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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 ↪ k {\displaystyle k\left\hookrightarrow k\left} between their coordinate rings, such that k {\displaystyle k\left} is integral over k {\displaystyle k\left}. 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 like y ∈ Y {\displaystyle y\in Y} has an affine neighbourhood V such that U = f − 1 {\displaystyle U=f^{-1}} is affine and f : U → V {\displaystyle f\colon U\to V} is a finite map.