Main theorem of elimination theory

In algebraic geometry, the main theorem of elimination theory states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. 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 any algebraic variety V defined over k, the projection map ${\displaystyle V\times \mathbb {P} _{k}^{n}\to V}$ 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.

A simple motivating example

The affine plane over a field k is the direct product ${\displaystyle A_{2}=L_{x}\times L_{y}}$  of two copies of k. Let

${\displaystyle \pi \colon L_{x}\times L_{y}\to L_{x}}$ 

be the projection

${\displaystyle (x,y)\mapsto \pi (x,y)=x.}$ 

This projection is not closed for the Zariski topology (as well as for the usual topology if ${\displaystyle k=\mathbb {R} }$  or ${\displaystyle k=\mathbb {C} }$ ), because the image by ${\displaystyle \pi }$  of the hyperbola H of equation ${\displaystyle xy-1=0}$  is ${\displaystyle L_{x}\setminus \{0\},}$  which is not closed, although H is closed, being an algebraic variety.

If one extend ${\displaystyle L_{y}}$  to a projective line ${\displaystyle P_{y},}$  the equation of the projective completion of the parabola becomes

${\displaystyle xy_{1}-y_{0}=0,}$ 

and contains

${\displaystyle {\overline {\pi }}(0,(1,0))=0,}$ 

where ${\displaystyle {\overline {\pi }}}$  is the prolongation of ${\displaystyle \pi }$  to ${\displaystyle L_{x}\times P_{y}.}$ 

This is commonly expressed by saying the origin of the affine plane is the projection of the point of the hyperbola that is at infinity, in the direction of the y-axis.

More generally, the image by ${\displaystyle \pi }$  of every algebraic set in ${\displaystyle L_{x}\times L_{y}}$  is either a finite number of points, or ${\displaystyle L_{x}}$  with a finite number of points removed, while the image by ${\displaystyle {\overline {\pi }}}$  of any algebraic set in ${\displaystyle L_{x}\times P_{y}}$  is either a finite number of points or the whole line ${\displaystyle L_{y}.}$  It follows that the image by ${\displaystyle {\overline {\pi }}}$  of any algebraic set is an algebraic set, that is that ${\displaystyle {\overline {\pi }}}$  is a closed map for Zariski topology.

The main theorem of elimination theory is a wide generalization of this property.

Classical formulation

For stating the theorem in terms of commutative algebra, one has to consider a polynomial ring ${\displaystyle R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]}$  over a commutative Noetherian ring R, and an homogeneous ideal I generated by homogeneous polynomials ${\displaystyle f_{1},\ldots ,f_{k}.}$  (In the original proof by Macaulay, k was equal to n, and R was a polynomial ring over the integers, whose indeterminates were all the coefficients of the${\displaystyle f_{i}\mathrm {s} .}$ )

Any ring homomorphism ${\displaystyle \varphi }$  from R into a field K, defines a ring homomorphism ${\displaystyle R[\mathbf {x} ]\to K[\mathbf {x} ]}$  (also denoted ${\displaystyle \varphi }$ ), by applying ${\displaystyle \varphi }$  to the coefficients of the polynomials.

The theorem is: there is an ideal r in R, uniquely determined by I, such that, for every ring homomorphism ${\displaystyle \varphi }$  from R into a field K, the homogeneous polynomials ${\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}$  have a nontrivial common zero (in an algebraic closure of K) if and only if ${\displaystyle \varphi (r)=\{0\}.}$ 

Moreover, r = 0 if k < n, and r is principal if k = n. In this latter case, a generator of r is called the resultant of ${\displaystyle f_{1},\ldots ,f_{k}.}$ 

Geometrical interpretation

In the preceding formulation, the polynomial ring ${\displaystyle R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]}$  defines a morphism of schemes (which are algebraic varieties if R if finitely generated over a field)

${\displaystyle \operatorname {Proj} (R[\mathbf {x} ])\to \operatorname {Spec} (R).}$ 

The theorem asserts that the image of the Zariski-closed set V(I) defined by I is the closed set V(r). Thus the morphism is closed.

See also

• Elimination of quantifiers
• Macaulay's 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.