Finite morphism

In algebraic geometry, a finite morphism between two affine varieties ${\displaystyle X,Y}$ is a dense regular map which induces isomorphic inclusion ${\displaystyle k\left[Y\right]\hookrightarrow k\left[X\right]}$ between their coordinate rings, such that ${\displaystyle k\left[X\right]}$ is integral over ${\displaystyle k\left[Y\right]}$.^[1] This definition can be extended to the quasi-projective varieties, such that a regular map ${\displaystyle f\colon X\to Y}$ between quasiprojective varieties is finite if any point ${\displaystyle y\in Y}$ has an affine neighbourhood V such that ${\displaystyle U=f^{-1}(V)}$ is affine and ${\displaystyle f\colon U\to V}$ is a finite map (in view of the previous definition, because it is between affine varieties).^[2]