Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

390 colin mclartyof the scheme. This sheaf assigns a ring O X (U) of ‘coordinate functions’ toeach open subset U ⊆ X, and to each inclusion of open subsets U ⊆ V of Xa ring morphismr U,V : O X (V ) → O X (U)The whole must be made of parts isomorphic to the spectra Spec(R) ofringsR with their coordinate functions. A scheme mapf : (X, O X ) → (Y, O Y )consists of a continuous function f : X → Y in the ordinary sense plus a greatmany ring morphisms in the opposite direction: By continuity of f ,eachopensubset V ⊂ Y has inverse image f −1 (V ) open in X, andtheschememapincludes a suitable ring morphismO Y (V ) → O X (f −1 (V ))for each open V ⊂ Y, showing how f −1 (V ) maps algebraically into V .This version of schemes dominates Grothendieck and Dieudonné (1971) andHartshorne (1977), though Grothendieck favored the functorial version in hiswork.An arithmetic scheme is a scheme pasted from finitely many parts defined byfinite lists of integer polynomials. Each integer polynomial ring Z[X 1 , ... X n ]is countable. Since each of its ideals is generated by a finite list of polynomialsthere are only countably many, thus countably many points and closed setsand functions on them. Altogether the Kroneckerian version of any arithmeticscheme is countable.14.3.4 Scheme cohomologySchemes were born for cohomology. In fact they were born and re-bornfor it. Jean-Pierre Serre introduced structure sheaves into algebraic geometryso as to produce the cohomology theory today called coherent cohomology.These structure sheaves were ‘the principle of the right definition’ of schemes(Grothendieck, 1958,p.106). Then Serre took the first step towards the soughtafter‘Weil cohomology’. Using ideas from differential geometry he definedcovers and he proved they gave good 1-dimensional Weil cohomology groupsH 1 (M) for algebraic spaces M. Notably, Serre proved his groups gave the firstnon-trivial step in the infinite series of a δ-functor as in our Section 14.2.4.¹⁹For Grothendieck the functorial pattern was decisive. An idea that gave thefirst step had to give every step.Hemadeitworkbyproducingthegeneral¹⁹ Serre (1958, esp.§1.2 and §3.6).