# Reduced scheme and covering with non affines reduced schemes

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!

• Since $X$ is an open cover of itself, it suffices to find a nonreduced scheme $X$ with reduced ring of global sections - this will imply that reducedness cannot be checked on global sections of an arbitrary cover. The answer here should be an example of this. Commented Nov 25, 2019 at 8:09

As Stahl says it suffices to have some non reduced scheme $$X$$ with global sections ring $$\mathcal{O}_X(X)$$ non reduce. The example given here is perfect: $$X=\operatorname{Proj}k[s,x_0,x_1]/(s^2)$$ is not reduce because with $$U=D_+(x_0)=\operatorname{Spec}\left(k\left(\frac{s}{x_0},\frac{x_1}{x_0}\right)/\left(\left(\frac{s}{x_0}\right)^2\right)\right)$$ one has $$\mathcal{O}_X(U)$$ not reduced but the global sections ring $$\mathcal{O}_X(X)$$ is reduced because $$\mathcal{O}_X(X)=k$$ se the link above.