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 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 sends Zariski-closed subsets to Zariski-closed subsets.

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. In fact, given a system of polynomial equations, which is homogeneous in some variables, the resultant eliminates these homogeneous variables by providing an equation in the other variables, which has, as solutions, the values of these other variables in the solutions of the original system.

A simple motivating example

The affine plane over a field k is the direct product of two copies of k. Let

be the projection

This projection is not closed for the Zariski topology (as well as for the usual topology if or ), because the image by of the hyperbola H of equation is which is not closed, although H is closed, being an algebraic variety.

If one extends to a projective line the equation of the projective completion of the parabola becomes

and contains

where is the prolongation of to

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 of every algebraic set in is either a finite number of points, or with a finite number of points removed, while the image by of any algebraic set in is either a finite number of points or the whole line It follows that the image by of any algebraic set is an algebraic set, that is that 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 over a commutative Noetherian ring R, and a homogeneous ideal I generated by homogeneous polynomials (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)

Any ring homomorphism from R into a field K, defines a ring homomorphism (also denoted ), by applying to the coefficients of the polynomials.

The theorem is: there is an ideal in R, uniquely determined by I, such that, for every ring homomorphism from R into a field K, the homogeneous polynomials have a nontrivial common zero (in an algebraic closure of K) if and only if

Moreover, if k < n, and is principal if k = n. In this latter case, a generator of is called the resultant of

Using above notation, one has first to characterize the condition that do not have any non-trivial common zero. This is the case if the maximal homogeneous ideal is the only homogeneous prime ideal containing Hilbert's Nullstellensatz asserts that this is the case if and only if contains a power of each or, equivalently, that 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 where m is a monomial of degree One has 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, have always a non-trivial common zero.

Otherwise, let be the degree of and suppose that the indices are chosen in order that The degree

is called Macaulay's degree or Macaulay's bound because Macaulay's has proved 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 is a principal ideal (although Macaulay matrix in degree D is not a square matrix when k > 2), which is generated by the resultant of This ideal is also generically a prime ideal, as it is prime if R is the ring of integer polynomials with the all coefficients of as indeterminates.

Geometrical interpretation

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). Thus the morphism is closed.

See also

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 20082012. Springer. ISBN 9783642394492.
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