×

On \(k\)-constructable sets, \(k\)-elementary formulae, and elimination theory. (English) Zbl 0186.25901

Using the terminology of A. Weil’s, “Foundations of algebraic geometry” [Providence, R.I.: AMS (1962; Zbl 0168.18701)], a subset of an abstract variety \(V\) is here defined to be constructable if it is the finite union of sets each of which is the intersection of a closed set and an open set; the \(k\)-constructable subsets of a variety \(V\) defined over \(k\) are similarly defined. The notion of constructable sets, at least for abstract varieties, is due to Chevalley, who has shown that the set-theoretic projection of a constructable subset of a product \(V\times W\) on a factor is constructable. Basing himself on the observation (Theorem A1) that a constructable subset of a variety \(V\) defined over \(k\) is \(k\)-constructable if and only if it is invariant under every automorphism of the universal domain \(\Omega\) over \(k\), the author obtains the corresponding “Projection Theorem” (Theorem A2) where reference is made throughout to a base field \(k\). For some reasons given, the author then treats this theorem ab initio, basing himself on some quite simple observations concerning \(k\)-constructable sets. In the affine case, the “Projection Theorems” can be reformulated as a (well-known) theorem of elimination theory; and conversely, from this case, the general theorem can be obtained by a simple “patching-together”.
In the second part of the paper, the author expands the notion of “elementary formula” due to Tarski. In the present setting, a \(k\)-elementary formula is a notion with reference to a field \(k\), a (fixed) sequence \(x_1,\ldots,x_n\) of letters, and variety \(V=V_1\times\cdots\times V_n\) with the \(V_i\) varieties defined over \(k\). A \(k\)-atomic formula is a formula of the form \((x_{i_1},\ldots,x_{i_n}) \in F\), where \(F\) is a \(k\)-closed subset of the partial product \(V_{i_1}\times\cdots\times V_{i_n}\) of \(V\). A \(k\)-elementary sentence is a formula built up in a finite number of steps from \(k\)-atomic formulae by negation \((\neg)\), conjunction \((\wedge)\), disjunction \((\vee)\), and quantification of the form \(\exists x_i(A)\), where \(A\) is a formula already constructed. A \(k\)-elementary sentence is a \(k\)-elementary formula involving no free variables. It should be realized that such formulae do not, at the start, have any meaning, but are merely arrays of symbols, though these arrays are eventually given an expected meaning. Every \(k\)-elementary formula then describes a definite locus \(L(A)\).
The main theorem now is the “Theorem of Elimination Theory” (Theorem A5), which reads: Let \(A\) be a \(k\)-elementary formula which is not a sentence, and let \(x_{i_1},\ldots,x_{i_n}\) be the free variables in it. Then there is a \(k\)-elementary formula \(B\) free of a signs having the same free variables and such that \(L(B)=L(A)\). By minor modifications indicated, the condition that \(A\) not be a sentence can be lifted. The “Elimination Theorem” is a device for moving often recurring logical statements (in particular, the kind that clog algebraic geometry) with ease.
The author has seen quite competent lectures spend hours proving what can be seen at a glance from Theorem A5; and papers subject to the same comment. The idea of applying the notion of elementary formulae to abstract varieties will then, it may be hoped, prove useful.
Reviewer: A. Seidenberg

MSC:

14-XX Algebraic geometry

Citations:

Zbl 0168.18701
PDFBibTeX XMLCite
Full Text: DOI Crelle EuDML