Localization Sequence

The localization sequence is the most important computational tool in algebraic K-theory. It relates the K-groups of a ring (or scheme) to those of a localization and the "fiber" at the removed points, producing a long exact sequence analogous to the Mayer--Vietoris sequence in topology.

Localization for abelian categories

Theorem3.1Quillen's Localization Theorem

Let $\mathcal{A}$ be an abelian category and $\mathcal{B} \subseteq \mathcal{A}$ a Serre subcategory (closed under subobjects, quotients, and extensions). Let $\mathcal{A}/\mathcal{B}$ be the Serre quotient category. Then there is a long exact sequence

$\cdots \to K_{n+1}(\mathcal{A}/\mathcal{B}) \xrightarrow{\partial} K_n(\mathcal{B}) \to K_n(\mathcal{A}) \to K_n(\mathcal{A}/\mathcal{B}) \xrightarrow{\partial} K_{n-1}(\mathcal{B}) \to \cdots$

extending to the left indefinitely and terminating on the right:

$\cdots \to K_0(\mathcal{B}) \to K_0(\mathcal{A}) \to K_0(\mathcal{A}/\mathcal{B}) \to 0.$

Localization for rings

Definition3.1Localization sequence for rings

Let $R$ be a Dedekind domain with fraction field $F$ . Let $\mathcal{M}_S(R)$ denote the category of finitely generated $R$ -modules supported on a set of primes $S$ . For a non-zero prime $\mathfrak{p}$ , devissage gives $K_n(\mathcal{M}_{\{\mathfrak{p}\}}(R)) \cong K_n(R/\mathfrak{p})$ .

The localization sequence takes the form:

$\cdots \to K_{n+1}(F) \xrightarrow{\partial} \bigoplus_{\mathfrak{p}} K_n(R/\mathfrak{p}) \to G_n(R) \to K_n(F) \xrightarrow{\partial} \bigoplus_{\mathfrak{p}} K_{n-1}(R/\mathfrak{p}) \to \cdots$

where the direct sum is over all non-zero primes $\mathfrak{p}$ of $R$ , and $\partial$ is the boundary map (or tame symbol in low degrees).

ExampleLocalization sequence for ℤ

For $R = \mathbb{Z}$ , $F = \mathbb{Q}$ , and residue fields $\mathbb{F}_p$ :

$\cdots \to K_{n+1}(\mathbb{Q}) \xrightarrow{\partial} \bigoplus_p K_n(\mathbb{F}_p) \to G_n(\mathbb{Z}) \to K_n(\mathbb{Q}) \to \cdots$

Since $\mathbb{Z}$ is regular, $G_n(\mathbb{Z}) = K_n(\mathbb{Z})$ . In low degrees:

$n = 0$ : $K_1(\mathbb{Q}) \xrightarrow{\partial} \bigoplus_p K_0(\mathbb{F}_p) \to K_0(\mathbb{Z}) \to K_0(\mathbb{Q}) \to 0$

This gives $\mathbb{Q}^{\times} \xrightarrow{v_p} \bigoplus_p \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z} \to 0$ , where $v_p$ is the $p$ -adic valuation. The cokernel of $\partial$ is $\mathbb{Z}$ (the class group of $\mathbb{Z}$ is trivial).

$n = 1$ : $K_2(\mathbb{Q}) \xrightarrow{\partial} \bigoplus_p \mathbb{F}_p^{\times} \to K_1(\mathbb{Z}) \to K_1(\mathbb{Q})$

The tame symbol $\partial\{a, b\} = (-1)^{v_p(a)v_p(b)} a^{v_p(b)} b^{-v_p(a)} \mod p$ . The sequence gives $\bigoplus_p \mathbb{F}_p^{\times} \to \{\pm 1\} \to \mathbb{Q}^{\times}$ which is $\bigoplus_p \mathbb{F}_p^{\times} \xrightarrow{(-1)^?} \{\pm 1\}$ , the product of signs.

Localization for schemes

Definition3.2Localization for schemes

For a Noetherian scheme $X$ with closed subscheme $Z \hookrightarrow X$ and open complement $U = X \setminus Z$ , there is a long exact sequence in G-theory:

$\cdots \to G_{n+1}(U) \xrightarrow{\partial} G_n(Z) \xrightarrow{i_*} G_n(X) \xrightarrow{j^*} G_n(U) \xrightarrow{\partial} G_{n-1}(Z) \to \cdots$

where $i: Z \hookrightarrow X$ is the closed immersion and $j: U \hookrightarrow X$ is the open immersion. The map $j^*$ is restriction of coherent sheaves, and $i_*$ is pushforward.

ExampleG-theory of the affine line

Consider $X = \mathbb{A}^1_k = \operatorname{Spec} k[t]$ , $Z = \{0\} = \operatorname{Spec} k$ , $U = \mathbb{A}^1_k \setminus \{0\} = \operatorname{Spec} k[t, t^{-1}]$ :

$\cdots \to G_{n+1}(k[t, t^{-1}]) \to G_n(k) \to G_n(k[t]) \to G_n(k[t, t^{-1}]) \to \cdots$

Since $k[t]$ is regular, $G_n(k[t]) = K_n(k[t]) = K_n(k)$ by homotopy invariance. The sequence becomes:

$\cdots \to G_{n+1}(k[t, t^{-1}]) \to K_n(k) \to K_n(k) \to G_n(k[t, t^{-1}]) \to \cdots$

The map $K_n(k) \to K_n(k)$ factors as $K_n(k) \to K_n(k[t]) \xrightarrow{t=0} K_n(k)$ , which is the identity. So the connecting map is zero, and we get short exact sequences $0 \to K_n(k) \to G_n(k[t, t^{-1}]) \to K_{n-1}(k) \to 0$ . These split, giving

$G_n(k[t, t^{-1}]) \cong K_n(k) \oplus K_{n-1}(k)$

which is the BHS decomposition for regular rings.

The boundary map

RemarkThe boundary map as a residue

The boundary map $\partial: K_n(F) \to K_{n-1}(k)$ in the localization sequence for a DVR $(R, \mathfrak{m}, k, F)$ generalizes classical residue maps:

$n = 1$ : $\partial: K_1(F) = F^{\times} \to K_0(k) = \mathbb{Z}$ is the valuation $v: F^{\times} \to \mathbb{Z}$ .
$n = 2$ : $\partial: K_2(F) \to K_1(k) = k^{\times}$ is the tame symbol: $\partial\{a, b\} = (-1)^{v(a)v(b)} \frac{a^{v(b)}}{b^{v(a)}} \mod \mathfrak{m}$ .
Higher $n$ : $\partial$ is a "higher tame symbol" that can be described in terms of Milnor K-theory or motivic cohomology.

The exactness of the localization sequence at $K_n(F)$ means: an element $\alpha \in K_n(F)$ lies in the image of $K_n(R) \to K_n(F)$ (i.e., has "no pole" at $\mathfrak{p}$ ) if and only if $\partial_\mathfrak{p}(\alpha) = 0$ .