[proofplan] We use the localization sequence for the inclusion of the exact category of finitely generated torsion $A$-modules into finitely generated $A$-modules. Devissage decomposes the torsion contribution into its local pieces at the nonzero prime ideals, and each local piece is measured by length over the discrete valuation ring $A_{\mathfrak p}$. The connecting homomorphism is then computed on the automorphism of the one-dimensional $F$-[vector space](/page/Vector%20Space) $F$ given by multiplication by $x\in F^\times$; locally this reduces to the calculation that a uniformizer contributes one copy of the residue field. Additivity gives the coefficient $v_{\mathfrak p}(x)$, and only finitely many coefficients are nonzero. [/proofplan] [step:Identify the target of the boundary with the direct sum of residue-field Grothendieck groups] Let $S=A\setminus\{0\}$, and let $\mathcal T(A)=\mathcal T_S(A)$ denote the exact category of finitely generated torsion $A$-modules. Since a Dedekind domain is regular and Noetherian, the [[Localization Exact Sequence in Low Degree](/theorems/8670)][citetheorem:8670] applied to $A$ and $S$ gives a connecting homomorphism \begin{align*} \partial:K_1(F)\longrightarrow K_0(\mathcal T(A)). \end{align*} Here $S^{-1}A=F$, and the quotient category is represented by finite-dimensional $F$-vector spaces. This is the boundary map appearing in the statement, after the devissage identification described below. For each nonzero prime ideal $\mathfrak p\subset A$, let $\mathcal T_{\mathfrak p}(A)$ be the full exact subcategory of finitely generated torsion $A$-modules whose support is contained in $\{\mathfrak p\}$. Since $A$ is a Dedekind domain, every nonzero prime ideal is maximal. If $M$ is a finitely generated torsion $A$-module, define its annihilator ideal by \begin{align*} \operatorname{Ann}_A(M):=\{a\in A: aM=0\}. \end{align*} This ideal is nonzero, and the quotient $A/\operatorname{Ann}_A(M)$ is Artinian, so it has only finitely many prime ideals. Hence $M$ has finite support among nonzero primes. For each such $\mathfrak p$, define the $\mathfrak p$-primary summand \begin{align*} M_{\mathfrak p}:=\{m\in M: \mathfrak p^r m=0 \text{ for some } r\ge 1\}. \end{align*} The primary decomposition theorem for finite torsion modules over a Dedekind domain gives a functorial finite direct-sum decomposition $M\cong \bigoplus_{\mathfrak p}M_{\mathfrak p}$. If \begin{align*} 0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0 \end{align*} is a short exact sequence in $\mathcal T(A)$, then applying the exact functor of $\mathfrak p$-primary parts gives short exact sequences \begin{align*} 0\longrightarrow M'_{\mathfrak p}\longrightarrow M_{\mathfrak p}\longrightarrow M''_{\mathfrak p}\longrightarrow 0 \end{align*} for every nonzero prime ideal $\mathfrak p$. Hence the primary decomposition is compatible with the exact structures, so the [Additivity Theorem for K Zero of an Exact Category][citetheorem:8642] gives the corresponding decomposition \begin{align*} K_0(\mathcal T(A))\cong \bigoplus_{\mathfrak p\ne 0}K_0(\mathcal T_{\mathfrak p}(A)). \end{align*} For a fixed nonzero prime ideal $\mathfrak p$, the category $\mathcal T_{\mathfrak p}(A)$ is a length category whose unique simple object up to isomorphism is $A/\mathfrak p$. The [Jordan Holder Computation of K Zero][citetheorem:8643] therefore identifies $K_0(\mathcal T_{\mathfrak p}(A))$ with $K_0(A/\mathfrak p)$. Since $A/\mathfrak p$ is a field, [K0 Of A Division Ring][citetheorem:8677] gives \begin{align*} K_0(A/\mathfrak p)\cong \mathbb Z, \end{align*} with generator the class of the one-dimensional $A/\mathfrak p$-vector space $A/\mathfrak p$. Thus the target of $\partial$ is identified with \begin{align*} \bigoplus_{\mathfrak p\ne 0}K_0(A/\mathfrak p)\cong \bigoplus_{\mathfrak p\ne 0}\mathbb Z. \end{align*} [/step] [step:Reduce the computation to multiplication by a power of a uniformizer in a DVR] By [K One Of A Field][citetheorem:8661], the determinant identifies $K_1(F)$ with $F^\times$. Under this identification, an element $x\in F^\times$ is represented by the automorphism \begin{align*} m_x:F\longrightarrow F \end{align*} of the one-dimensional $F$-vector space $F$, where $m_x(y)=xy$ for every $y\in F$. Fix a nonzero prime ideal $\mathfrak p\subset A$. Since $A$ is a Dedekind domain, the localization $A_{\mathfrak p}$ is a discrete valuation ring. Let $\pi_{\mathfrak p}\in A_{\mathfrak p}$ be a uniformizer, and let \begin{align*} v_{\mathfrak p}:F^\times\longrightarrow \mathbb Z \end{align*} be the normalized valuation determined by $v_{\mathfrak p}(\pi_{\mathfrak p})=1$ and $v_{\mathfrak p}(u)=0$ for every unit $u\in A_{\mathfrak p}^{\times}$. Let $\operatorname{pr}_{\mathfrak p}:K_0(\mathcal T(A))\to K_0(\mathcal T_{\mathfrak p}(A))$ be the projection induced by the direct-sum decomposition above. Let $\mathcal T_{\mathrm{fl}}(A_{\mathfrak p})$ denote the exact category of finite-length $A_{\mathfrak p}$-modules. The exact localization functor \begin{align*} L_{\mathfrak p}:\mathcal T_{\mathfrak p}(A)\longrightarrow \mathcal T_{\mathrm{fl}}(A_{\mathfrak p}) \end{align*} is given on objects by $L_{\mathfrak p}(M)=A_{\mathfrak p}\otimes_A M$. This notation distinguishes the localized module $A_{\mathfrak p}\otimes_A M$ from the $\mathfrak p$-primary summand $M_{\mathfrak p}$ defined earlier. Since every object of $\mathcal T_{\mathfrak p}(A)$ is killed by a power of $\mathfrak p$, localization at $\mathfrak p$ is faithful on it, and every finite-length $A_{\mathfrak p}$-module is obtained from its underlying $A$-module; hence $L_{\mathfrak p}$ is an exact equivalence. We define the local boundary component \begin{align*} \partial_{\mathfrak p}:K_1(F)\longrightarrow K_0(\mathcal T_{\mathrm{fl}}(A_{\mathfrak p})) \end{align*} to be the composite of $\partial$ with $\operatorname{pr}_{\mathfrak p}$ and $K_0(L_{\mathfrak p})$. The localization map $A\to A_{\mathfrak p}$ sends $S=A\setminus\{0\}$ to the nonzero elements of $A_{\mathfrak p}$, and its induced map on total quotient rings is the identity map $F\to F$. The exact functor $A_{\mathfrak p}\otimes_A -$ sends every torsion module supported away from $\mathfrak p$ to zero and restricts to the equivalence $L_{\mathfrak p}$ on $\mathcal T_{\mathfrak p}(A)$. Thus projection to the $\mathfrak p$-primary summand followed by $L_{\mathfrak p}$ is exactly the torsion part of the functorial localization square induced by $A\to A_{\mathfrak p}$. By [naturality of the connecting homomorphism](/theorems/4538) in [Localization Exact Sequence in Low Degree][citetheorem:8670], the composite $K_0(L_{\mathfrak p})\circ\operatorname{pr}_{\mathfrak p}\circ\partial$ is the boundary homomorphism for the discrete valuation ring $A_{\mathfrak p}\subset F$. In $F^\times$ there is a factorization \begin{align*} x=u_{\mathfrak p}\pi_{\mathfrak p}^{v_{\mathfrak p}(x)} \end{align*} with $u_{\mathfrak p}\in A_{\mathfrak p}^{\times}$. Multiplication by the unit $u_{\mathfrak p}$ restricts to an automorphism of the lattice $A_{\mathfrak p}\subset F$. Its kernel and cokernel in the torsion category are both zero, so its local connecting class is zero. Hence the $\mathfrak p$-component of $\partial(x)$ is determined by the integer power $\pi_{\mathfrak p}^{v_{\mathfrak p}(x)}$. [/step] [step:Compute the local boundary of a uniformizer] We compute in the discrete valuation ring $A_{\mathfrak p}$. Let \begin{align*} \kappa(\mathfrak p):=A_{\mathfrak p}/\mathfrak p A_{\mathfrak p} \end{align*} denote its residue field. Multiplication by $\pi_{\mathfrak p}$ defines an injective $A_{\mathfrak p}$-[linear map](/page/Linear%20Map) \begin{align*} \mu_{\pi_{\mathfrak p}}:A_{\mathfrak p}\longrightarrow A_{\mathfrak p} \end{align*} given by $\mu_{\pi_{\mathfrak p}}(a)=\pi_{\mathfrak p}a$. Its cokernel is \begin{align*} A_{\mathfrak p}/\pi_{\mathfrak p}A_{\mathfrak p}\cong \kappa(\mathfrak p). \end{align*} By the boundary convention fixed in the statement for the localization sequence [Localization Exact Sequence in Low Degree][citetheorem:8670], the connecting homomorphism sends the class of multiplication by $\pi_{\mathfrak p}$ to the class of this cokernel: \begin{align*} \partial_{\mathfrak p}(\pi_{\mathfrak p})=[\kappa(\mathfrak p)]. \end{align*} Because $\mathfrak p$ is maximal, localization induces an isomorphism $A/\mathfrak p\cong A_{\mathfrak p}/\mathfrak p A_{\mathfrak p}=\kappa(\mathfrak p)$. Under devissage, the class $[\kappa(\mathfrak p)]$ therefore corresponds to the positive generator $1\in\mathbb Z$. For an integer $n\ge 0$, multiplication by $\pi_{\mathfrak p}^{n}$ has cokernel \begin{align*} A_{\mathfrak p}/\pi_{\mathfrak p}^{n}A_{\mathfrak p}. \end{align*} This module has a filtration with $n$ successive quotients isomorphic to $\kappa(\mathfrak p)$, namely the filtration by the submodules \begin{align*} \pi_{\mathfrak p}^{i}A_{\mathfrak p}/\pi_{\mathfrak p}^{n}A_{\mathfrak p} \end{align*} for $0\le i\le n$. By the [Additivity Theorem for K Zero of an Exact Category][citetheorem:8642], its class in $K_0(\mathcal T_{\mathrm{fl}}(A_{\mathfrak p}))$ is $n[\kappa(\mathfrak p)]$, and \begin{align*} \partial_{\mathfrak p}(\pi_{\mathfrak p}^{n})=n[\kappa(\mathfrak p)]. \end{align*} For $n<0$, the homomorphism property of $\partial_{\mathfrak p}$ gives \begin{align*} \partial_{\mathfrak p}(\pi_{\mathfrak p}^{n})=-\partial_{\mathfrak p}(\pi_{\mathfrak p}^{-n})=n[\kappa(\mathfrak p)]. \end{align*} [guided] The local calculation is the point where the sign and the valuation normalization enter. We work in the discrete valuation ring $A_{\mathfrak p}$, whose maximal ideal is generated by the chosen uniformizer $\pi_{\mathfrak p}$. Define the residue field \begin{align*} \kappa(\mathfrak p):=A_{\mathfrak p}/\mathfrak p A_{\mathfrak p}. \end{align*} Since $\mathfrak p A_{\mathfrak p}=\pi_{\mathfrak p}A_{\mathfrak p}$, multiplication by $\pi_{\mathfrak p}$ gives an injective $A_{\mathfrak p}$-linear map \begin{align*} \mu_{\pi_{\mathfrak p}}:A_{\mathfrak p}\longrightarrow A_{\mathfrak p} \end{align*} defined by $\mu_{\pi_{\mathfrak p}}(a)=\pi_{\mathfrak p}a$. The quotient by the image is \begin{align*} \operatorname{coker}(\mu_{\pi_{\mathfrak p}})=A_{\mathfrak p}/\pi_{\mathfrak p}A_{\mathfrak p}\cong \kappa(\mathfrak p). \end{align*} The boundary convention fixed in the theorem for [Localization Exact Sequence in Low Degree][citetheorem:8670] is precisely the convention that this cokernel contributes positively. Hence \begin{align*} \partial_{\mathfrak p}(\pi_{\mathfrak p})=[\kappa(\mathfrak p)]. \end{align*} Since $\mathfrak p$ is maximal in the Dedekind domain $A$, localization induces an isomorphism $A/\mathfrak p\cong A_{\mathfrak p}/\mathfrak p A_{\mathfrak p}=\kappa(\mathfrak p)$, so this class is the same residue-field generator used in the global devissage identification. Now consider a nonnegative integer $n$. Multiplication by $\pi_{\mathfrak p}^{n}$ is again injective, and its cokernel is \begin{align*} A_{\mathfrak p}/\pi_{\mathfrak p}^{n}A_{\mathfrak p}. \end{align*} Why does this represent $n$ copies of the residue field? For each integer $i$ with $0\le i\le n$, define the submodule \begin{align*} N_i:=\pi_{\mathfrak p}^{i}A_{\mathfrak p}/\pi_{\mathfrak p}^{n}A_{\mathfrak p} \end{align*} of $A_{\mathfrak p}/\pi_{\mathfrak p}^{n}A_{\mathfrak p}$. These submodules satisfy $N_0=A_{\mathfrak p}/\pi_{\mathfrak p}^{n}A_{\mathfrak p}$, $N_n=0$, and $N_{i+1}\subset N_i$ for every $0\le i<n$. For each $i$ with $0\le i<n$, the quotient of two consecutive terms is \begin{align*} \pi_{\mathfrak p}^{i}A_{\mathfrak p}/\pi_{\mathfrak p}^{i+1}A_{\mathfrak p}\cong A_{\mathfrak p}/\pi_{\mathfrak p}A_{\mathfrak p}\cong \kappa(\mathfrak p), \end{align*} where the first isomorphism is induced by multiplication by $\pi_{\mathfrak p}^{i}$. The [Additivity Theorem for K Zero of an Exact Category][citetheorem:8642] applied successively to the short exact sequences coming from this filtration therefore gives \begin{align*} [A_{\mathfrak p}/\pi_{\mathfrak p}^{n}A_{\mathfrak p}]=n[\kappa(\mathfrak p)]. \end{align*} Thus \begin{align*} \partial_{\mathfrak p}(\pi_{\mathfrak p}^{n})=n[\kappa(\mathfrak p)] \end{align*} for $n\ge 0$. If $n<0$, then $\pi_{\mathfrak p}^{n}$ is the inverse of $\pi_{\mathfrak p}^{-n}$ in $F^\times$, and $\partial_{\mathfrak p}$ is a [group homomorphism](/page/Group%20Homomorphism). Therefore \begin{align*} \partial_{\mathfrak p}(\pi_{\mathfrak p}^{n})=-\partial_{\mathfrak p}(\pi_{\mathfrak p}^{-n})=n[\kappa(\mathfrak p)]. \end{align*} This proves that the local boundary records exactly the normalized valuation. [/guided] [/step] [step:Assemble the local computations into the global divisor] For the fixed prime $\mathfrak p$, the previous steps give \begin{align*} \partial_{\mathfrak p}(x)=v_{\mathfrak p}(x)[\kappa(\mathfrak p)]. \end{align*} Under the devissage identification $[\kappa(\mathfrak p)]\mapsto 1$, this is the integer $v_{\mathfrak p}(x)$. It remains only to check that the resulting family belongs to the direct sum. Choose nonzero elements $a,b\in A$ such that \begin{align*} x=ab^{-1}. \end{align*} Since $A$ is noetherian of Krull dimension one, the principal ideals $(a)$ and $(b)$ are contained in only finitely many nonzero prime ideals. For every nonzero prime ideal $\mathfrak p$ outside this finite set, both $a$ and $b$ are units in $A_{\mathfrak p}$, so $v_{\mathfrak p}(x)=0$. Hence \begin{align*} (v_{\mathfrak p}(x))_{\mathfrak p\ne 0}\in \bigoplus_{\mathfrak p\ne 0}\mathbb Z. \end{align*} Combining the identifications $K_1(F)\cong F^\times$ and $K_0(A/\mathfrak p)\cong\mathbb Z$ with the local computation at every nonzero prime ideal $\mathfrak p$, the boundary map is \begin{align*} x\longmapsto (v_{\mathfrak p}(x))_{\mathfrak p}. \end{align*} This is the asserted identification of the localization boundary with the divisor valuation map. [/step]