Editing Scheme (mathematics) (section)

=== Affine line over the integers ===
The affine space <math>\mathbb{A}^1_{\mathbb{Z}} = \{a\ \text{for}\ a\in \mathbb{Z}\}</math> is a variety with coordinate ring <math>\mathbb{Z}[x]</math>, the polynomials with integer coefficients. The corresponding scheme is <math>Y=\operatorname{Spec}(\mathbb{Z}[x])</math>, whose points are all of the prime ideals <math>\mathfrak{p}\subset \mathbb{Z}[x]</math>. The closed points are maximal ideals of the form <math>\mathfrak{m}=(p, f(x))</math>, where <math>p  </math> is a prime number, and <math>f(x)  </math> is a non-constant polynomial with no integer factor and which is irreducible modulo <math>p  </math>. Thus, we may picture <math>Y  </math> as two-dimensional, with a "characteristic direction" measured by the coordinate <math>p  </math>, and a "spatial direction" with coordinate <math>x </math>.
[[File:SpecZx.png|alt=Spec Z[x]|center|376x376px]]
A given prime number <math>p   </math> defines a "vertical line", the subscheme <math>V(p)</math> of the prime ideal <math>\mathfrak{p}=(p)   </math>: this contains <math>\mathfrak{m}=(p, f(x))</math> for all <math>f(x)</math>, the "characteristic <math>p   </math> points" of the scheme. Fixing the <math>x</math>-coordinate, we have the "horizontal line" <math>x=a   </math>, the subscheme <math>V(x-a)</math> of the prime ideal <math>\mathfrak{p}=(x-a)   </math>. We also have the line <math>V(bx-a)</math> corresponding to the rational coordinate <math>x=a/b   </math>, which does not intersect <math>V(p)</math> for those <math>p   </math> which divide <math>b   </math>. 

A higher degree "horizontal" subscheme like <math>V(x^2+1)    </math> corresponds to <math>x</math>-values which are roots of <math>x^2+1   </math>, namely <math>x=\pm \sqrt{-1}   </math>. This behaves differently under different <math>p   </math>-coordinates. At <math>p=5</math>, we get two points <math>x=\pm 2\ \text{mod}\ 5    </math>, since <math>(5,x^2+1)=(5,x-2)\cap(5,x+2)   </math>. At <math>p=2</math>, we get one [[Ramification point|ramified]] double-point <math>x=1\ \text{mod}\ 2   </math>, since <math>(2,x^2+1)=(2,(x-1)^2)   </math>. And at <math>p=3</math>, we get that <math>\mathfrak{m}=(3, x^2+1)</math> is a prime ideal corresponding to <math>x=\pm \sqrt{-1}   </math> in an extension field of <math>\mathbb{F}_3   </math>; since we cannot distinguish between these values (they are symmetric under the [[Galois group]]), we should picture <math>V(3, x^2+1)</math> as two fused points. Overall, <math>V(x^2+1)    </math> is a kind of fusion of two Galois-symmetric horizonal lines, a curve of degree 2. 

The residue field at <math>\mathfrak{m}=(p, f(x))</math> is <math>k(\mathfrak{m})=\Z[x]/\mathfrak{m} = \mathbb{F}_p[x]/(f(x))\cong \mathbb{F}_{p}(\alpha)</math>, a field extension of <math>\mathbb{F}_p  </math> adjoining a root <math>x=\alpha  </math> of <math>f(x)  </math>; this is a finite field with <math>p^d  </math>elements, <math>d=\operatorname{deg}(f)  </math>. A polynomial <math>r(x)\in\Z[x]  </math> corresponds to a function on the scheme <math>Y</math> with values <math>r(\mathfrak{m}) = r \ \mathrm{mod}\ \mathfrak{m}</math>, that is <math>r(\mathfrak{m}) = r(\alpha)\in \mathbb{F}_p(\alpha)  </math>. Again each <math>r(x)\in\Z[x]  </math> is determined by its values <math>r(\mathfrak{m})</math> at closed points; <math>V(p)</math> is the vanishing locus of the constant polynomial <math>r(x)=p</math>; and <math>V(f(x))</math> contains the points in each characteristic <math>p   </math> corresponding to Galois orbits of roots of <math>f(x)</math> in the algebraic closure <math>\overline{\mathbb{F}}_p</math>.

The scheme <math>Y</math> is not [[Proper scheme|proper]], so that pairs of curves may fail to [[Bézout's theorem|intersect with the expected multiplicity]]. This is a major obstacle to analyzing [[Diophantine equations]] with [[Diophantine geometry|geometric tools]]. [[Arakelov theory]] overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding to [[Valuation (algebra)|valuations]].