Talk:Main theorem of elimination theory

Sketch of proof
(Put this to the article page, when it becomes complete.)

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
 * $$Z_d = \{ y \in \operatorname{Spec} R | I_y \not\supset (x_0, \dots, x_n)^d \}$$

where $$I_y$$ is 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 $$x_i$$ (imagine 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.

Move?
Why isn't this part of elimination theory? 31.50.156.122 (talk) 18:05, 3 July 2019 (UTC)


 * Because the theorem can appear outside the context of elimination theory; namely in algebraic geometry. -- Taku (talk) 19:00, 3 July 2019 (UTC)