Finite normalization commutes with principal localization

Given: a field $k$, a finite-type integral domain $A$ over $k$, its integral closure $B$ in $\mathrm{Frac}(A)$, and an element $0\mathrm{\ne }f\in A$.

Let $A\to B$ be a homomorphism of commutative rings, $S\subseteq A$ multiplicative and $b\in B$: if $b$ is integral over $A$ then $b\mathrm{/}1$ is integral over $S^{-1}A$. Moreover, if $A$ is a domain, $S\subseteq A\setminus \{0\}$, $K$ is a field extension of $\mathrm{Frac}(A)$ and $\bar{A}$ is the integral closure of $A$ in $K$, then the integral closure of $S^{-1}A$ in $K$ is exactly $S^{-1}\bar{A}$ (Integrality and integral closure commute with localisation, Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions).

For a commutative ring $R$ and $f\in R$ the powers $S_f=\{f^n:n\in \mathbb{N}\}$ form a multiplicative subset, and the principal localisation is $R_f=S_f^{-1}R$ with elements written $r\mathrm{/}{f}^n$; the element $f$ becomes a unit, and an element $r\mathrm{/}1$ is zero exactly when $f^{n}r=0$ for some $n$ (Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$, Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions).

The integral closure of a finite-type domain over a field $k$ in its fraction field is a finite module over the domain (A finite-type domain over a field has finite normalization, Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Integral closure in an extension ring and integrally closed domains).

If $A$ is a domain and $0\mathrm{\ne }f\in A$, then $A_f$ is a domain containing $A$ and the identity on $A$ exhibits $\mathrm{Frac}(A)$ as a field containing $A_f$, and $\mathrm{Frac}(A_f)=\mathrm{Frac}(A)$: each element of $\mathrm{Frac}(A_f)$ is a fraction of elements of $A_f$, hence lies in $\mathrm{Frac}(A)$, while each $a\mathrm{/}b$ with $a,b\in A$, $b\mathrm{\ne }0$, equals $(a\mathrm{/}1)\mathrm{/}(b\mathrm{/}1)$ with $a\mathrm{/}1,b\mathrm{/}1\in A_f$ (The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain, Zero divisor, and integral domain: a commutative ring with $1\mathrm{\ne }0$ and no zero divisors, Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$).

If $B$ is a finite $A$-module with generators $b_1,\dots ,b_m$ and $A\subseteq A_f$, then $B_f=A_f(b_1\mathrm{/}1)+\cdots +A_f(b_m\mathrm{/}1)$ is a finite $A_f$-module: every element of $B_f$ is $b\mathrm{/}{f}^n$ with $b={\sum }_i{a}_i{b}_i$, and $b\mathrm{/}{f}^n={\sum }_i(a_i\mathrm{/}{f}^n)(b_i\mathrm{/}1)$ (Module finiteness is transitive along a tower of algebras, Generated submodule, cyclic and finitely generated modules, module basis and free module, Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$).