Editing Scheme (mathematics) (section)

==Examples==
Here and below, all the rings considered are commutative.

=== Affine space ===
Let {{mvar|k}} be an algebraically closed field. The affine space <math>\bar X = \mathbb{A}^n_k</math> is the algebraic variety of all points <math>a=(a_1,\ldots,a_n)</math> with coordinates in {{mvar|k}}; its coordinate ring is the polynomial ring <math>R = k[x_1,\ldots,x_n]</math>. The corresponding scheme <math>X = \mathrm{Spec}(R)</math> is a topological space with the Zariski topology, whose closed points are the maximal ideals <math>\mathfrak{m}_a = (x_1-a_1,\ldots,x_n-a_n)</math>, the set of polynomials vanishing at <math>a</math>. The scheme also contains a non-closed point for each non-maximal prime ideal <math>\mathfrak{p}\subset R </math>, whose vanishing defines an irreducible subvariety <math>\bar V=\bar V(\mathfrak{p})\subset \bar X</math>; the topological closure of the scheme point <math>\mathfrak{p}</math> is the subscheme <math>V(\mathfrak{p})=\{\mathfrak{q}\in X \ \ \text{with}\ \ \mathfrak{p}\subset\mathfrak{q}\}</math>, specially including all the closed points of the subvariety, i.e. <math>\mathfrak{m}_a</math> with <math>a\in \bar V</math>, or equivalently <math>\mathfrak{p}\subset\mathfrak{m}_a</math>. 

The scheme <math>X</math> has a basis of open subsets given by the complements of hypersurfaces, 
<math display="block">U_f = X\setminus V(f) = \{\mathfrak{p}\in X\ \ \text{with}\ \ f\notin \mathfrak{p}\} </math>
for irreducible polynomials <math>f\in R</math>. This set is endowed with its coordinate ring of regular functions 
<math display="block">\mathcal{O}_X(U_f) = R[f^{-1}] = \left\{\tfrac{r}{f^m}\ \ \text{for}\ \ r\in R, \ m\in \mathbb{Z}_{\geq 0}\right\}.</math>
This induces a unique sheaf <math>\mathcal{O}_X</math> which gives the usual ring of rational functions regular on a given open set <math>U</math>. 

Each ring element <math>r=r(x_1,\ldots,x_n)\in R</math>, a polynomial function on <math>\bar X</math>, also defines a function on the points of the scheme <math>X</math> whose value at <math>\mathfrak{p}</math> lies in the quotient ring <math>R/\mathfrak{p}</math>, the ''residue ring''. We define <math>r(\mathfrak{p})</math> as the image of <math>r</math> under the natural map <math>R\to R/\mathfrak{p}</math>. A maximal ideal <math>\mathfrak{m}_a</math> gives the ''residue field'' <math>k(\mathfrak{m}_a)=R/\mathfrak{m}_a\cong k</math>, with the natural isomorphism <math>x_i\mapsto a_i</math>, so that <math>r(\mathfrak{m}_a)</math> corresponds to the original value <math>r(a)</math>. 

The vanishing locus of a polynomial <math>f = f(x_1,\ldots,x_n)</math> is a [[hypersurface]] subvariety <math>\bar V(f) \subset \mathbb{A}^n_k</math>, corresponding to the [[principal ideal]] <math>(f)\subset R</math>. The corresponding scheme is <math display="inline"> V(f)=\operatorname{Spec}(R/(f))</math>, a closed subscheme of affine space. For example, taking {{mvar|k}} to be the complex or real numbers, the equation <math> x^2=y^2(y+1)</math> defines a [[singular point of an algebraic variety#Definition|nodal cubic curve]] in the affine plane <math>\mathbb{A}^2_k</math>, corresponding to the scheme <math>V = \operatorname{Spec} k[x,y]/(x^2-y^2(y+1))</math>.

=== 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>. 

=== 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]].  

=== Arithmetic surfaces ===
If we consider a polynomial <math>f \in \mathbb{Z}[x,y]</math> then the affine scheme <math>X = \operatorname{Spec}(\mathbb{Z}[x,y]/(f))</math> has a canonical morphism to <math>\operatorname{Spec}\mathbb{Z}</math> and is called an [[arithmetic surface]]. The fibers <math>X_p = X \times_{\operatorname{Spec}(\mathbb{Z})}\operatorname{Spec}(\mathbb{F}_p)</math> are then algebraic curves over the finite fields <math>\mathbb{F}_p</math>. If <math>f(x,y) = y^2 - x^3 + ax^2 + bx + c</math> is an [[elliptic curve]], then the fibers over its discriminant locus, where  <math display="block">\Delta_f = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2 = 0 \ \text{mod}\ p,</math>are all singular schemes.<ref>{{Cite web |title=Elliptic curves |url=https://homepages.warwick.ac.uk/~maskal/MA426_EllipticCurves_2018.pdf |page=20}}</ref> For example, if <math>p</math> is a prime number and <math display="block">X = \operatorname{Spec}
    \frac{\mathbb{Z}[x,y]}{(y^2 - x^3 - p)}</math> then its discriminant is <math>-27p^2</math>. This curve is singular over the prime numbers <math>3, p</math>.

=== Non-affine schemes ===
* For any commutative ring ''R'' and natural number ''n'', '''projective space''' <math>\mathbb{P}^n_R</math> can be constructed as a scheme by gluing ''n'' + 1 copies of affine ''n''-space over ''R'' along open subsets. This is the fundamental example that motivates going beyond affine schemes. The key advantage of projective space over affine space is that <math>\mathbb{P}^n_R</math> is [[proper morphism|proper]] over ''R''; this is an algebro-geometric version of compactness. Indeed, [[complex projective space]] <math>\C\mathbb{P}^n</math> is a compact space in the classical topology, whereas <math>\C^n</math> is not.
* A [[homogeneous polynomial]] ''f'' of positive degree in the polynomial ring {{math|1=''R''[''x''<sub>0</sub>, ..., ''x''<sub>''n''</sub>]}} determines a closed subscheme {{math|1=''f'' = 0}} in projective space <math>\mathbb{P}^n_R</math>, called a [[projective hypersurface]]. In terms of the [[Proj construction]], this subscheme can be written as <math display="block"> \operatorname{Proj} R[x_0,\ldots,x_n]/(f).</math> For example, the closed subscheme {{math|1=''x''<sup>3</sup> + ''y''<sup>3</sup> = ''z''<sup>3</sup>}} of <math>\mathbb{P}^2_\Q</math> is an [[elliptic curve]] over the [[rational number]]s.
* The '''line with two origins''' (over a field ''k'') is the scheme defined by starting with two copies of the affine line over ''k'', and gluing together the two open subsets A<sup>1</sup> − 0 by the identity map. This is a simple example of a non-separated scheme. In particular, it is not affine.{{sfn|Hartshorne|1997|loc=Example II.4.0.1}}
* A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let <math>X=\mathbb{A}^n\smallsetminus\{0\}</math>, say over the complex numbers <math>\C</math>; then ''X'' is not affine for ''n'' ≥ 2. (However, the affine line minus the origin is isomorphic to the affine scheme <math>\mathrm{Spec}\,\C[x,x^{-1}]</math>. To show ''X'' is not affine, one computes that every regular function on ''X'' extends to a regular function on <math>\mathbb{A}^n</math> when ''n'' ≥ 2: this is analogous to [[Hartogs's lemma]] in complex analysis, though easier to prove. That is, the inclusion <math>f:X\to\mathbb{A}^n</math> induces an isomorphism from <math>O(\mathbb{A}^n)=\C[x_1,\ldots,x_n] </math> to <math>O(X)</math>. If ''X'' were affine, it would follow that ''f'' is an isomorphism, but ''f'' is not surjective and hence not an isomorphism. Therefore, the scheme ''X'' is not affine.{{sfn|Hartshorne|1997|loc=Exercises I.3.6 and III.4.3}}
* Let ''k'' be a field. Then the scheme <math display="inline">\operatorname{Spec}\left(\prod_{n=1}^\infty k\right)</math> is an affine scheme whose underlying topological space is the [[Stone–Čech compactification]] of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the [[ultrafilter]]s on the positive integers, with the ideal <math display="inline">\prod_{m \neq n} k</math> corresponding to the principal ultrafilter associated to the positive integer ''n''.{{sfn|Arapura|2011|loc=section 1}} This topological space is [[Krull dimension|zero-dimensional]], and in particular, each point is an [[irreducible component]]. Since affine schemes are [[quasi-compact]], this is an example of a non-Noetherian quasi-compact scheme with infinitely many irreducible components. (By contrast, a [[Noetherian scheme]] has only finitely many irreducible components.)

=== Examples of morphisms ===
{{expand section|date=March 2024}}
It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.