The '''main theorem of elimination theory''' states that a [[projective scheme]] is [[proper scheme|proper]].
== Sketch of proof ==
We need to show that <math>p: \mathbf{P}_R \to \operatorname{Spec}R</math> is closed for a ring ''R''. Thus, let <math>X \subset \mathbf{P}_R</math> be a closed subset, defined by a homogeneous ideal ''I'' of <math>R[x_0, \dots, x_n]</math>. Let
:<math>Z_d = \{ y \in \operatorname{Spec} R | I_y \not\supset (x_0, \dots, x_n)^d \}</math>
where <math>I_y</math> is
Then:
:<math>\textstyle p(X) = \cap_d Z_d</math>.
Thus, it is enough to prove <math>Z_d</math> is closed. Let ''M'' be the matrix whose entries are coefficients of monomials of degree ''d'' in <math>x_i</math> in
:<math>x_0^{i_0} \cdots x_n^{i_n} f</math>
with homogeneous polynomials ''f'' in ''I'' and <math>i_0 + \dots + i_n + \operatorname{deg}f = d</math>. Then the number of columns of ''M'', denoted by ''q'', is the number of monomials of degree ''d'' in <math>x_i</math> (imagine a system of equations.) We allow ''M'' to have infinitely many rows.
Then <math>y \in Z_d \Leftrightarrow M(y)</math> has rank <math>< q \Leftrightarrow </math> all the <math>q \times q</math>-minors vanish at ''y''.
