As the title says, could you provide a local ring which is nonreduced and its $Spec$ is reducible?
It's one of the exercise in the Wedhorn's book.
To get a local ring I only need to do localization at a prime ideal, and to make it nonreduced I need to quotient some ideal. But it seems hard to ensure the reducibility of the Spec of that ring when the first two conditions are satisfied...
I have tried something like $(k[X,Y,Z]/(X^2,Y^2))_{(X,Y,Z)}$. It satisfies the first two condition, but it has a generic point $(X,Y)$ hence irreducible.
Could you provide some examples for this?