# Structure sheaf is ample

I have the following question. It is a well-known fact that if $$X=\operatorname{Spec} A$$ is the spectrum of a ring then $$\mathcal{O}_{X}$$ is ample. Let us suppose that $$X$$ is any scheme and $$\mathcal{A}$$ is a quasi-coherent sheaf of algebras. Is it the structure sheaf of $$\underline{\operatorname{Spec}}(\mathcal{A})$$ ample?

• Ampleness seems to me to be a relative condition. Often we refer to ampleness over a field for example, in which case this well known fact is false, since it would imply all affine varieties are also projective. If you mean $\mathcal{O}_X$ is ample for $X$ over the base scheme $X$, that's probably true, but you should be a little more specific I think.
• Do you know an example where $\mathcal{O}_X$ is not ample? If so, what happens if you take $\mathcal{A}=\mathcal{O}_X$? Commented Apr 12, 2020 at 20:45
Solution: Let us suppose that $$X$$ is a scheme such that $$\mathcal{O}_{X}$$ is not ample. Let us consider the spectrum of the structural sheaf, $$\operatorname{Spec} \mathcal{O}_{X}=X$$. Therefore, the structure sheaf of $$\operatorname{Spec}\mathcal{O}_{X}$$ is not ample.