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In a vector space over a field $k$, an affine line is a translate of a one-dimensional vector subspace. Its direction is that vector subspace, so multiplying $v$ by a nonzero scalar preserves the direction. Over a finite field of size $q$ it contains exactly $q$ distinct points. This usage concerns affine subsets of vector spaces, rather than the affine line as an affine scheme.

On an affine variety or affine scheme with coordinate ring $A$, the global section functor and $M\mapsto \tilde{M}$ are inverse equivalences between modules and quasi-coherent sheaves. On every principal open subset, sections are the appropriate localization of a module. Localization detects exactness at maximal ideals, so these functors are exact even for modules which are not finitely generated.

Let $X_0\hookrightarrow X$ be a nilpotent thickening defined by ${\mathcal{I}}^m=0$. If $X$ is an affine scheme, its closed subscheme $X_0$ is an affine scheme. Conversely, suppose $X_0$ is an affine scheme. The underlying topological spaces are identical, so $X$ is a quasi-compact scheme. For any quasi-coherent sheaf $\mathcal{F}$, each subquotient ${\mathcal{I}}^j\mathcal{F}/{\mathcal{I}}^{j+1}\mathcal{F}$ is naturally a quasi-coherent sheaf on $X_0$. By sheaf cohomology under a closed inclusion and vanishing of quasi-coherent cohomology on an affine scheme, its first sheaf cohomology on $X$ vanishes. The long exact sequence in sheaf cohomology and descending mathematical induction through this finite filtration prove $H^1(X,\mathcal{F})=0$. The Serre criterion for affineness now makes $X$ an affine scheme.

An affine open subscheme is an open subset of a scheme with its restricted structure sheaf, which is itself an affine scheme. Every scheme has an open cover by such subschemes. On an affine scheme, the principal open subschemes give an open basis.

The affine plane over a field $k$ is the affine scheme $\mathrm{Spec}k[x,y]$.

The two-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

Let $P$ have an affine open neighbourhood $U$, and let $Y=X\setminus U$. Evaluation gives ${\mathcal{I}}_Y\twoheadrightarrow k_P$ with kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. If first sheaf cohomology of every coherent ideal sheaf vanishes, a global section $f$ of ${\mathcal{I}}_Y$ can be chosen with $f(P)=1$. Then $X_f\subseteq U$ is a principal open subset of an affine variety and is affine.

For an affine scheme $X=\mathrm{Spec}A$, its ring of global sections is $A$. A morphism of schemes between affine schemes is determined contravariantly by its map on these rings. Consequently a morphism between affine schemes is an isomorphism of schemes exactly when that ring map is an isomorphism. The affine hypothesis is essential, as the punctured affine plane shows.

A morphism of schemes into an affine scheme is uniquely determined by its homomorphism on global sections. Conversely, a ring homomorphism $A\to \Gamma (X,{\mathcal{O}}_X)$ restricts on every affine open subscheme of $X$ to a map into $\mathrm{Spec}A$; equality on affine covers of overlaps glues these maps. No quasi-compactness or affineness assumption on $X$ is needed.

Restricting the functor of points to affine schemes gives a fully faithful functor from schemes to covariant set-valued functors on commutative rings. Given a natural transformation, apply it to the inclusions of an affine open cover of $X$. Naturality on affine open subschemes of overlaps makes the resulting maps to $Y$ agree, so they glue. For an arbitrary map from an affine scheme to $X$, cover its inverse images of these charts by affine open subschemes and apply naturality again. This proves that the glued map induces the prescribed natural transformation, and uniqueness follows from the original cover. In particular, the restricted functor of points determines the scheme up to an isomorphism of schemes.

A line bundle $L$ is ample when some positive tensor power $L^{\otimes m}$ is very ample.

A point of a topological space is closed when its singleton is a closed set. In a T1 space every point is closed. The closed points of an affine scheme $\mathrm{Spec}A$ are its maximal ideals; the closure of the point corresponding to $\mathfrak{p}$ is $V(\mathfrak{p})$.

For a commutative ring $R$, remove the section $[1:0:0]$ from ${\mathbb{P}}_R^2$. The two remaining standard charts have coordinate rings $R[a,b]$ and $R[a/b,b^{-1}]$ inside the Laurent polynomial ring $R[a,b,b^{-1}]$. Their intersection consists of constants: their monomial exponent regions are $i\ge 0,j\ge 0$ and $i\ge 0,j\le -i$, respectively. The Čech cohomology in degree one is free over $R$ on the missing monomials $a^i{b}^{-k}$ with $1\le k<i$. Equivalently its basis consists of the classes of $x_0^{m+n}/(x_1^m{x}_2^n)$ for $m,n\ge 1$. Degree zero is $R$ and every degree at least two is zero. The charts and their intersection are affine schemes, so this computes sheaf cohomology. If $R$ is nonzero, the first sheaf cohomology is nonzero and this scheme is not an affine scheme.

The dual-number ring over a field $k$ consists of elements $a+b\epsilon$ with ${\epsilon }^2=0$. The element $\epsilon$ is a nonzero nilpotent element; its ideal is the ring's only prime ideal. Hence its affine scheme has the same one-point topology as $\mathrm{Spec}k$ while its structure sheaf differs.

Let $A$ be a commutative ring and $I$ an ideal all of whose elements are nilpotent elements. Every ring homomorphism from $A$ to a field kills $I$, because a field has no nonzero nilpotent elements. Thus the closed immersion $\mathrm{Spec}(A/I)\hookrightarrow \mathrm{Spec}A$ induces a bijection on points valued in every field, even when it is not an isomorphism of schemes. The dual numbers provide the smallest example. Testing the functor of points on all affine schemes does detect this information, by affine test schemes determine scheme morphisms.

An algebra over a commutative ring $A$ is finitely generated if it is a quotient ring of a polynomial ring $A[t_1,\dots ,t_n]$ in finitely many variables. Equivalently, finitely many elements generate it under addition, multiplication, and the action of $A$. The induced map of affine schemes $\mathrm{Spec}B\to \mathrm{Spec}A$ is a morphism of finite type exactly when this holds. Finite generation as an algebra does not require finite generation as an $A$-module.

For a closed subscheme with ideal sheaf $\mathcal{I}$ in a locally Noetherian scheme, the completion is the formal scheme on $|Y|$ with topological structure sheaf ${\underleftarrow{\lim }}_n({\mathcal{O}}_X/{\mathcal{I}}^{n+1}){|}_{|Y|}$. On a Noetherian affine scheme $\mathrm{Spec}A$, it is $\mathrm{Spf}({\underleftarrow{\lim }}_{n}A/I^{n+1})$. Its adic topology remembers every infinitesimal neighbourhood.

// Target: commutative-algebra.bigb

For a morphism of schemes $X\to S$ with $S$ an integral scheme and generic point $\eta$, the generic fiber is its scheme-theoretic fiber over $\eta$. On affine schemes $S=\mathrm{Spec}A$ and $X=\mathrm{Spec}B$, it is $\mathrm{Spec}(B{\otimes }_A\mathrm{Frac}A)$. It can be empty, and it need not be a reduced scheme or an irreducible scheme without further assumptions.

A generic point is a point whose closure is the whole space. An integral scheme has a unique generic point; on an affine scheme associated to an integral domain, it corresponds to the zero prime ideal. Its local ring is the fraction field of the domain. On every nonempty affine chart of an integral scheme these fields identify with the same function field.

A global regular function is a global section of the structure sheaf of a scheme, defined on the entire scheme. These functions form its ring $H^0(X,{\mathcal{O}}_X)$. On an affine scheme this recovers the defining ring; on a positive-dimensional projective complete intersection it consists only of constants over the base field.

For $X=V_{+}(I)\subseteq {\mathbb{P}}^n$, the homogeneous coordinate ring is the graded ring $k[x_0,\dots ,x_n]/I$.