Lefschetz’s principle and local functors

Abstract

The notion of an w-local functor is used to formulate and prove a theorem which is claimed to encompass Lefschetz's principle in algebraic geometry. In his foundational work, A. Weil formulates Lefschetz's principle as follows: For a given value of the characteristic p [=zero or a prime], every result involving only a finite number of points and of varieties, which has been proved for some choice of the universal [i.e. an algebraically closed field of characteristic p of infinite transcendence degree over the prime field] remains valid without restriction; there is but one algebraic geometry of characteristic p for each value of p, not one algebraic geometry for each universal domain ([6, p. 306]). The purpose of this paper in applied logic is to give a proof of this statement, a proof which is motivated by the typical example with which Weil follows the above statement. Weil states a theorem which he proves for the universal of complex numbers (using complex analytic methods); this is followed by an argument proving the theorem for any universal of characteristic zero, an argument which involves successive extensions of an isomorphism of finitely-generated subfields of two universal domains. This type of argument is one commonly found in logic and usually called the argument. (See [1] for an expository paper on the use of the method.) The theorem of ?1 of this paper may be regarded as a formalization of this argument to prove a general result which may be said to include Lefschetz's principle as stated above. In fact our theorem is just a special case of a theorem of Feferman [3] and our only contribution is the observation that this result has relevance for algebraic geometry. The significance-theoretical, if not practical-of our general theorem lies in the fact that the back-and-forth argument, which enables one to transfer a result from one universal to another, is done once and for all. Then to apply the general theorem to a particular result of algebraic geometry only requires a check that the particular result falls Received by the editors July 28, 1971. AMS (MOS) subject class flcations (1970). Primary 14A99, 02H15.