For a scheme $X$ to be reduced there is a criterion that says that $X$ is reduced if there is a covering $(U_i)$ by affines open subsets such that for all $i$, $\mathcal{O}_X(U_i)$ is reduced. I guess that it doesn't works anymore when the $U_i$ are not affines. Does someone has a example.
I tried with the classical $\operatorname{Spec}(k[X]/(X^2))$ but all its open sets are affine... something like $\operatorname{Spec}k[x,y]/(x^2)$ maybe?
Thanks!