Editing Scheme (mathematics) (section)

=== Spec of the integers ===
The ring of integers <math>\mathbb{Z}</math> can be considered as the coordinate ring of the scheme <math>Z = \operatorname{Spec}( \mathbb{Z} ) </math>. The Zariski topology has closed points <math>\mathfrak{m}_p = (p) </math>, the principal ideals of the prime numbers <math>p\in\mathbb{Z}</math>; as well as the generic point <math>\mathfrak{p}_0 = (0) </math>, the zero ideal, whose [[Dense set|closure is the whole scheme]]. Closed sets are finite sets, and open sets are their complements, the cofinite sets; any infinite set of points is dense.
[[File:SpecZ.png|alt=Spec(Z)|center|300x300px]]
The basis open set corresponding to the irreducible element <math>p \in \mathbb{Z}</math> is <math>U_p = Z\smallsetminus\{ \mathfrak{m}_p \}</math>, with coordinate ring <math>\mathcal{O}_Z (U_p) = \mathbb{Z}[p^{-1}] = \{\tfrac{n}{p^m}\ \text{for}\ n\in\mathbb{Z}, \ m\geq 0\}</math>. For the open set <math>U = Z\smallsetminus\{\mathfrak{m}_{p_1},\ldots,\mathfrak{m}_{p_\ell}\}</math>, this induces <math>\mathcal{O}_Z (U) = \mathbb{Z}[p_1^{-1},\ldots,p_\ell^{-1}]</math>.

A number <math>n\in \mathbb{Z}</math> corresponds to a function on the scheme <math>Z</math>, a function whose value at <math>\mathfrak{m}_p</math> lies in the residue field <math>k(\mathfrak{m}_p)=\mathbb{Z}/(p) = \mathbb{F}_p</math>, the [[finite field]] of integers modulo <math>p  </math>'':'' the function is defined by <math>n(\mathfrak{m}_p) = n \ \text{mod}\ p</math>, and also <math>n(\mathfrak{p}_0)=n</math> in the generic residue ring <math>\mathbb{Z}/(0) = \mathbb{Z}</math>. The function <math>n</math> is determined by its values at the points <math>\mathfrak{m}_p</math> only, so we can think of <math>n</math> as a kind of "regular function" on the closed points, a very special type among the arbitrary functions <math>f</math> with <math>f(\mathfrak{m}_p)\in \mathbb{F}_p</math>.  

Note that the point <math>\mathfrak{m}_p</math> is the vanishing locus of the function <math>n=p  </math>, the point where the value of <math>p  </math> is equal to zero in the residue field. The field of "rational functions" on <math>Z</math> is the fraction field of the generic residue ring, <math>k(\mathfrak{p}_0)=\operatorname{Frac}(\mathbb{Z}) = \mathbb{Q}</math>. A fraction <math>a/b</math> has "poles" at the points <math>\mathfrak{m}_p</math> corresponding to prime divisors of the denominator. 

This also gives a geometric interpretaton of [[Bezout's lemma]] stating that if the integers <math>n_1,\ldots, n_r</math> have no common prime factor, then there are integers <math>a_1,\ldots,a_r</math> with <math>a_1 n_1+\cdots + a_r n_r = 1</math>. Geometrically, this is a version of the weak [[Hilbert's Nullstellensatz|Hilbert Nullstellensatz]] for the scheme <math>Z</math>: if the functions <math>n_1,\ldots, n_r</math> have no common vanishing points <math>\mathfrak{m}_p</math> in <math>Z</math>, then they generate the unit ideal <math>(n_1,\ldots,n_r) = (1) </math> in the coordinate ring <math>\Z</math>. Indeed, we may consider the terms <math>\rho_i = a_i n_i</math> as forming a kind of [[partition of unity]] subordinate to the covering of <math>Z</math> by the open sets <math>U_i = Z\smallsetminus V(n_i)</math>.