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An affine algebraic curve is an affine variety of Krull dimension one. An irreducible affine curve has a finitely generated one-dimensional domain as its coordinate ring. Quotienting this ring by a nonzero element gives a zero-dimensional Noetherian ring, hence an Artinian ring. Consequently a nonzero regular function has finitely many zeros. No nonsingularity assumption is part of the definition.

An open subset of an algebraic variety equipped with an identification with an affine variety. Such charts make quasi-coherent sheaves and regular maps accessible through their local coordinate rings.

For an affine variety with coordinate ring $A$ and an $A$-module $M$, localization defines sections on principal open subsets by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of stalk values which locally come from one fraction $m/f^r$. These data form a quasi-coherent sheaf, with global sections $M$ and stalk $M_{{\mathfrak{m}}_P}$ at $P$. Sheaf tensor products correspond to tensor products over $A$; pullback of a sheaf of modules corresponds to extension of scalars, and affine direct image to restriction of scalars.

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.

A classical variety is affine if finitely many global regular functions generate the unit ideal and their nonvanishing opens are affine. Localization of global sections on a principal open identifies their coordinate rings with $A_{f_i}$, where $A=\Gamma (X,{\mathcal{O}}_X)$. Choose a finite-type subalgebra of $A$ containing the functions, their unit-ideal coefficients and numerators of generators of every $A_{f_i}$. Its affine variety has the same principal affine charts, whose isomorphisms glue globally. This avoids assuming finite generation of $A$ before affineness is known.

For a classical algebraic variety $X$ and affine variety $Y$, a homomorphism on the indicated rings gives a morphism by evaluating the images of affine coordinate generators at each point of $X$. The relations defining $Y$ remain zero. Local fractions pull back to local regular fractions, proving regularity; the coordinate ring generators also prove uniqueness.

For a field $k$, a homomorphism of $k$-algebras is a unital ring homomorphism preserving scalar multiplication by every element of $k$. For unital algebras it is equivalently a ring homomorphism commuting with their structural maps from $k$. Pullback by a morphism of algebraic varieties gives such a homomorphism between their coordinate rings.

For an affine variety with coordinate ring $A$ acted on by a finite group $G$, the affine quotient is $\mathrm{Spec}{A}^G$. Elements of $A$ satisfy monic orbit polynomials over $A^G$, so the quotient map is a finite morphism. Over an algebraically closed field, invariant functions separate finite orbits; in characteristic not dividing $|G|$, interpolate and average a function separating two orbits. The quotient fibres are consequently the orbits.

The morphism ${\mathbb{A}}^1\to V(x^4-y^3)$ given by $t\mapsto (t^3,t^4)$ is bijective on points over an algebraically closed field, but is not an isomorphism. At a nonzero curve point, $t=y/x$ is the unique inverse parameter, while the origin has unique preimage zero. The induced map on coordinate rings has image $k[t^3,t^4]\subset k[t]$, which lacks the monomial $t$: all its monomial exponents have the form $3a+4b$, with $a,b\ge 0$. Nor can $t$ be regular at the cusp, since an equality $t=P(t^3,t^4)/Q(t^3,t^4)$ with $Q(0)\ne 0$ would produce a nonzero degree-one term in $tQ$ and none in $P$. Thus pointwise bijectivity does not guarantee an algebraic inverse, even though the inverse is regular away from one singular point.

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.

For affine varieties $X,Y$ over an algebraically closed field $k$, their product in a category is affine, with coordinate ring $A{\otimes }_{k}B$, where $A=k[X]$ and $B=k[Y]$. Presenting $A=k[x]/J$ and $B=k[y]/L$ identifies this ring with $k[x,y]/(J,L)$. If both varieties are irreducible, this is an integral domain: express a nonzero tensor $F$ using linearly independent second factors. The points $x$ where its specialization $F_x$ is nonzero form a nonempty open subset of $X$. For two nonzero tensors $F,G$, choose $x$ in the intersection of these open subsets; then $F_x{G}_x\ne 0$ in the domain $B$. The Hilbert Nullstellensatz identifies $(J,L)$ with the defining ideal of $X\times Y$. The universal property follows from the universal property of the tensor product of modules as a tensor product of commutative algebras.

A regular function $f$ on an entire affine variety is a polynomial function, even if the variety is reducible. Put $J=\{g\in A(X):gf\in A(X)\}$, an ideal of its coordinate ring. Near a point $p$, write $f=a/b$ with $b$ nonzero, and choose a principal open subset $D(h)$ containing $p$ inside that neighbourhood. Then $(hb)f=ha$ on all of $X$: on $D(h)$ this is the fraction identity, and off $D(h)$ both sides vanish. Hence $hb\in J$ and $(hb)(p)\ne 0$. No point is a common zero of $J$, so the weak Hilbert Nullstellensatz implies $J=A(X)$. Thus $1\in J$ and $f\in A(X)$. This proof does not require a fraction field for a reducible coordinate ring.

In an affine variety with coordinate ring $A$, every dense open subset $U$ contains a dense principal open subset. Write its closed complement as $V(I)$. Density says that $I$ is contained in none of the finitely many minimal prime ideals of the reduced ring $A$. Prime avoidance supplies $s\in I$ outside all these primes. Thus $D(s)\subseteq U$, and $s$ is a non-zero-divisor. Noetherian Zariski topology also makes $U$ a finite union of principal opens; one can add $D(s)$ to this cover. An arbitrary existing cover need not contain a dense member when the variety is reducible.

The supremum of the lengths of strict chains of nonempty irreducible closed subsets. It is the maximum of the dimensions of the irreducible components. For an affine algebraic set it equals the Krull dimension of its coordinate ring; for a quasi-projective algebraic set it is the supremum of these dimensions over affine open subsets.

For a normal variety with affine coordinate ring $A$ and a Weil divisor $D$, the fractional ideal of functions whose principal Weil divisor is at least $D$ isFor an effective prime divisor this is its height-one prime ideal. If the divisor is Cartier locally, this ideal is locally principal. A nonprincipal height-one prime ideal in a normal local ring can therefore detect a non-Cartier divisor.

A nonempty closed subset which, with its induced topology, is an irreducible topological space. In an affine algebraic set these subsets correspond, with reversed inclusion, to prime ideals of the coordinate ring.

The algebraic sethas the two irreducible componentsEach is irreducible because the linear change $u=x-z$, $v=x+z$ identifies its coordinate ring withan integral domain.

An algebraic variety whose underlying topological space with its Zariski topology is an irreducible topological space: it is nonempty and cannot be a union of two proper closed subsets. For a reduced affine variety, this is equivalent to its coordinate ring being an integral domain.

An affine algebraic group is an affine variety $G$ whose multiplication and inversion are regular maps. Equivalently, its coordinate ring $k[G]$ is a commutative Hopf algebra, with comultiplication induced by multiplication in $G$.

For a quasi-compact separated scheme and a global regular function $f$, global sections on $X_f$ are the localization of global sections by $f$. Choose a finite affine open cover. Its intersections are affine, so the sheaf gluing axiom describes global sections as the kernel of the difference map between two finite products of affine coordinate rings. Exactness of localization preserves this kernel, and localization commutes with those finite products. Localizing the same cover gives the claimed isomorphism. This applies before knowing whether $X$ itself is affine.