I’m reading Qing Liu’s Algebraic Geometry and Arithmetic Curves. I’m confusing my double definitions of projective morphism over a ring $A$.
In definition 2.3.42, projective scheme over $A$ is an $A$-scheme that is isomorphic to a closed subscheme of $\operatorname{Proj} A[x_0,…,x_n]$.
In definition 3.1.11, $X \rightarrow \operatorname{Spec A}$ is projective if it factors in to a closed immersion $X \rightarrow \operatorname{Proj} A[x_0,…,x_n]$ followed by the canonical morphism $\operatorname{Proj} A[x_0,…,x_n] \rightarrow \operatorname{Spec} A$.
So I think A scheme $X$ can be said projective in definition 3.1.11 sense.
So I think this two definitions should be coincide.
I can understand if $X$ is projective in the sense of definition 3.1.11, $X$ is projective in the sense of 2.3.42.
But I can’t understand the converse, when $X$ is projective in the sense of 2.3.42, why does $f:X \rightarrow \operatorname{Spec} A$ have a decomposition two morphisms.
Best Answer
Given a scheme $S$, an $S$-scheme is a scheme $X$ together with a morphism $X \to S$ (often referred to as the structure map), and a morphism of $S$-schemes is a morphism of schemes $X \to Y$ such that the structure map $X \to S$ factors as $X \to Y \to S$. So, phrased in this language, the second definition literally just says that $X$ can be realized as a closed $A$-subscheme of some projective space over $A$. The factorization of $X \to \operatorname{Spec} A$ as a composition $X \to \operatorname{Proj} A[x_0, \dots, x_n] \to \operatorname{Spec} A$ is encoded in the fact that a projective scheme over $A$ is (by definition) an $A$-scheme, and the isomorphism to a closed subscheme of projective space is a morphism of $A$-schemes.