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In [[algebraic geometry]], the '''main theorem of elimination theory''' states that any [[projective scheme]] is [[proper scheme|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 (mathematics)|field]], denote by <math>\mathbb{P}_k^n</math> 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 <math>\mathbb{P}_k^n \times \mathbb{P}_k^m \to \mathbb{P}_k^m</math> sends [[Zariski-closed]] subsets to Zariski-closed subsets. Since Zariski-closed subsets in projective spaces are related to [[homogeneous polynomial]]s, 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 {{mvar|k}} is the [[direct product]] <math>A_2=L_x\times L_y</math> of two copies of {{mvar|k}}. Let
:<math>\pi\colon L_x\times L_y \to L_x</math>
be the projection
:<math>(x,y)\mapsto \pi(x,y)=x.</math>
This projection is not [[closed map|closed]] for the [[Zariski topology]] (as well as for the usual topology if <math>k= \R</math> or <math>k= \C</math>), because the image by <math>\pi</math> of
the [[hyperbola]] {{mvar|H}} of equation <math>xy-1=0</math> is <math>L_x\setminus \{0\},</math> which is not closed, although {{mvar|H}} is closed, being an [[algebraic variety]].
If one extend <math>L_y</math> to a projective line <math>P_y,</math> the equation of the [[projective completion]] of the parabola becomes
:<math>xy_1-y_0=0,</math>
and contains
:<math>\overline\pi(0,(1,0))=0,</math>
where <math>\overline\pi</math> is the prolongation of <math>\pi</math> to <math>L_x\times P_y.</math>
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 {{mvar|y}}-axis.
More generally, the image by <math>\pi</math> of every algebraic set in <math>L_x\times L_y</math> is either a finite number of points, or <math>L_x</math> with a finite number of points removed, while the image by <math>\overline\pi</math> of any algebraic set in <math>L_x\times P_y</math> is either a finite number of points to the whole line <math>L_y.</math> It follows that the image by <math>\overline\pi</math> of any algebraic set is an algebraic set, that is that <math>\overline\pi</math> 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]] <math>R[\mathbf x]=R[x_1, \ldots, x_n]</math> over a commutative [[Noetherian ring]] {{mvar|R}}, and an [[homogeneous ideal]] {{mvar|I}} generated by [[homogeneous polynomial]]s <math>f_1,\ldots, f_k.</math> (In the original proof by [[Francis Sowerby Macaulay|Macaulay]], {{mvar|k}} was equal to {{mvar|n}}, and {{mvar|R}} was a polynomial ring over the integers, whose indeterminates were all the coefficients of the<math>f_i\mathrm s.</math>)
Any [[ring homomorphism]] <math>\varphi</math> from {{mvar|R}} into a field {{mvar|K}}, defines a ring homomorphism <math>R[\mathbf x] \to K[\mathbf x]</math> (also denoted <math>\varphi</math>), by applying <math>\varphi</math> to the coefficients of the polynomials.
The theorem is: there is an ideal {{mvar|r}} in {{mvar|R}}, uniquely determined by {{mvar|I}}
such that, for every ring homomorphism <math>\varphi</math> from {{mvar|R}} into a field {{mvar|K}}, the homogeneous polynomials <math>\varphi(f_1),\ldots, \varphi(f_k)</math> have a nontrivial common zero (in an algebraic closure of {{mvar|K}}) if and only if <math>\phi(r)=\{0\}.</math>
Moreover, {{math|1=''r'' = 0}} if {{math|1=''k'' < ''n''}}, and {{mvar|r}} is [[principal ideal|principal]] if {{math|1=''k'' = ''n''}}. In this latter case, a generator of {{mvar|r}} is called the [[Macaulay's resultant|resultant]] of <math>f_1,\ldots, f_k.</math>
===Geometrical interpretation===
In the preceding formulation, the [[polynomial ring]] <math>R[\mathbf x]=R[x_1, \ldots, x_n]</math> defines a morphism of [[Scheme (mathematics)|scheme]] (which are algebraic varieties if {{mvar|R}} if finitely generated over a field)
:<math>\operatorname{Proj}(R[\mathbf x]) \to \operatorname{Spec}(R).</math>
The theorem asserts that the image of the Zariski-closed set {{math|''V''(''I'')}} defined by {{mvar|I}} is the closed set {{math|''V''(''r'')}}. Thus the morphism is closed.
==See also==
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