CHAPTER 13 The Rising Sea : Grothendieck on Simplicity and Generality Colin
نویسنده
چکیده
In 1949, Andre Weil published striking conjectures linking number theory to topology and a striking strategy for a proof [Weil, 1949]. Around 1953, Jean-Pierre Serre took on the project and soon recruited Alexander Grothendieck. Serre created aseries of concise elegant tools which Grothendieck and coworkers simplified into thousands of pages of category theory. Some have complained of this style, but Miles Reid, for exarnple, says "Grothendieck himself can't necessarily be blamed" for the growth of category theory "since his Own use of categories was very successful in solving problems" [Reid, 1990, p. 116]1 This chapter describes the methods Grothendieck made standard in algebraic geometry by 1958, pursuing the Weil conjectures: Abelian categories, derived functor cohomology, and schemes. It touches on Grothendieck topologies, toposes, and etale cohomology, which arose around 1958, to see how Grothendieck himself relates them to the earlier ideas2 Grothendieck describes two styles in mathematics. Wben he thinks of a theorem to be proved as a nut to be opened, so as to reach "the nourishing flesh protected by the shell," then the hammer and chisel principle is: "put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks-and you are satisfied." He goes on to say:3
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At a time when mathematical fashion despises generality (seen as gratuitous “generalities”, i.e. vacuities) I affirm the principle force in all my work has been the quest for the “general.” In truth I prefer to accent “unity” rather than “generality.” But for me these are two aspects of one quest. Unity represents the profound aspect, and generality the superficial aspect. (Grothendieck, 87, p....
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