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

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

Then:

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

Thus, it is enough to prove ${\displaystyle Z_{d}}$  is closed. Let M be the matrix with

Then ${\displaystyle y}$  is a closed point of ${\displaystyle Z_{d}\Leftrightarrow M(y)}$  has rank ${\displaystyle <q\Leftrightarrow }$  all the ${\displaystyle q\times q}$ -minors vanish at y.