 Isomorphic local rings determine isomorphic neighborhoods

A $k$-algebra isomorphism between the local rings at closed points extends to a morphism on some neighborhoods: represent the images of finitely many affine coordinate generators by fractions, and shrink where their denominators are nonzero and their finitely many relations hold. Extend its inverse the same way. The two compositions agree with the identity as germs; finitely many coordinate equalities make them identities after shrinking again. Taking inverse images of these neighborhoods then gives mutually inverse actual neighborhood isomorphisms, not just a formal-completion isomorphism.