How Grothendieck simplified algebraic geometry

Abstract

Philosophers, and not only philosophers of mathematics, have reason to notice what Grothendieck saw in mathematics and especially how close the highest level theory can be to concrete motivations. Plenty of geometers will say that, far from simplifying anything, Grothendieck made algebraic geometry more complicated, abstract, and difficult. And plenty of philosophers suspect mathematicians like abstraction and difficulty for their own sake. But in fact Grothendieck's ideas grew surprisingly directly from concrete geometric problems known to Riemann and Poincaré. They spread because they made Weil's conjectures in arithmetic easier to understand and finally prove. The archetypal post-Grothendieck textbook Algebraic Geometry by Hartshorne is easier to read, and covers more general theorems, and treats more concrete geometric problems, than the previous leading advanced text Foundations of Algebraic Geometry by Weil. This article describes what Grothendieck simplified and how. Grothendieck says the elegant mathematician Jean-Pierre Serre uses the "method of hammer and chisel," seeking swift, powerful tools to cut through hard problems. And he calls Serre's influence indispensable to his own work. But Grothendieck's ideal is the "method of the rising sea,'' allowing time to develop a general framework to soften a problem before trying to solve it.