This is a question about proposition 3.36 in Qing Liu's book.
Proposition 3.36(b) Let $X$ be a quasi-projective scheme over $A$ and $F$ a finite set of points of $X$. Then $F$ is contained in an affine open subset of $X$.
I understand every steps except for the last sentence. Why $D_+(f)\cap \hat{X}$ is an affine open set of $X$? Is this something really trivial to see?
I'm also wondering if it is true that any closed subscheme of $\mathbb P_A^n$ has a canonical strucure $\operatorname{Proj}(A[T_0,\dots,T_n]/I)$?
Best Answer
$D_+(f)\cap \widehat{X}$ is a closed subset of $D_+(f)$ and an open subset of $X$. Since $D_+(f)$ is affine and closed subsets of affines are affine, $D_+(f)\cap \widehat{X}$ is affine. Since it's open in $X$, it's an affine open inside $X$.
As for your second question, every closed subscheme of $\operatorname{Proj} R$ is indeed $\operatorname{Proj} R/I$ for some graded ideal $I$. This isn't unique without extra conditions, though: one needs to require that $I$ be saturated to get uniqueness. This is covered in most books.