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 whose entries are coefficients of monomials of degree d in

${\displaystyle x_{i}}$  in ${\displaystyle x_{0}^{i_{0}}\cdots x_{n}^{i_{n}}f}$ 

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

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