Main theorem of elimination theory

In algebraic geometry, the main theorem of elimination theory states that any projective scheme is proper. A version of this theorem predates the existence of modern algebraic geometry. It can be stated, proved, and applied in the following more classical setting. Let k be a field, denote by ${\displaystyle \mathbb {P} _{k}^{n}}$ the n-dimensional projective space over k. The main theorem of elimination theory is the statement that for any n and m the projection map ${\displaystyle \mathbb {P} _{k}^{n}\times \mathbb {P} _{k}^{m}\to \mathbb {P} _{k}^{m}}$ sends Zariski-closed subsets to Zariski-closed subsets. Since Zariski-closed subsets in projective spaces are related to homogeneous polynomials, it's possible to state the theorem in that language directly, as was customary in the elimination theory.

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

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

where ${\displaystyle I_{y}}$  is 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, denoted by q, is the number of monomials of degree d in ${\displaystyle x_{i}}$  (imagine 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.

See also

• Elimination of quantifiers
• Resultant
• Gröbner basis

References

• Mumford, David (1999). The Red Book of Varieties and Schemes. Springer. ISBN 9783540632931.
• Eisenbud, David (2013). Commutative Algebra: with a View Toward Algebraic Geometry. Springer. ISBN 9781461253501.
• Milne, James S. (2014). "The Work of John Tate". The Abel Prize 2008–2012. Springer. ISBN 9783642394492.