One frequently finds in the history of mathematics that a definition thought to be completely general was in fact too restrictive to treat certain problems of interest. The notion of “number,” for instance, has been expanded again and again—most notably to incorporate irrationalities and complex numbers, the former arising from problems in geometry and the latter needed in order to describe solutions to arbitrary algebraic equations. In a similar way, algebraic geometry, which was once understood as the study of algebraic varieties, or solution sets of algebraic equations in some finite-dimensional space, has grown to encompass more general objects known as “schemes.” As a very meager example one might consider the two equations x + y = 0 and (x + y)2 = 0. The two equations have the same set of solutions in the plane, so they describe the same variety; but the schemes attached to the two objects are completely different. The reformulation of algebraic geometry in the language of schemes was a tremendous project spearheaded by Alexander Grothendieck in the 1960s. As the above example suggests, the scheme-theoretic viewpoint tends to emphasize the algebraic aspects of the subject (equations) rather than the traditionally geometric ones (solution sets of equations). This viewpoint has made a reality of the long-hoped-for unification of ALGEBRAIC NUMBER THEORY [IV.1] and algebraic geometry, and, indeed, much recent progress in number theory would have been impossible without the geometric insight supplied by the theory of schemes.
Even schemes are not enough to handle all the problems of current interest, and still more general notions (stacks, “noncommutative varieties,” derived categories of sheaves, etc.) are brought to bear when necessary. These can appear exotic, but to our successors they will no doubt be second nature, just as schemes are to us. For more on algebraic geometry in general, see ALGEBRAIC GEOMETRY [IV.4]. Schemes are discussed at greater length in ARITHMETIC GEOMETRY [IV.5].