Main theorem of elimination theory

The main theorem of elimination theory states that a projective scheme is proper.

Sketch of proof

We need to show that $p:\mathbf {P} _{R}\to \operatorname {Spec} R$ is closed for a ring R. Thus, let $X\subset \mathbf {P} _{R}$ be a closed subset, defined by a homogeneous ideal I of $R[x_{0},\dots ,x_{n}]$. Let

$Z_{d}=\{y\in \operatorname {Spec} R|I(p^{-1}(y))\not \supset (x_{0},\dots ,x_{n})^{d}\}$.

Then:

$\textstyle p(X)=\cap _{d}Z_{d}$.

Thus, it is enough to prove $Z_{d}$ is closed. Let M be the matrix whose entries are coefficients of monomials of degree d in $x_{i}$ in

$x_{0}^{i_{0}}\cdots x_{n}^{i_{n}}f$

with homogeneous polynomials f in I and $i_{0}+\dots +i_{n}+\operatorname {deg} f=d$. Then the number of columns of M, denoted by q, is the number of monomials of degree d in $x_{i}$ (image a system of equations.) We allow M to have infinitely many rows.

Then $y\in Z_{d}\Leftrightarrow M(y)$ has rank $<q\Leftrightarrow$ all the $q\times q$-minors vanish at y.