Classical algebraic prevarieties, regular maps, and varieties

Fix an algebraically closed field $k$. A classical algebraic prevariety over $k$ is a quasi-compact locally ringed space over $k$ that is covered by open subspaces isomorphic, as locally ringed spaces over $k$, to affine algebraic sets with their regular-function sheaves. Equivalently, it admits a finite such affine cover. A map is regular when it is a morphism of these locally ringed spaces over $k$; equivalently, this can be checked on affine charts. Thus its maps on structure sheaves respect the fixed $k$-algebra structures.

A prevariety is separated when the equalizer of every pair of regular maps into it is closed. A (classical) algebraic variety is a separated prevariety. In the comparison below, “irreducible classical variety” means a nonempty variety whose underlying topological space is irreducible.