A finite morphism is a morphism whose inverse image of every affine open set is affine and whose corresponding coordinate ring is finite as a module over the base coordinate ring. A nonconstant morphism between smooth projective curves is finite, and its degree is the number of points in a generic fiber counted with multiplicity.
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In algebraic geometry, a **finite morphism** is a type of morphism between algebraic varieties (or schemes) that is analogous to a finite extension of fields in algebra.