In algebraic geometry, a finite morphism between two affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion k [ Y ] ↪ k [ X ] {\displaystyle k\left[Y\right]\hookrightarrow k\left[X\right]} between their coordinate rings, such that k [ X ] {\displaystyle k\left[X\right]} is integral over k [ Y ] {\displaystyle k\left[Y\right]} . This definition can be extended to the quasi-projective varieties, such that a regular map f : X → Y {\displaystyle f\colon X\to Y} between quasiprojective varieties is finite if any point y ∈ Y {\displaystyle y\in Y} has an affine neighbourhood V such that U = f − 1 ( V ) {\displaystyle U=f^{-1}(V)} is affine and f : U → V {\displaystyle f\colon U\to V} is a finite map (in view of the previous definition, because it is between affine varieties).