How Grothendieck Simplified Algebraic Geometry

Abstract

256 Notices of the AMS Volume 63, Number 3 The idea of scheme is childishly simple—so simple, so humble, no one before me dreamt of stooping so low.…It grew by itself from the sole demands of simplicity and internal coherence. [A. Grothendieck, Récoltes et Semailles (R&S), pp. P32, P28] Algebraic geometry has never been really simple. It was not simple before or after David Hilbert recast it in his algebra, nor when André Weil brought it into number theory. Grothendieck made key ideas simpler. His schemes give a bare minimal definition of space just glimpsed as early as Emmy Noether. His derived functor cohomology pares insights going back to Bernhard Riemann down to an agile form suited to étale cohomology. To be clear, étale cohomology was no simplification of anything. It was a radically new idea, made feasible by these simplifications. Grothendieck got this heritage at one remove from the original sources, largely from Jean-Pierre Serre in shared pursuit of the Weil conjectures. Both Weil and Serre drew deeply and directly on the entire heritage. The original ideas lie that close to Grothendieck's swift reformulations. Generality As the Superficial Aspect Grothendieck's famous penchant for generality is not enough to explain his results or his influence. Raoul Bott put it better fifty-four years ago describing the Grothendieck-Riemann-Roch theorem. Riemann-Roch has been a mainstay of analysis for one hundred fifty years, showing how the topology of a Riemann surface affects analysis on it. Mathematicians from Richard Dedekind to Weil generalized it to curves over any field in place of the complex numbers. This makes theorems of arithmetic follow from topological and analytic reasoning over the field í µí´½ í µí± of integers modulo a prime í µí±. Friedrich Hirzebruch generalized the complex version to work in all dimensions. Grothendieck proved it for all dimensions over all fields, which was already a feat, and he went further in a signature way. Beyond single varieties he proved it for suitably continuous families of varieties. Thus: Grothendieck has generalized the theorem to the point where not only is it more generally applica-ble than Hirzebruch's version but it depends on a simpler and more natural proof. (Bott [6]) This was the first concrete triumph for his new cohomol-ogy and nascent scheme theory. Recognizing that many mathematicians distrust generality, he later wrote: I prefer to accent " unity " rather than " general-ity. " But for me these are two aspects of one quest. Unity represents the profound aspect, and generality the superficial. [16, p. PU 25]