Editing Scheme (mathematics) (section)

=== 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>.