A morphism from an open subset of a classical affine variety to an affine variety

Let $U$ be open in an affine algebraic set $X$ and let $Y$ be an affine algebraic set. A set map $\varphi :U\to Y$ is a morphism over $k$ if $s\circ \varphi \in {\mathcal{O}}_X(U)$ for every $s\in {\mathcal{O}}_Y(Y)$. For a map whose target is an open subset of an affine algebraic set, the phrase locally regular morphism means a continuous map pulling regular functions on each target open back to regular functions on its inverse image. An isomorphism between open subsets of affine algebraic sets is a bijection for which the map and its inverse are locally regular morphisms. Continuity and this local pullback property for the affine-target definition will be established in the next lemma; they are not assumed in the affine-target test.

Source comparison: Milne, Algebraic Geometry, v6.10, §3d and Proposition 3.26, pp. 64–67. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.