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It is well known that André Weil declared, in the 1940's, that elimination theory must be eliminated from algebraic geometry. I would like to understand his mathematical reasons to adopt such an attitude against elimination theory. (N.B. Later Abhyankar answered back that eliminators of elimination theory have to be eliminated !. ).


I was expecting a deep reason why André Weil , in his "Foundations of Algebraic Geometry (1946)", expressed his wish to see elimination theory completely eliminated from algebraic geometry (page 31). He writes himself that his book is first of all intended to give a "detailed and connected treatment of intersection-multiplicities" (For this, it is of course necessary to lay down "foundations of algebraic geometry"). Now, at these times (let us say between the two World Wars),the use of elimination theory in intersection theory led to contradictions .The main reason was lack of well established foundations (cf. the Introduction in Weil's book). André Weil did the job for intersection-multiplicities (and put the first foundations for our present algebraic geometry (opening the way to Grothendieck)).As for elimination theory, he quite simply hoped to see it eliminated ! (That avoids contradictions !) It is Jean-Pierre Jouanolou who did the work for elimination theory(during the 1980's). Thanks to him, elimination theory is now a full part of algebraic geometry, à la Grothendieck.

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    $\begingroup$ Probably it was some combination of Chevalley's work on specialization (rendering the explicit use of elimination theory unnecessary) and Weil's inability to resist a play on words. $\endgroup$ – Steven Landsburg Oct 3 '13 at 21:58
  • $\begingroup$ I wouldn't say "the use of elimination theory in intersection theory led to contradictions". Whatever mistakes or contradictions happened, they are not to be imputed to the theory itself but to the sloppiness of its users. $\endgroup$ – Abdelmalek Abdesselam Oct 15 '13 at 16:03
  • $\begingroup$ In a recent publication "Résultant, Discriminant"(Enseign.Math.58(2012)), Michel Demazure attributes the famous "Il faut éliminer la théorie de l'élimination" to Dieudonné ! $\endgroup$ – Al-Amrani Nov 30 '13 at 11:05
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I think the usual interpretation is this (see S. Landsburg's comment):

The classical proof that $\mathbb{P}^n$ is proper uses elimination theory: we need to prove that for a ring $R$, $\mathbb{P}^n_R\to Spec(R)$ is closed. Say $R = k[t_1, \ldots, t_k]$, then a closed subset of $\mathbb{P}^n_R$ is given by equations in $t_1, \ldots, t_k, x_0, \ldots, x_{n}$, homogeneous in the $x_i$, and finding the image of this subset means eliminating the $x_i$.

Using Chevalley's valuation-theoretic approach ("valuative criterion of properness"), we can replace this rather elaborate argument by a very short and conceptual one. Hence we eliminated elimination theory.

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