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 $\mathbb {P} _{k}^{n}$ 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 $V\times \mathbb {P} _{k}^{n}\to V$ 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 $A_{2}=L_{x}\times L_{y}$ of two copies of k. Let

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

be the projection

$(x,y)\mapsto \pi (x,y)=x.$

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

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

$xy_{1}-y_{0}=0,$

and contains

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

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

This is commonly expressed by saying the 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 $\pi$ of every algebraic set in $L_{x}\times L_{y}$ is either a finite number of points, or $L_{x}$ with a finite number of points removed, while the image by ${\overline {\pi }}$ of any algebraic set in $L_{x}\times P_{y}$ is either a finite number of points or the whole line $L_{y}.$ It follows that the image by ${\overline {\pi }}$ of any algebraic set is an algebraic set, that is that ${\overline {\pi }}$ 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 $R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]$ over a commutative Noetherian ring R, and an homogeneous ideal I generated by homogeneous polynomials $f_{1},\ldots ,f_{k}.$ (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$f_{i}\mathrm {s} .$)

Any ring homomorphism $\varphi$ from R into a field K, defines a ring homomorphism $R[\mathbf {x} ]\to K[\mathbf {x} ]$ (also denoted $\varphi$), by applying $\varphi$ 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 $\varphi$ from R into a field K, the homogeneous polynomials $\varphi (f_{1}),\ldots ,\varphi (f_{k})$ have a nontrivial common zero (in an algebraic closure of K) if and only if $\varphi (r)=\{0\}.$

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 $f_{1},\ldots ,f_{k}.$

Geometrical interpretation

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

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

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.