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

‘there is no ontology here’ 387polynomials. In general there are many ways to choose these parts and no onecovering is intrinsic to the scheme.A scheme map, on this approach, is simply any natural transformation betweenthe functors. This is the start of many pay-offs as geometrically importantideas find quite direct functorial expression. It may not even seem plausiblethat such an abstract definition of maps could have geometric content untilyou have seen it work; as we do not have space to show here. In fact,though, it gives exactly the same patterns of maps as the more evidentlygeometrical definition in the following section. The functorial apparatuslooked heavy to some algebraic geometers but it merely made a centralfact of algebra explicit: Every integer polynomial has a (possibly empty) setof solutions in every commutative ring, and every ring morphism preservessolutions. The techniques spread as geometers learned the practical valueof using arbitrary commutative rings to prove theorems on the ring Z ofintegers.Grothendieck and Dieudonné (1971) used this version of schemes inthe introduction. Such a scheme is ‘basically a structured set’ (Mumford,1988, p.113) but strictly it is a structured proper class. There are technicalmeans to avoid proper classes—by specific tricks in specific situations, orgenerally by adding Grothendieck universes to set theory (Artin et al., 1972,pp. 185ff.). But standard textbook presentations use functors on all commutativerings and these are proper classes in either ZF or categorical settheory.14.3.3 Schemes as Kroneckerian spacesThe other approach to schemes constructs a space from the data. If the equationX 2 + Y 2 = 1 (14.3)is to define a space then intuitively a coordinate function on that space shouldbe given by a polynomial in X and Y, while two polynomials P(X,Y),Q(X,Y)give the same function if they agree all over the space. To a first approximationwe would say P(X,Y) andQ(X,Y) give the same function on thisspace ifP(a, b) = Q(a, b) for all 〈a, b〉 ∈Z 2 with a 2 + b 2 − 1 = 0But this puts too much stress on integer solutions. In the case of equations withno integer solutions this poor definition would make every two polynomialsgive the same function trivially. So we actually use a stronger condition takenfrom the defining polynomial: Polynomials P(X,Y) and Q(X,Y) are called