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.

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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 extends <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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==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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==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=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)}}

[[Category:AlgebraicTheorems in algebraic geometry]]

{{algebraic-geometry-stub}}

[[Category:Algebraic geometry]]