Under AC, the Picard-to-class-group map is injective on normal Noetherian integral schemes

Given: a normal Noetherian integral scheme $X$ and an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$ whose class in $\mathrm{Pic}(X)$ lies in the kernel of the canonical homomorphism $\mathrm{Pic}(X)\to \mathrm{Cl}(X)$.

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

$\mathrm{A}\mathrm{C}\Rightarrow \mathrm{D}\mathrm{C}$, and DC includes a prescribed initial point (AC implies DC implies countable choice).

Assume DC. For a normal Noetherian integral scheme $X$, the associated Weil divisor satisfies $\mathrm{cyc}:\mathrm{CaDiv}(X)\to \mathrm{Div}(X)$, and $\mathrm{cyc}(D+E)=\mathrm{cyc}(D)+\mathrm{cyc}(E)$ for Cartier divisors $D,E$, while $\mathrm{cyc}({\mathrm{div}}_C(f))={\mathrm{div}}_W(f)$ for $f\in K(X{)}^{\times }$; there is a canonical homomorphism $\mathrm{Pic}(X)\to \mathrm{Cl}(X)$ carrying $[{\mathcal{O}}_X(D)]$ to the class of $\mathrm{cyc}(D)$ (The Cartier-to-Weil map respects addition and principal divisors).

Assume DC. Let $X$ be a normal Noetherian scheme and let $D$ be a Cartier divisor represented by local equations $f_i\in {\mathcal{K}}_X(U_i{)}^{\times }$. For every prime divisor $Z$ with generic point $\xi$ and every index $i$ with $\xi \in U_i$, the coefficient of $\mathrm{cyc}(D)$ at $Z$ is $v_{\xi }(f_{i,\xi })$, the value of the normalized valuation of the discrete valuation ring ${\mathcal{O}}_{X,\xi }$, and this is independent of $i$ and of the local-equation datum (Cartier divisors on a normal Noetherian scheme give Weil divisors).

Let $X$ be an integral scheme with generic point $\eta$ and $\mathcal{L}$ invertible. Then $K_X(\mathcal{L})=\mathcal{L}\otimes {\mathcal{K}}_X$ is the constant sheaf with value the stalk ${\mathcal{L}}_{\eta }$, a one-dimensional $K(X)$-vector space, so it is nonzero; a rational section of $\mathcal{L}$ is by definition a nonzero element of this vector space (Rational section line bundle).

Let $X$ be integral, $\mathcal{L}$ invertible and $s$ a rational section of $\mathcal{L}$. Then $D={\mathrm{div}}_C(s)$ is a well-defined Cartier divisor on $X$, there is a canonical isomorphism ${\mathcal{O}}_X(D)\to \mathcal{L}$ carrying $1_D$ to $s$, and for every Cartier divisor $D$ the canonical section $1_D$ satisfies ${\mathrm{div}}_C(1_D)=D$ (Rational sections of line bundles are Cartier divisors).

For $f\in K(X{)}^{\times }$ the principal Cartier divisor ${\mathrm{div}}_C(f)$ is represented by the single global equation $f$, and principal Cartier divisors form a subgroup of $\mathrm{CaDiv}(X)$ (Principal cartier divisor).

Cartier divisors on $X$ form an abelian group and are represented on open covers by meromorphic units with unit ratios; a divisor represented by unit equations is the zero divisor, and a divisor whose restriction to every member of an open cover is zero is zero (Cartier divisor).

The Picard group $\mathrm{Pic}(X)$ is the group of isomorphism classes of invertible ${\mathcal{O}}_X$-modules, with identity $[{\mathcal{O}}_X]$ (Picard group of a scheme).

Assume DC. On a normal Noetherian integral scheme $X$ one has $\mathrm{\Gamma }(X,{\mathcal{K}}_X^{\times })=K(X{)}^{\times }$, the map ${\mathrm{div}}_W:K(X{)}^{\times }\to \mathrm{Div}(X)$ is a group homomorphism with image the subgroup $P(X)$ of principal Weil divisors, and $\mathrm{Cl}(X)=\mathrm{Div}(X)\mathrm{/}P(X)$; consequently a Weil divisor has zero class exactly when it is of the form ${\mathrm{div}}_W(f)$ for some $f\in K(X{)}^{\times }$ (Principal weil divisor and class group).

For a prime divisor $Z$ with generic point $\xi$ the order of vanishing is ${\mathrm{ord}}_Z(f)=v_{\xi }(f_{\xi })$, and in the case of an integral normal locally Noetherian scheme $v_{\xi }(g)=0$ for $g\in K(X{)}^{\times }$ if and only if $g$ is a unit of ${\mathcal{O}}_{X,\xi }$ (Order codimension one rational function).

Assume AC. Every commutative Noetherian integrally closed domain satisfies $(S_2)$ (normal domain implies s two).

Assume AC. If $R$ is a commutative Noetherian domain satisfying $(S_2)$, then inside its fraction field $K$ one has $R={\bigcap }_{\mathrm{ht}\mathfrak{p}=1}{R}_{\mathfrak{p}}$; for a field the empty intersection is interpreted as $K=R$ (r one s two intersection of height one localisations).

Assume AC. A domain $A$ is integrally closed if and only if every localisation $A_{\mathfrak{p}}$ at a prime ideal is integrally closed (A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are).

An integral scheme is nonempty, reduced and irreducible; equivalently every nonempty affine open subscheme is the spectrum of a domain (Integral schemes).

A Noetherian normal scheme has a finite affine open cover by spectra of Noetherian rings; normal means every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain, and the local ring at the generic point of a prime divisor is a one-dimensional local ring (Locally Noetherian and Noetherian schemes, Weil divisor normal noetherian scheme).

On an integral scheme, $D\mapsto [{\mathcal{O}}_X(D)]$ is a homomorphism $\mathrm{CaDiv}(X)\to \mathrm{Pic}(X)$ with kernel exactly the principal Cartier divisors; hence every principal Cartier divisor has trivial associated invertible sheaf (On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group).

Every point of an open subset of an affine spectrum has a distinguished-open neighbourhood contained in that subset; localizations of Noetherian rings are Noetherian. (Every point of a Zariski-open set has a distinguished-open neighbourhood inside it, Every quotient and every localisation of a Noetherian ring is Noetherian)