Proof of the Localization Sequence for Dedekind Domains

We give a detailed proof of the localization sequence in the classical case of a Dedekind domain, illustrating the key ideas without the full generality of the Q-construction. This case already captures the essential interaction between K-theory, valuations, and arithmetic.

Setup and statement

Theorem3.7Localization for Dedekind Domains

Let $R$ be a Dedekind domain with fraction field $F$ . There is an exact sequence:

$K_1(R) \xrightarrow{\alpha} K_1(F) \xrightarrow{\partial} \bigoplus_{\mathfrak{p}} K_0(R/\mathfrak{p}) \xrightarrow{\beta} K_0(R) \xrightarrow{\gamma} K_0(F) \to 0$

where the sum is over all non-zero primes $\mathfrak{p}$ of $R$ . The maps are:

$\alpha$ : induced by the inclusion $R \hookrightarrow F$
$\partial$ : the boundary (tame symbol / valuation) map
$\beta$ : sends $[R/\mathfrak{p}]$ to the class of the ideal $\mathfrak{p}$ in $K_0$
$\gamma$ : induced by the inclusion $R \hookrightarrow F$ (rank map)

Proof

Step 1: Exactness at $K_0(F)$ (surjectivity of $\gamma$ ).

Since $F$ is a field, $K_0(F) = \mathbb{Z}$ generated by $[F]$ . The map $\gamma: K_0(R) \to K_0(F)$ sends $[P] \mapsto \operatorname{rk}(P)$ . Since $[R] \mapsto 1 = [F]$ , the map $\gamma$ is surjective.

Step 2: Identify $\ker(\gamma) = \operatorname{im}(\beta)$ .

The kernel of the rank map $\gamma$ is $\widetilde{K}_0(R)$ . For a Dedekind domain, every finitely generated projective $R$ -module has the form $P \cong R^{n-1} \oplus \mathfrak{a}$ for some ideal $\mathfrak{a}$ , so $\widetilde{K}_0(R) \cong \operatorname{Cl}(R)$ , the ideal class group.

The map $\beta: \bigoplus_\mathfrak{p} \mathbb{Z} \to K_0(R)$ sends the generator of the $\mathfrak{p}$ -summand to $[\mathfrak{p}] - [R] \in \widetilde{K}_0(R)$ . Since every ideal class is represented by a product of prime ideals, $\beta$ surjects onto $\widetilde{K}_0(R) = \ker(\gamma)$ .

Step 3: Identify $\ker(\beta) = \operatorname{im}(\partial)$ .

An element $\sum_\mathfrak{p} n_\mathfrak{p} \cdot [\mathfrak{p}]$ lies in $\ker(\beta)$ iff $\prod_\mathfrak{p} \mathfrak{p}^{n_\mathfrak{p}}$ is a principal ideal $(a)$ for some $a \in F^{\times}$ . Define the boundary map:

$\partial: K_1(F) = F^{\times} \to \bigoplus_\mathfrak{p} \mathbb{Z}, \quad a \mapsto \sum_\mathfrak{p} v_\mathfrak{p}(a) \cdot [\mathfrak{p}]$

where $v_\mathfrak{p}$ is the $\mathfrak{p}$ -adic valuation. Then $\operatorname{im}(\partial) = \{(n_\mathfrak{p}) : \prod \mathfrak{p}^{n_\mathfrak{p}} = (a) \text{ is principal}\} = \ker(\beta)$ .

Step 4: Identify $\ker(\partial) = \operatorname{im}(\alpha)$ .

The kernel of $\partial$ consists of $a \in F^{\times}$ with $v_\mathfrak{p}(a) = 0$ for all $\mathfrak{p}$ , i.e., $a \in R^{\times}$ (units of $R$ ). The map $\alpha: K_1(R) \to K_1(F) = F^{\times}$ sends $R^{\times} \hookrightarrow F^{\times}$ (plus the contribution from $SK_1$ ).

For commutative $R$ : $K_1(R) = R^{\times}$ (since $SK_1 = 0$ for Dedekind domains by Bass--Milnor--Serre). So $\operatorname{im}(\alpha) = R^{\times} = \ker(\partial)$ .

More precisely, $\alpha$ factors as $K_1(R) = R^{\times} \oplus SK_1(R) \to F^{\times} = K_1(F)$ . The $SK_1(R)$ part maps to $SK_1(F) = 0$ (since $F$ is a field). The $R^{\times}$ part embeds into $F^{\times}$ . So $\ker(\partial) = R^{\times} = \operatorname{im}(\alpha)$ .

Conclusion: All four terms in the sequence are exact, proving the theorem. $\square$

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Applications

ExampleClass group from the localization sequence

For $R = \mathcal{O}_F$ (the ring of integers of a number field $F$ ), the localization sequence gives:

$\mathcal{O}_F^{\times} \hookrightarrow F^{\times} \xrightarrow{(v_\mathfrak{p})} \bigoplus_\mathfrak{p} \mathbb{Z} \to K_0(\mathcal{O}_F) \to \mathbb{Z} \to 0.$

This extracts:

$\ker(v) = \mathcal{O}_F^{\times}$ : the unit group (Dirichlet's theorem gives its rank).
$\operatorname{coker}(v) = \operatorname{Cl}(\mathcal{O}_F)$ : the class group.
The exact sequence $0 \to \mathcal{O}_F^{\times} \to F^{\times} \to \bigoplus_\mathfrak{p} \mathbb{Z} \to \operatorname{Cl}(\mathcal{O}_F) \to 0$ is the principal ideal theorem in a different guise.

For $F = \mathbb{Q}(\sqrt{-5})$ : $\mathcal{O}_F^{\times} = \{\pm 1\}$ , $\operatorname{Cl}(\mathcal{O}_F) = \mathbb{Z}/2$ , and $K_0(\mathcal{O}_F) \cong \mathbb{Z} \oplus \mathbb{Z}/2$ .

ExampleThe tame symbol as boundary map

Extending to $K_2$ , the localization sequence for a DVR $(R, \mathfrak{m}, k)$ with fraction field $F$ gives:

$K_2(R) \to K_2(F) \xrightarrow{\partial} K_1(k) = k^{\times}$

The boundary map $\partial$ is the tame symbol. For $a, b \in F^{\times}$ :

$\partial(\{a, b\}) = (-1)^{v(a)v(b)} \frac{a^{v(b)}}{b^{v(a)}} \bmod \mathfrak{m}$

where $v$ is the discrete valuation. This formula shows:

If $a, b \in R^{\times}$ (both units), then $\partial(\{a, b\}) = 1$ (the symbol comes from $K_2(R)$ ).
If $a = \pi$ (uniformizer) and $b \in R^{\times}$ , then $\partial(\{\pi, b\}) = b \bmod \mathfrak{m} \in k^{\times}$ .
If $a = b = \pi$ , then $\partial(\{\pi, \pi\}) = -1 \in k^{\times}$ .

RemarkHigher localization and Gersten's conjecture

The localization sequence generalizes to higher K-groups and leads to the Gersten resolution: for a regular local ring $R$ of dimension $d$ with fraction field $F$ , there is an exact complex

$0 \to K_n(R) \to K_n(F) \to \bigoplus_{\operatorname{ht}(\mathfrak{p})=1} K_{n-1}(k(\mathfrak{p})) \to \cdots \to \bigoplus_{\operatorname{ht}(\mathfrak{p})=d} K_{n-d}(k(\mathfrak{p})) \to 0.$

Gersten's conjecture asserts this complex is exact. It was proved by Quillen for smooth varieties over a field and by Panin for equi-characteristic regular local rings. The Gersten resolution is fundamental for the construction of the Brown--Gersten--Quillen spectral sequence.