Residue Maps and the Gersten Complex

Residue maps in Milnor K-theory generalize the valuation map and the tame symbol to higher degrees. They fit into the Gersten complex, which provides a flasque resolution of the K-theory sheaf and connects Milnor K-theory to Chow groups and cycle class maps.

Residue maps for discrete valuations

Definition4.5Residue homomorphism

Let $F$ be a field with a discrete valuation $v: F^{\times} \to \mathbb{Z}$ , valuation ring $\mathcal{O}_v$ , and residue field $k(v) = \mathcal{O}_v / \mathfrak{m}_v$ . The residue homomorphism (or tame symbol) is

$\partial_v: K_n^M(F) \to K_{n-1}^M(k(v))$

defined on symbols by choosing a uniformizer $\pi$ (with $v(\pi) = 1$ ):

$\partial_v(\{u_1, \ldots, u_n\}) = 0 \quad \text{if all } v(u_i) = 0$

$\partial_v(\{\pi, u_2, \ldots, u_n\}) = \{\bar{u}_2, \ldots, \bar{u}_n\}$

where $\bar{u}_i = u_i \bmod \mathfrak{m}_v \in k(v)^{\times}$ . The general formula is:

$\partial_v(\{a_1, \ldots, a_n\}) = (-1)^{\sum_{i<j} v(a_i)v(a_j)} \left\{\frac{a_1^{v(a_2) \cdots}}{(\text{powers of } \pi)}, \ldots \right\}$

extended by multilinearity from the cases above. Crucially, $\partial_v$ is independent of the choice of uniformizer $\pi$ .

ExampleResidue in degree 2

For $n = 2$ , the residue $\partial_v: K_2^M(F) \to k(v)^{\times}$ recovers the classical tame symbol:

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

For $F = \mathbb{Q}$ and $v = v_p$ ( $p$ -adic valuation):

$\partial_p(\{p, u\}) = u \bmod p$ for $u \in \mathbb{Z}_p^{\times}$
$\partial_p(\{p, p\}) = -1 \in \mathbb{F}_p^{\times}$
$\partial_p(\{u_1, u_2\}) = 1$ for $u_1, u_2 \in \mathbb{Z}_p^{\times}$

The Gersten complex

Definition4.6Gersten complex for Milnor K-theory

Let $X$ be a smooth variety over a field $k$ with function field $F = k(X)$ . For each point $x \in X^{(p)}$ of codimension $p$ , let $k(x)$ be its residue field. The Gersten complex for $K_n^M$ is:

$0 \to K_n^M(F) \xrightarrow{\oplus \partial_x} \bigoplus_{x \in X^{(1)}} K_{n-1}^M(k(x)) \xrightarrow{\oplus \partial_x} \bigoplus_{x \in X^{(2)}} K_{n-2}^M(k(x)) \to \cdots \to \bigoplus_{x \in X^{(n)}} \mathbb{Z} \to 0$

where $\partial_x$ is the residue map along the discrete valuation associated to the codimension-1 point $x$ . The fact that $\partial \circ \partial = 0$ (this complex is indeed a complex) is a non-trivial result, requiring the Weil reciprocity law and its higher analogues.

RemarkExactness: the Gersten conjecture

The Gersten conjecture for Milnor K-theory states that the Gersten complex is exact (except at the leftmost term). This was proved:

For smooth varieties over a field: by Kerz (2009), building on earlier work of Rost and Gabber.
For discrete valuation rings of mixed characteristic: partial results by various authors.

When exact, the Gersten complex provides a flasque resolution of the Zariski sheaf $\mathcal{K}_n^M$ (the sheafification of $U \mapsto K_n^M(\mathcal{O}(U))$ ). This means:

$H^p_{\text{Zar}}(X, \mathcal{K}_n^M) = H^p(\text{Gersten complex})$

and in particular, $H^0_{\text{Zar}}(X, \mathcal{K}_n^M) = K_n^M(F) \cap (\text{no residues})$ is the "unramified" Milnor K-theory.

Connection to Chow groups

ExampleMilnor K-theory and Chow groups

The Gersten complex for $K_n^M$ on a smooth variety $X$ of dimension $d$ gives:

$H^n_{\text{Zar}}(X, \mathcal{K}_n^M) = \operatorname{coker}\left(\bigoplus_{x \in X^{(n-1)}} k(x)^{\times} \xrightarrow{\partial} \bigoplus_{x \in X^{(n)}} \mathbb{Z}\right) = CH^n(X)$

the Chow group of codimension- $n$ cycles modulo rational equivalence. Here:

The group $\bigoplus_{x \in X^{(n)}} \mathbb{Z}$ is the group of codimension- $n$ cycles.
The map $\partial$ from $\bigoplus_{x \in X^{(n-1)}} k(x)^{\times}$ gives rational equivalence: $\partial_x(f) = \operatorname{div}(f)$ for $f \in k(x)^{\times}$ .

This recovers the classical identification and places Chow groups within the K-theory framework.

ExampleThe Gersten complex on a curve

For a smooth curve $C$ over a field $k$ with function field $F = k(C)$ :

The Gersten complex for $K_1^M$ : $0 \to F^{\times} \xrightarrow{\oplus v_x} \bigoplus_{x \in C^{(1)}} \mathbb{Z} \to 0$

gives $H^0 = k(C)^{\times} \cap (\text{no poles}) = k^{\times}$ (constant functions) and $H^1 = \operatorname{Pic}(C)$ (the Picard group = divisor class group).

The Gersten complex for $K_2^M$ : $0 \to K_2^M(F) \xrightarrow{\oplus \partial_x} \bigoplus_{x \in C^{(1)}} k(x)^{\times} \to 0$

gives $H^0 =$ unramified $K_2$ and $H^1 = CH^1(C) = \operatorname{Pic}(C)$ . The Weil reciprocity law $\prod_{x \in C} \partial_x(\{f, g\}) = 1$ ensures that $\partial \circ \partial = 0$ (here the complex has only two terms, so this is the statement that the image of $K_2$ has trivial degree).

Weil reciprocity

RemarkWeil reciprocity and the sum of residues

For a smooth projective curve $C$ over a field $k$ and $f, g \in k(C)^{\times}$ , the Weil reciprocity law states:

$\sum_{x \in C} v_x(\partial_x(\{f, g\})) = 0 \quad \text{in } \bigoplus_{x} \mathbb{Z}$

or more precisely, $\prod_{x \in C} N_{k(x)/k}(\partial_x(\{f, g\})) = 1 \in k^{\times}$ .

This generalizes to higher Milnor K-theory: for $\alpha \in K_n^M(k(C))$ ,

$\sum_{x \in C} \operatorname{cores}_{k(x)/k}(\partial_x(\alpha)) = 0 \in K_{n-1}^M(k)$

where $\operatorname{cores}$ is the corestriction (norm) map. This is the key identity ensuring the Gersten complex is a complex, and its higher-dimensional generalization is one of the deepest parts of the theory.