Finite morphism

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.

For a finite generically separable morphism between smooth projective curves, the field trace on rational differentials identifies the differential sheaf with the trace-dual module. Locally, for $A=k[[s]]\subset B=k[[u]]$ with $s=f(u)$, the trace-dual fractional ideal is $f^{\prime }(u{)}^{-1}B$. Multiplying it by $ds=f^{\prime }(u)du$ gives $Bdu$. For a monogenic finite extension with monic polynomial $P$, the identity $\mathrm{Tr}(g(u)/P^{\prime }(u))=[T^{e-1}](g(T)\mathrm{mod}P(T))$ proves the dual-module formula: the top-coefficient pairing on the basis $1,u,\dots ,u^{e-1}$ has invertible antitriangular matrix. Tensoring locally with an invertible sheaf yields ${\pi }_{\ast }({\mathcal{M}}^{\vee }\otimes {\Omega }_C^1)\cong \mathcal{H}om({\pi }_{\ast }\mathcal{M},{\Omega }_D^1)$.

A finite morphism is of finite type and separated: on affine charts the multiplication map $B{\otimes }_{A}B\to B$ is surjective and represents its closed diagonal. Each element of a finite algebra satisfies a monic equation by the determinant trick. In a valuative lifting problem, its image in the fraction field is therefore integral over the valuation ring and belongs to that ring, since valuation rings are integrally closed. This gives a lift, unique because the valuation ring embeds in its fraction field. The valuative criterion for properness proves properness.