Definability, Constructibility and Transfer

Throughout, we will fix an algebraically closed field K. Classical algebraic geometry studies varieties defined over K, where in this proposal, a variety means a solution set of some polynomial equation system over K. Therefore, the study of a variety V is equivalent to the study of the maximal spectrum of its coordinate ring A(V ). Grothendieck realized that a relative version of the concept of a variety, termed scheme, would allow for the infinitesimal study of varieties, and at the same time, greatly facilitate the study of algebraic families of varieties (if f : X → S is morphism of schemes, then the collection of all its fibers forms an algebraic family of varieties). To fully exploit this new point of view, the classical notion of point as a K-rational solution has to be replaced by allowing arbitrary L-rational solutions (up to some obvious congruence relation), where L is any extension field of K. Put differently, this leads to the study of all prime ideals of the coordinate ring, or, for that matter, of any ring. The main principle of modern algebraic geometry could be phrased as follows.