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.

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.