Main theorem of elimination theory: Difference between revisions

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{{short description|The image of a projective variety by a projection is also a variety}}

In [[algebraic geometry]], the '''main theorem of elimination theory''' states that every [[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 {{math|''k''}} be a [[field (mathematics)|field]], denote by <math>\mathbb{P}_k^n</math> the {{math|''n''}}-dimensional [[projective space]] over {{math|''k''}}. The main theorem of elimination theory is the statement that for any {{math|''n''}} and any [[algebraic variety]] {{mvar|V}} defined over {{math|''k''}}, the projection map <math>V \times \mathbb{P}_k^n \to V</math> sends [[Zariski-closed]] subsets to Zariski-closed subsets.

The main theorem of elimination theory is a corollary and a generalization of [[Francis Sowerby Macaulay|Macaulay's]] theory of [[multivariate resultant]]. The resultant of {{mvar|n}} [[homogeneous polynomial]]s in {{mvar|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.

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:<math>(x,y)\mapsto \pi(x,y)=x.</math>

This projection is not [[closed map|closed]] for the [[Zariski topology]] (as well asnor 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 extendextends <math>L_y</math> to a projective line <math>P_y,</math> the equation of the [[projective completion]] of the parabolahyperbola becomes

:<math>xy_1-y_0=0,</math>

and contains

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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 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 or 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.

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==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 ana [[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.

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Using above notation, one has first to characterize the condition that <math>\varphi(f_1),\ldots, \varphi(f_k)</math> do not have any non-trivial common zero. This is the case if the maximal homogeneous ideal <math>\mathfrak m = \langle x_1, \ldots, x_n\rangle</math> is the only homogeneous prime ideal containing <math>\varphi(I)=\langle \varphi(f_1),\ldots, \varphi(f_k)\rangle.</math> [[Hilbert's Nullstellensatz]] asserts that this is the case if and only if <math>\varphi(I)</math> contains a power of each <math>x_i,</math> or, equivalently, that <math>\mathfrak m^d \subseteq \varphi(I)</math> for some positive integer {{mvar|d}}.

For this study, [[Francis Sowerby Macaulay|Macaulay]] introduced a matrix that is now called ''Macaulay matrix'' in degree {{mvar|d}}. Its rows are indexed by the [[monomial]]s of degree {{mvar|d}} in <math>x_1, \ldots, x_n,</math> and its columns are the vectors of the coefficients on the [[monomial basis]] of the polynomials of the form <math>m\varphi(f_i),</math> where {{mvar|m}} is a monomial of degree <math>d-\deg(f_i).</math> One has <math>\mathfrak m^d \subseteq \varphi(I)</math> if and only if the rank of the Macaulay matrix equals the number of its rows.

If {{math|''k'' < ''n''}}, the rank of the Macaulay matrix is lower than the number of its rows for every {{mvar|d}}, and, therefore, <math>\varphi(f_1),\ldots, \varphi(f_k)</math> have always a non-trivial common zero.

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==Geometrical interpretation==

In the preceding formulation, the [[polynomial ring]] <math>R[\mathbf x]=R[x_1, \ldots, x_n]</math> defines a [[morphism of schemes|morphism]] of [[Scheme (mathematics)|schemes]] (which are algebraic varieties if {{mvar|R}} ifis finitely generated over a field)

:<math>\mathbb{P}^{n-1}_R = \operatorname{Proj}(R[\mathbf x]) \to \operatorname{Spec}(R).</math>

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*{{cite book|last=Eisenbud|first=David|title=Commutative Algebra: with a View Toward Algebraic Geometry|publisher=Springer|year=2013|isbn=9781461253501|author-link=David Eisenbud}}

*{{cite book|last=Milne|first=James S.|title=The Abel Prize 2008&ndash;2012|chapter=The Work of John Tate|publisher=Springer|year=2014|isbn=9783642394492|author-link=James Milne (mathematician)}}

{{algebraic-geometry-stub}}

[[Category:Theorems in algebraic geometry]]