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.
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
The main theorem of elimination theory is a corollary and a generalization of Macaulay's theory of multivariate resultant.
The resultant of n homogeneous polynomials in n variables is the value of a polynomial function of the coefficients, which takes the value zero if and only if the polynomials have a common non-trivial zero over some field containing the coefficients.
This belongs to elimination theory, as computing the resultant amounts to eliminate variables between polynomial equations.
The affine plane over a field k is the direct product
the equation of the projective completion of the hyperbola becomes and contains where
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.
with a finite number of points removed, while the image by
is a closed map for Zariski topology.
The main theorem of elimination theory is a wide generalization of this property.
For stating the theorem in terms of commutative algebra, one has to consider a polynomial ring
(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
from R into a field K, defines a ring homomorphism
in R, uniquely determined by I, such that, for every ring homomorphism
have a nontrivial common zero (in an algebraic closure of K) if and only if
This is the case if the maximal homogeneous ideal
Hilbert's Nullstellensatz asserts that this is the case if and only if
for some positive integer d. For this study, Macaulay introduced a matrix that is now called Macaulay matrix in degree d. Its rows are indexed by the monomials of degree d in
and its columns are the vectors of the coefficients on the monomial basis of the polynomials of the form
if and only if the rank of the Macaulay matrix equals the number of its rows.
If k < n, the rank of the Macaulay matrix is lower than the number of its rows for every d, and, therefore,
and suppose that the indices are chosen in order that
have a non-trivial common zero if and only if the rank of the Macaulay matrix in degree D is lower than the number to its rows.
In other words, the above d may be chosen once for all as equal to D. Therefore, the ideal
whose existence is asserted by the main theorem of elimination theory, is the zero ideal if k < n, and, otherwise, is generated by the maximal minors of the Macaulay matrix in degree D. If k = n, Macaulay has also proved that
In the preceding formulation, the polynomial ring
defines a morphism of schemes (which are algebraic varieties if R is finitely generated over a field) The theorem asserts that the image of the Zariski-closed set V(I) defined by I is the closed set V(r).