $K=\mathbb Q$ and $L=\mathbb C$ .