Algebraic Zariski Main for Quasi-Finite Morphisms: Examples

Given: A field $k$, the polynomial ring $R=k[t]$, the principal localisation $S=R_t=k[t,t^{-1}]$ of $R$ at $t$, and the Axiom of Choice.

The map $R\to S$ is quasi-finite at $\mathfrak{q}$ when $S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$ is finite over $\kappa (\mathfrak{p})$, the map is quasi-finite when it is of finite type and quasi-finite at every prime, and the fibre of $\mathrm{Spec}(S)\to \mathrm{Spec}(R)$ over $\mathfrak{p}$ is $\mathrm{Spec}(S{\otimes }_R\kappa (\mathfrak{p}))$, the local ring at the prime over $\mathfrak{p}$ being $S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$ (Quasi-finiteness at a prime of a finite-type algebra).

An $R$-algebra $A$ is of finite type over $R$ when $A=R[a_1,\dots ,a_n]$ for finitely many elements, and module-finite over $R$ when it is finitely generated as an $R$-module (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras).

For $f\in R$ the principal localisation is $R_f=S_f^{-1}R$ with $S_f=\{1,f,f^2,\dots \}$, and its elements may be written $r\mathrm{/}{f}^n$; in particular $R_1$ is canonically isomorphic to $R$ (Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$).

For $f\in R$ the localisation map $R\to R_f$ induces a homeomorphism from $\mathrm{Spec}(R_f)$ onto the distinguished open subset $D(f)$ (The spectrum of a principal localisation is the distinguished open D(f)).

For a unital ring map $A\to C$ and a multiplicative subset $M\subseteq A$ there is a ring isomorphism $(M^{-1}A){\otimes }_{A}C\cong {\bar{M}}^{-1}C$, with no flatness, finite-generation or nonzero-ring hypothesis (Presentations and localization under base extension).

For multiplicative subsets $S,T\subseteq R$, with $\bar{T}$ the image of $T$ in $S^{-1}R$ and $U$ generated by $S\cup T$, there is an isomorphism ${\bar{T}}^{-1}(S^{-1}R)\cong U^{-1}R$ (Localising twice is localising once at the multiplicative set generated by both denominator sets).

For a prime ideal $\mathfrak{p}$ of $R$ there is a canonical field isomorphism $R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}\cong \mathrm{Frac}(R\mathrm{/}\mathfrak{p})$, the residue field $\kappa (\mathfrak{p})$ ($R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}\cong \mathrm{Frac}(R\mathrm{/}\mathfrak{p})$ is the residue field at $\mathfrak{p}$).

Contraction along the localisation map $R\to S^{-1}R$ is an inclusion-preserving bijection from $\mathrm{Spec}(S^{-1}R)$ onto the primes of $R$ disjoint from $S$, with inverse $\mathfrak{p}\mapsto S^{-1}\mathfrak{p}$ (Prime ideals of a localization are exactly the primes disjoint from the denominator set).

Assume the Axiom of Choice. If $R\to S$ is of finite type and quasi-finite at every prime of $S$ and $S^{\mathrm{\prime }}$ is the integral closure of the image of $R$ in $S$, then there are a finite $R$-subalgebra $T\subseteq S^{\mathrm{\prime }}$ and finitely many $g_1,\dots ,g_n\in T$ with the contraction map $\mathrm{Spec}(S)\to \mathrm{Spec}(T)$ a homeomorphism onto the open set $U=D_T(g_1)\cup \cdots \cup D_T(g_n)$, $T_{g_i}\cong S_{g_i}$, and $T_g\cong S_g$ for every $g\in T$ with $D_T(g)\subseteq U$ (A quasi-finite algebra factors openly through a finite algebra).

An element $b$ of a commutative ring $B$ is integral over a subring $A\subseteq B$ when it is a root of a monic polynomial in $A[X]$ (Integral elements over a commutative ring and algebraic integers).

For a unital ring map $R\to S$ the relative integral closure ${\mathrm{Int}}_R(S)$ is the subring of $S$ consisting of the elements integral over the map; it contains the image of $R$ and is the integral closure of that image in $S$ (Integral elements subalgebra of an arbitrary ring map).

In the localisation $S^{-1}R$ two fractions are equal, $r\mathrm{/}s=r^{\mathrm{\prime }}\mathrm{/}{s}^{\mathrm{\prime }}$, if and only if $v(r{s}^{\mathrm{\prime }}-r^{\mathrm{\prime }}s)=0$ for some $v\in S$, and every $s\in S$ maps to a unit (Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions).

If $R$ is an integral domain then so is $R[x]$; in particular $k[t]$ is a domain (A polynomial ring over an integral domain is an integral domain).

The Axiom of Choice (AC) is the statement that every family of nonempty sets has a choice function (The Axiom of Choice).

Given: A ring map $R\to S$ such that $S$ is module-finite over $R$, i.e. a finite $R$-algebra, with $A=\mathrm{Im}(R)\subseteq S$ the image of the structure map, and the Axiom of Choice.

An $R$-algebra $A$ is module-finite over $R$ when it is finitely generated as an $R$-module (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras).

An element $b$ of a commutative ring $B$ is integral over a subring $A\subseteq B$ when it is a root of a monic polynomial in $A[X]$ (Integral elements over a commutative ring and algebraic integers).

The Axiom of Choice (AC) is the statement that every family of nonempty sets has a choice function (The Axiom of Choice).

Let $A\subseteq B$ be commutative rings with $A\mathrm{\ne }0$ and $b\in B$. Then $b$ is integral over $A$ if and only if there exists a faithful $A[b]$-module that is finitely generated over $A$, faithfulness meaning that $rM=0$ implies $r=0$ for $r\in A[b]$ (Integrality and finite-module characterizations for one element).

For a unital ring map $R\to S$ the relative integral closure ${\mathrm{Int}}_R(S)$ is the subring of $S$ of elements integral over the map; it contains the image of $R$ and is exactly the integral closure of that image in $S$ (Integral elements subalgebra of an arbitrary ring map).

A proper ideal $P⊊R$ is prime when $ab\in P$ implies $a\in P$ or $b\in P$ (Prime ideals and maximal ideals in a commutative ring).

For $f\in R$ the principal localisation is $R_f=S_f^{-1}R$ with $S_f=\{1,f,f^2,\dots \}$; in particular $R_1$ is canonically isomorphic to $R$ (Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$).

For every ring homomorphism $\phi :R\to A$ contraction defines a continuous map $\mathrm{Spec}(A)\to \mathrm{Spec}(R)$, and $\mathrm{Spec}$ is a contravariant functor, so the identity ring map induces the identity on spectra (The prime-spectrum construction is a contravariant functor to topological spaces).

The subsets $V(I)$ of $\mathrm{Spec}(R)$ contain $\mathrm{Spec}(R)$ and $\mathrm{\varnothing }$ and define a topology on $\mathrm{Spec}(R)$ (The vanishing sets define the Zariski topology on the prime spectrum).

For $f\in R$ one has $D(0)=\mathrm{\varnothing }$ and $D(1)=\mathrm{Spec}(R)$, and $D(fg)=D(f)\cap D(g)$ (Distinguished-subset identities).

Assume the Axiom of Choice. For a finite type map $R\to S$ quasi-finite at $\mathfrak{q}\in \mathrm{Spec}(S)$ there is $g\in S^{\mathrm{\prime }}\setminus \mathfrak{q}$ with $S^{\mathrm{\prime }}\to S$ inducing an isomorphism $S_g^{\mathrm{\prime }}\cong S_g$ (Algebraic Zariski Main localization at a quasi-finite prime).

Assume the Axiom of Choice. For a finite type map $R\to S$ quasi-finite at every prime, with $S^{\mathrm{\prime }}$ the integral closure of the image of $R$ in $S$, there are a finite $R$-subalgebra $T\subseteq S^{\mathrm{\prime }}$ and finitely many $g_1,\dots ,g_n\in T$ with the contraction map $\mathrm{Spec}(S)\to \mathrm{Spec}(T)$ a homeomorphism onto the open set $U=D_T(g_1)\cup \cdots \cup D_T(g_n)$, $T_{g_i}\cong S_{g_i}$, and $T_g\cong S_g$ whenever $D_T(g)\subseteq U$ (A quasi-finite algebra factors openly through a finite algebra).

Given: A field $k$, the polynomial ring $R=k[t]$, the principal localisation $B=k[t,t^{-1}]=R_t$ of $R$ at $t$, the product ring $S=R\times B$ with the diagonal $R$-algebra structure $r\mapsto (r,r)$, the elements $u=(0,t^{-1})$, $e=(1,0)$ and $\bar{t}=(t,t)$ of $S$, and the Axiom of Choice.

For a commutative ring $R$ the polynomial ring $R[x]$ is the set of finitely supported coefficient functions with convolution product, the indeterminate $x$ being the coefficient function supported at $1$ with value $1_R$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

If $R$ is an integral domain then $R[x]$ is an integral domain; in particular $k[t]$ is a domain with $0\mathrm{\ne }1$ (A polynomial ring over an integral domain is an integral domain).

A field is a commutative unital ring with $0\mathrm{\ne }1$ in which every nonzero element has a multiplicative inverse (Field).

The product ring $R\times B$ carries componentwise operations and identity $(1_R,1_B)$, and $(u,v)$ is a unit of $R\times B$ if and only if $u$ and $v$ are units, with $(u,v{)}^{-1}=(u^{-1},v^{-1})$ (The product ring $R\times S$ with componentwise operations, its identity $(1_R,1_S)$ and its units $R^{\times }\times S^{\times }$).

For $f\in R$ the principal localisation is $R_f=S_f^{-1}R$ with $S_f=\{1,f,f^2,\dots \}$, its elements may be written $r\mathrm{/}{f}^n$, $R_1$ is canonically isomorphic to $R$, and $R_0$ is the zero ring (Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$).

In a localisation $S^{-1}R$ the classes are fractions $r\mathrm{/}s$, two fractions are equal exactly when $u(r{s}^{\mathrm{\prime }}-r^{\mathrm{\prime }}s)=0$ for some $u\in S$, every $s\in S$ maps to a unit with $(s\mathrm{/}1{)}^{-1}=1\mathrm{/}s$, and if $0\in S$ the localisation is the zero ring (Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions).

Let $A$ be a commutative $R$-algebra. For $a_1,\dots ,a_n\in A$ its image under the evaluation homomorphism is written $R[a_1,\dots ,a_n]$ and is the smallest subring of $A$ containing the image of $R$ and the $a_i$; $A$ is of finite type over $R$ when $A=R[a_1,\dots ,a_n]$ for finitely many elements, and module-finite over $R$ when $A$ is finitely generated as an $R$-module (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras).

For $0\mathrm{\ne }f\in R[x]$ the degree $\mathrm{deg}f$ is the largest $n$ with $a_n\mathrm{\ne }0$ and $\mathrm{lc}(f)=a_{\mathrm{deg}f}$; the zero polynomial has no degree (Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree).

For nonzero $f,g\in R[x]$ the coefficient of $x^{\mathrm{deg}f+\mathrm{deg}g}$ in $fg$ is $\mathrm{lc}(f)\mathrm{lc}(g)$, and if $fg\mathrm{\ne }0$ then $\mathrm{deg}(fg)\le \mathrm{deg}f+\mathrm{deg}g$ (Degree inequalities for sums and products over a commutative ring).

An element $b$ of a commutative ring $B$ is integral over a subring $A\subseteq B$ when it is a root of a monic polynomial in $A[X]$ (Integral elements over a commutative ring and algebraic integers).

For a unital ring map $R\to S$ with image $\phi (R)$, the relative integral closure ${\mathrm{Int}}_R(S)$ is the set of $s\in S$ satisfying a monic equation $s^d+\phi (a_{d-1})s^{d-1}+\cdots +\phi (a_0)=0$, and it is a subring of $S$ containing $\phi (R)$ (Integral elements subalgebra of an arbitrary ring map).

A domain $A$ is integrally closed when every element of $\mathrm{Frac}(A)$ that is integral over $A$ already lies in $A$ (Integral closure in an extension ring and integrally closed domains).

For an integral domain $D$ the field of fractions is $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$, with elements fractions $a\mathrm{/}b$ for $a,b\in D$, $b\mathrm{\ne }0$ (The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain).

If $R$ is an integrally closed domain then the polynomial ring $R[x]$ is integrally closed (Polynomial rings over normal domains are normal).

Let $A\subseteq B$ be commutative rings with $A\mathrm{\ne }0$ and let $b\in B$. Then $b$ is integral over $A$ if and only if $A[b]$ is finitely generated as an $A$-module (Integrality and finite-module characterizations for one element).

For a prime $\mathfrak{p}$ of $R$ there is a canonical field isomorphism $R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}\cong \mathrm{Frac}(R\mathrm{/}\mathfrak{p})$; this is the residue field $\kappa (\mathfrak{p})$ ($R_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{R}_{\mathfrak{p}}\cong \mathrm{Frac}(R\mathrm{/}\mathfrak{p})$ is the residue field at $\mathfrak{p}$).

For commutative rings $R,S$, a unital ring map $\phi :R\to S$ and $s\in S$ there is a unique unital ring homomorphism ${\mathrm{ev}}_{\phi ,s}:R[x]\to S$ extending $\phi$ on constants and sending $x$ to $s$, given by ${\mathrm{ev}}_{\phi ,s}({\sum }_i{a}_i{x}^i)={\sum }_i\phi (a_i)s^i$ (Universal property of $R[x]$: a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism).

If $g\in R[x]$ is monic and $f\in R[x]$, there are unique $q,r\in R[x]$ with $f=qg+r$ and $r=0$ or $\mathrm{deg}r<\mathrm{deg}g$ (Division by a monic polynomial over a commutative ring).

A ring homomorphism with kernel $I$ induces an isomorphism from $R\mathrm{/}I$ onto its image (First isomorphism theorem for rings: $R\mathrm{/}\ker f\cong \mathrm{im}f$).

For a family of $R$-modules $(M_i)$ and an $R$-module $N$ there is a natural isomorphism ${⨁}_i(M_i{\otimes }_{R}N)\cong ({⨁}_i{M}_i){\otimes }_{R}N$; in particular tensoring distributes over finite direct sums (Tensor products commute with arbitrary direct sums).

For every $R$-module $N$ the multiplication map $R{\otimes }_{R}N\to N$, $r\otimes n\mapsto rn$, is an isomorphism (The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$).

For a unital ring map $A\to C$ and a multiplicative subset $M\subseteq A$ there is a ring isomorphism $(M^{-1}A){\otimes }_{A}C\cong {\bar{M}}^{-1}C$, where $\bar{M}$ is the image of $M$ in $C$ (Presentations and localization under base extension).

For a finite-type map $R\to S$ and $\mathfrak{q}\in \mathrm{Spec}(S)$ with $\mathfrak{p}=\mathfrak{q}\cap R$, the map is quasi-finite at $\mathfrak{q}$ when $S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$ is finite over $\kappa (\mathfrak{p})$, and quasi-finite when it is finite type and quasi-finite at every prime; the fibre over $\mathfrak{p}$ is $\mathrm{Spec}(S{\otimes }_R\kappa (\mathfrak{p}))$, the prime $\mathfrak{q}$ determines a prime $\bar{\mathfrak{q}}$ of the fibre, and the local ring of the fibre at that prime is $S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$, either description being usable as the definition (Quasi-finiteness at a prime of a finite-type algebra).

For $f\in R$ the principal distinguished subset is $D(f)=\{\mathfrak{p}\in \mathrm{Spec}(R):f\mathrm{\notin }\mathfrak{p}\}$ (Principal distinguished subsets of the prime spectrum).

For a multiplicative subset $S\subseteq R$ contraction along the localisation map $R\to S^{-1}R$ is a homeomorphism from $\mathrm{Spec}(S^{-1}R)$ onto $\{\mathfrak{p}\in \mathrm{Spec}(R):\mathfrak{p}\cap S=\mathrm{\varnothing }\}$ (The spectrum of a localisation is the subspace of primes disjoint from the denominator set).

Let $R$ be a commutative ring, $I⊴R$ an ideal and $\pi :R\to R\mathrm{/}I$ the quotient map. Then contraction along $\pi$ is a homeomorphism from $\mathrm{Spec}(R\mathrm{/}I)$ onto the closed subset $V(I)\subseteq \mathrm{Spec}(R)$ (The spectrum of a quotient is a closed subspace).

Contraction along a ring map is a continuous map, $\mathrm{Spec}({\mathrm{id}}_R)={\mathrm{id}}_{\mathrm{Spec}(R)}$ and $\mathrm{Spec}(\psi \circ \phi )=\mathrm{Spec}(\phi )\circ \mathrm{Spec}(\psi )$ (The prime-spectrum construction is a contravariant functor to topological spaces).

The Axiom of Choice states that every family of nonempty sets has a choice function (The Axiom of Choice).

Assume the Axiom of Choice. If $R\to S$ is finite type and quasi-finite at every prime of $S$, with $S^{\mathrm{\prime }}$ the integral closure of the image of $R$ in $S$, then there are a finite $R$-subalgebra $T\subseteq S^{\mathrm{\prime }}$ that is module-finite over $R$ and finitely many $g_1,\dots ,g_n\in T$ with $U=D_T(g_1)\cup \cdots \cup D_T(g_n)$ open, the contraction map $\mathrm{Spec}(S)\to \mathrm{Spec}(T)$ a homeomorphism onto $U$, and $T_{g_i}\to S_{g_i}$ an isomorphism for every $i$; moreover for every $g\in T$ with $D_T(g)\subseteq U$ the inclusion induces an isomorphism $T_g\to S_g$ (A quasi-finite algebra factors openly through a finite algebra).

Assume the Axiom of Choice. If $R\to S$ is finite type and quasi-finite at every prime of $S$, then for every $\mathfrak{q}\in \mathrm{Spec}(S)$ there are a finite $R$-subalgebra $T\subseteq S^{\mathrm{\prime }}$ of the relative integral closure, module-finite over $R$, and an element $g\in T$ with $g\mathrm{\notin }\mathfrak{q}$ such that $T_g\cong S_g$ (Quasi-finite algebras are source locally localizations of finite algebras).

A proper ideal $P⊊R$ is prime when $ab\in P$ implies $a\in P$ or $b\in P$; in particular every prime ideal is proper and a unit of $R$ lies in no prime ideal (Prime ideals and maximal ideals in a commutative ring).

A two-sided ideal $I⊴R$ of a ring $R$ is an additive subgroup of $(R,+)$ with $ri\in I$ for every $r\in R$ and $i\in I$; in a commutative ring the left, right and two-sided ideals agree (Left, right and two-sided ideals).