Proof of the Nesterenko--Suslin--Totaro Isomorphism

The Nesterenko--Suslin--Totaro theorem identifies Milnor K-theory with motivic cohomology on the diagonal, providing the crucial link $K_n^M(F) \cong H^n_{\text{mot}}(F, \mathbb{Z}(n))$ . This result is fundamental for the entire motivic approach to algebraic K-theory.

Statement

Theorem6.6Nesterenko--Suslin--Totaro

For any field $F$ , there is a natural isomorphism:

$K_n^M(F) \xrightarrow{\;\sim\;} H^n_{\text{mot}}(\operatorname{Spec} F, \mathbb{Z}(n)) = CH^n(\operatorname{Spec} F, n).$

The map sends the Milnor symbol $\{a_1, \ldots, a_n\} \in K_n^M(F)$ to the cycle class of the point $(a_1, \ldots, a_n) \in (\mathbb{A}^1 \setminus \{0\})^n \cap \Delta^n$ (after appropriate identification).

The cycle map

Definition6.8The cycle map from Milnor K-theory

Recall that $CH^n(\operatorname{Spec} F, n) = H_n(z^n(\operatorname{Spec} F, \bullet))$ where $z^n(\operatorname{Spec} F, m)$ consists of codimension- $n$ cycles on $\operatorname{Spec} F \times \Delta^m = \Delta^m_F$ meeting all faces properly.

For $m = n$ : a codimension- $n$ cycle on $\Delta^n_F \cong \mathbb{A}^n_F$ is a 0-cycle, i.e., a formal sum of closed points.

Define $\phi: (F^{\times})^n \to z^n(\operatorname{Spec} F, n)$ by:

$\phi(a_1, \ldots, a_n) = \left[\left(\frac{a_1}{1 + a_1}, \ldots, \frac{a_n}{1 + a_n}\right)\right]$

using coordinates $t_i / (t_0 + \cdots + t_n)$ on $\Delta^n$ , adjusted so that the point avoids all faces. Specifically, identify $\Delta^n = \{(s_1, \ldots, s_n) : s_i \geq 0, \sum s_i \leq 1\}$ and the point lies at the interior point corresponding to $(a_1, \ldots, a_n)$ via a suitable parametrization.

Proof

The proof proceeds in several steps, combining the work of Nesterenko--Suslin (1990) and Totaro (1992).

Step 1: Constructing the map $K_n^M(F) \to CH^n(F, n)$ . The multilinear map $\phi: (F^{\times})^n \to z^n(F, n)$ defined above descends to a well-defined group homomorphism $K_n^M(F) \to CH^n(F, n)$ . We must verify:

(a) Multilinearity: $\phi(a_1, \ldots, a_i b_i, \ldots, a_n) = \phi(a_1, \ldots, a_i, \ldots) + \phi(a_1, \ldots, b_i, \ldots)$ modulo boundaries. This uses an explicit 1-parameter family (a cycle on $\Delta^{n+1}$ ) interpolating between the two sides.

(b) Steinberg relation: $\phi(a_1, \ldots, a_i, 1 - a_i, \ldots, a_n) = 0$ in $CH^n(F, n)$ . This is shown by constructing an explicit cycle $Z \in z^n(F, n+1)$ with $\partial Z = \phi(a_1, \ldots, a_i, 1-a_i, \ldots)$ .

Step 2: The Steinberg relation via geometry. Consider the map $f: \mathbb{A}^1 \to \mathbb{A}^2$ defined by $t \mapsto (t, 1-t)$ . The image is the line $x + y = 1$ . For elements $a, 1-a \in F^{\times}$ , the point $(a, 1-a)$ lies on this line. The key observation: the line $x + y = 1$ in $(\mathbb{A}^1 \setminus \{0\})^2$ is isomorphic to $\mathbb{A}^1 \setminus \{0, 1\}$ , and the cycle associated to $(a, 1-a)$ is the boundary of the cycle on $\Delta^3_F$ defined by $\{(t, s, 1-t-s) : t + s + (1-t-s) = 1, t(1-t) \neq 0\}$ .

More precisely, Totaro shows that the graph of the map $\Delta^1 \to (\mathbb{G}_m)^2$ given by $u \mapsto (a/(1-u+au), (1-a)/(1-u+au))$ provides the required null-homotopy, assuming it meets all faces properly.

Step 3: Surjectivity. We must show every class in $CH^n(F, n)$ is in the image of $K_n^M(F)$ . A 0-cycle on $\Delta^n_F$ meeting all faces properly is a formal sum $\sum n_i [P_i]$ where $P_i$ are closed points with coordinates avoiding all faces. By moving lemma arguments (or direct construction), one shows that any such cycle is homologous to a sum of cycles of the form $\phi(a_1, \ldots, a_n)$ .

For closed points $P$ with residue field $L = F(P) \neq F$ , the cycle $[P]$ corresponds via the norm map $N_{L/F}: K_n^M(L) \to K_n^M(F)$ to a symbol in $K_n^M(L)$ pushed down to $K_n^M(F)$ .

Step 4: Injectivity. Suppose $\alpha = \sum n_j \{a_{j,1}, \ldots, a_{j,n}\} \in K_n^M(F)$ maps to zero in $CH^n(F, n)$ , i.e., $\phi(\alpha) = \partial Z$ for some $Z \in z^n(F, n+1)$ .

The cycle $Z$ is a 0-cycle on $\Delta^{n+1}_F$ meeting all faces properly. Its boundary is:

$\partial Z = \sum_{i=0}^{n+1} (-1)^i Z|_{\text{face}_i}.$

Each face restriction gives a 0-cycle on $\Delta^n_F$ , and the alternating sum equals $\phi(\alpha)$ .

Nesterenko--Suslin show that the "specialization" of $Z$ to its faces can be analyzed using the Gersten resolution for Milnor K-theory, and the resulting symbol relations in $K_n^M$ force $\alpha = 0$ . The key technical tool is the residue map for Milnor K-theory along divisors on $\mathbb{A}^{n+1}_F$ , which provides the link between geometric boundaries and algebraic relations. $\square$

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Significance

RemarkThe diagonal of the motivic E₂ page

The isomorphism $K_n^M(F) \cong H^n_{\text{mot}}(F, \mathbb{Z}(n))$ identifies the diagonal of the motivic spectral sequence:

$E_2^{n,n} = H^{n-n}_{\text{mot}}(F, \mathbb{Z}(-(-n))) = H^0_{\text{mot}}(F, \mathbb{Z}(n)) = ...$

Wait, let us use the correct indexing. In the spectral sequence $E_2^{p,q} = H^{p-q}(F, \mathbb{Z}(-q)) \Rightarrow K_{-p-q}(F)$ , the term $E_2^{-n, -n} = H^0(F, \mathbb{Z}(n)) = K_n^M(F)$ contributes to $K_{2n}$ ... actually the indexing conventions vary. The essential point is:

The Nesterenko--Suslin--Totaro theorem ensures that Milnor K-theory appears naturally in the motivic spectral sequence, and the edge map $K_n^M(F) \to K_n(F)$ factors through motivic cohomology. Combined with the Bloch--Kato conjecture, this gives a complete description of the mod- $\ell$ motivic spectral sequence for fields in terms of Galois cohomology.

ExampleApplication to function fields

For the function field $F = k(X)$ of a smooth variety $X/k$ :

$K_n^M(k(X)) \cong CH^n(\operatorname{Spec} k(X), n)$

This links symbols $\{f_1, \ldots, f_n\} \in K_n^M(k(X))$ (with $f_i \in k(X)^{\times}$ ) to 0-cycles on $\Delta^n_{k(X)}$ . The residue maps $\partial_v: K_n^M(k(X)) \to K_{n-1}^M(k(v))$ along divisorial valuations $v$ correspond to boundary maps in the localization sequence for higher Chow groups, providing the geometric realization of the Gersten complex.