Let $k$ be an algebraically closed field and $R=k[x]$. For $\lambda_0 \in k$, what is the localization of the $R$-module $$M=\bigoplus_{\lambda \in k} k[x]/(x-\lambda)$$ at the maximal ideal $(x-\lambda_0)$?
My motivation is in investigating an example of a quasi-coherent sheaf on an affine scheme whose support is not closed. I've found it stated that the support of the quasi-coherent $\mathscr{O}_{\mathbb{A}_k^1}$-module $\widetilde{M}$ is not a closed subset of $\mathbb{A}_k^1$, but I am running into problems when trying to compute the stalks of this sheaf. Thanks in advance for your help!
Localization commutes with direct sums, so the problem is to determine $k[x]/(x - \lambda)$ localized at $(x - \lambda_0)$. If $\lambda \neq \lambda_0$ then one of the elements we'll try to invert is $x - \lambda$, so what happens then? If $\lambda = \lambda_0$ then something similarly silly will happen, although the result will be different. You could also do this by commuting the quotient and the localization.
For this to really display a counterexample, we should also check the generic point. It's also worth thinking about why we can't use $\prod$ instead of $\bigoplus$.