Quasi-finiteness at a prime of a finite-type algebra

Let $R\to S$ be a ring map of finite type (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras), let $\mathfrak{q}\in \mathrm{Spec}(S)$ be a prime, and let $\mathfrak{p}=\mathfrak{q}\cap R$ be its contraction to $R$. Write $\kappa (\mathfrak{p})=R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}\cong \mathrm{Frac}(R\mathrm{/}\mathfrak{p})$ for the residue field ($R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}\cong \mathrm{Frac}(R\mathrm{/}\mathfrak{p})$ is the residue field at $\mathfrak{p}$).

The map $R\to S$ is quasi-finite at $\mathfrak{q}$ when the $\kappa (\mathfrak{p})$-algebra

$S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$

is finite over $\kappa (\mathfrak{p})$, that is, finitely generated as a $\kappa (\mathfrak{p})$-module, equivalently finite-dimensional over $\kappa (\mathfrak{p})$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis); the quotient is a $\kappa (\mathfrak{p})$-algebra because $\mathfrak{p}{S}_{\mathfrak{q}}$ is the extension of the ideal $\mathfrak{p}$. The map $R\to S$ is quasi-finite when it is of finite type and quasi-finite at every prime of $S$.

The fibre form. The fibre of $\mathrm{Spec}(S)\to \mathrm{Spec}(R)$ over $\mathfrak{p}$ is $\mathrm{Spec}(S{\otimes }_R\kappa (\mathfrak{p}))$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums). The prime $\mathfrak{q}$ determines a prime $\bar{\mathfrak{q}}$ of the fibre, and the local ring of the fibre at that prime is $S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$. Indeed the quotient and tensor laws identify $S{\otimes }_R\kappa (\mathfrak{p})\cong S_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{p}}$ (locally on the base use $M{\otimes }_{R}R\mathrm{/}I\cong M\mathrm{/}IM$ naturally over the local ring $R_{\mathfrak{p}}$ and Localisation commutes with quotient rings: $S^{-1}R\mathrm{/}{S}^{-1}I\cong {\bar{S}}^{-1}(R\mathrm{/}I)$), and localising that $\kappa (\mathfrak{p})$-algebra at the prime $\bar{\mathfrak{q}}$ and quotienting by $\mathfrak{p}$ recovers $S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$. Either description may be used as the definition: the two are related by these canonical identifications, and all items on this page use whatever form makes the step at hand shortest.

Conventions kept here. (i) The definition is phrased only at a prime of $S$ and never requires a chosen closed point, so it applies to nonreduced rings, to noninjective maps, and to primes of arbitrarily large residue field. (ii) A fibre may be empty or may have infinitely many primes; quasi-finiteness is a condition at one prime at a time, and the map is quasi-finite only when the condition holds at all primes of $S$. (iii) This is distinct from the classical closed-point convention of Quasi-finite classical morphisms, which tests only closed-point fibres of classical varieties over an algebraically closed field; the algebraic notion above is the one used by the Zariski Main Theorem on this page.