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

388 colin mclartycongruent modulo X 2 + Y 2 − 1, writtenif and only ifP(X, Y) ≡ Q(X, Y) (mod X 2 + Y 2 − 1)Q(X, Y) − P(X, Y) is divisible by X 2 + Y 2 − 1A coordinate function on the space is a congruence class of polynomials. Forexample X 2 Y defines a coordinate function on this space, the same functionas Y − Y 3 , since light calculation shows(X 2 Y) − (Y − Y 3 ) = Y.(X 2 + Y 2 − 1)Altogether the ring of coordinate functions is the ring of integer polynomialsin two variables, Z[X, Y], modulo X 2 + Y 2 − 1, i.e. it is the quotientringZ[X, Y]/(X 2 + Y 2 − 1)These ‘functions’ are constructed from integer polynomials just as Kroneckerwould construct irrationals as in Section 14.1.1.The next step is to use the functions to define points of this space. Themotivation is that each point p should have a set of functionsp ⊆ Z[X, Y]/(X 2 + Y 2 − 1)which take value 0 at p. This set should be an ideal, closed under addition andunder multiplication by arbitrary functions. That is, if the functions P 1 (X, Y)and P 2 (X, Y) are both construed as taking value 0 at p then P 1 (X, Y) +P 2 (X, Y) must also be; and so must the product R(X, Y) · P 1 (X, Y) for anypolynomial R(X, Y). Further, the ideal should be prime, in the sense thatwhenever a product lies in then at least one factor already lies in it.¹⁷ Moreformally:If P 1 (X, Y) · P 2 (X, Y) ∈ then either P 1 (X, Y) ∈ or P 2 (X, Y) ∈ .Scheme theory demands no more than that for a point. It says there is apoint for each prime ideal. Textbooks say the points are the prime ideals. Soour space has a point for each integer solution a, b ∈ Z to Equation (14.3), butalso a point for the mod 7 solution 2, 5 ∈ Z/(7). It has one point combining¹⁷ This expresses the idea that if a product is 0 then at least one factor must be—which does not holdin every ring—but this definition makes it hold for function values at points of a scheme.