Classical algebraic prevarieties, regular maps, and varieties

Fix an algebraically closed field $k$. The affine models used here are polynomial zero sets $Z\subseteq k^n$, including empty and reducible sets. Their closed subsets are common zero loci of polynomial families, relative to $Z$. Intersections combine equations, and finite unions use products of equations, so these are the closed sets of a topology.

For an open $U\subseteq Z$, let ${\mathcal{O}}_Z(U)$ consist of functions $U\to k$ locally equal to $a\mathrm{/}b$ for polynomials $a,b$ with $b$ nonvanishing on the chosen neighbourhood. Pointwise operations and restriction make these sets a sheaf of $k$-algebras: quotient expressions persist on restriction, and functions that agree locally glue uniquely with the same local quotient property. At a point z its stalk is the ring of germs of such functions. Evaluation at z maps it onto k, with kernel the germs vanishing at z. A germ nonzero at z is invertible: shrink to where its numerator and denominator are both nonzero and interchange them. Thus the kernel is the unique maximal ideal. These affine models are locally ringed spaces in the sense of A locally ringed space, with residue fields canonically k.

A classical algebraic prevariety over k is a quasi-compact locally ringed space with a structure sheaf of k-algebras, covered by open subspaces isomorphic over k to these affine models. Here quasi-compact means every open cover has a finite subcover. Equivalently it admits a finite affine-model cover. To justify the converse, each affine model is quasi-compact: polynomial principal opens $D_Z(h)=\{z:h(z)\mathrm{\ne }0\}$ form a basis. Given an open cover, take all h whose principal opens lie in a member. Their principal opens cover Z. The ideal generated by these h in $k[t_1,\dots ,t_n]$ is finitely generated by Hilbert basis theorem: if $R$ is Noetherian then $R[x]$ is Noetherian, iterated from the field k. Express finitely many generators using finitely many of the original h. Those finitely many h have the same common zero locus as the whole ideal, hence their principal opens still cover Z and yield a finite subcover. The empty model has the empty subcover. A finite union of quasi-compact subspaces is quasi-compact.

A map is regular when it is a morphism of locally ringed spaces as in Morphisms of locally ringed spaces whose sheaf maps respect k. For these function sheaves its sheaf map is necessarily pullback of functions: the local stalk map induces a k-map between residue fields k, hence the identity, so the value of a pulled-back section at x is its value at the image of x. Conversely, continuous pullback of local regular functions gives a local stalk map because a function vanishing at the image vanishes after pullback.

Regularity can therefore be checked on affine charts: near every source point choose a target affine chart containing its image and an open neighbourhood in a source affine chart mapped into it; the target coordinate functions must pull back to locally polynomial quotients. This condition also implies continuity. Indeed zero sets of locally polynomial quotients are locally polynomial zero sets, hence closed; pulling back polynomial equations consequently gives closed sets. Substitution then pulls every local quotient on the target back to a local quotient on the source, using this continuity to restrict to its domain and its nonvanishing denominator. Thus the chart condition and the locally ringed-space definition agree.

A prevariety is separated when the equalizer of every pair of regular maps into it, from any classical prevariety, is closed. A classical algebraic variety is a separated prevariety. Here varieties may be reducible or empty; an irreducible classical variety is nonempty and irreducible in the sense of Irreducible topological spaces and irreducible subsets in the subspace topology. These definitions and the local verifications above use no Axiom of Choice.