# A particular example of a non-reduced scheme (with a reduced ring of global sections)?

I am searching for a scheme $X$ which can be obtained by gluing two affine schemes (along open subsets) such that: 1) X is non-reduced; 2) $\Gamma(X,\mathcal O_X)$ is a reduced ring. Any ideas? Thanks in advance.

## 1 Answer

Hint:

Can you construct a ''fat'' projective line of some sort? Namely, remember that $\text{Spec } k[x]$ is the affine line and we get projective space by gluing together two affine lines. Can you do something similar by fattening up the lines you glue?

My example was the following. Take $\text{Proj } k[x,y,z]/z^2$. I believe that working on affines, this should produce something with only $k$ as global sections. I haven't worked it out in detail however.

• What is your specific example? A fat projective line as a fat ring of global sections, basically because you cannot glue along the fat variable. – Martin Brandenburg Oct 24 '14 at 14:43
• I am also thinking in this direction, but I find it difficult to prevent the nilpotent sections to be glued together and form a nilpotent global section. – AYK Oct 24 '14 at 14:45
• @MartinBrandenburg See my example here. I hope it works, I haven't worked it through in detail as I wrote. – user101036 Oct 24 '14 at 14:50
• $\mathrm{Proj}(k[x,y,z]/(z^2)) = D_+(x) \cup D_+(y)$ (since $D_+(z)=\emptyset$) with $D_+(x) = \mathrm{Spec}(k[y/x,z/x]/(z/x)^2)$ and $D_+(y) = \mathrm{Spec}(k[x/y,z/y]/(z/y)^2)$. I don't think that the ring of global sections contains $z$. It has a good chance to be $k$. – Martin Brandenburg Oct 24 '14 at 15:05
• I have computed this and barring mistakes, I believe that this has global sections $k$. – user101036 Oct 24 '14 at 15:06