Milnor conjecture: $K^M_n(F)/2 \cong H^n(F,\mathbb{Z}/2)$

Read-back of milnor_conjecture.

Hypotheses. $F$ is an arbitrary field whose underlying type lives in the lowest universe (Type, not an arbitrary universe). The only extra assumption is that $2 \neq 0$ in $F$, i.e. $\operatorname{char} F \neq 2$. Finally, $n \ge 0$ is an arbitrary natural number, so $n = 0$ is included.

The group $K_n(F)$ (MilnorK F n). Write $F^\times$ additively. Let $T_n = (F^\times)^{\otimes_{\mathbb Z} n}$ be its $n$-fold tensor power over $\mathbb Z$. Let $S_n \subseteq T_n$ be the subgroup ($\mathbb Z$-span) generated by the pure tensors $a_1 \otimes \cdots \otimes a_n$ with $a_i \in F^\times$ for which some adjacent pair of positions $(i, i+1)$ satisfies $a_i + a_{i+1} = 1$ in $F$. Then

$$K_n(F) := T_n / S_n .$$

For $a = (a_1,\dots,a_n) \in (F^\times)^n$, the symbol $\{a_1,\dots,a_n\} \in K_n(F)$ (symbol a) is the image of $a_1 \otimes \cdots \otimes a_n$.

• For $n = 0$: $T_0 \cong \mathbb Z$ and there are no adjacent pairs, so $S_0 = 0$ and $K_0(F) \cong \mathbb Z$. Under this isomorphism the empty symbol corresponds to $1$.
• For $n = 1$: there are no adjacent pairs, so $K_1(F) \cong F^\times$ (written additively), with $\{a\} \leftrightarrow a$.

The group $H^n(F)$ (H F n).

• Let $G_F = \operatorname{Gal}(F^{\mathrm{sep}}/F)$, where $F^{\mathrm{sep}}$ is Mathlib's chosen separable closure and $G_F$ carries the Krull topology (a profinite, hence locally compact, group).
• $H^n(F)$ is Mathlib's continuous cohomology $H^n_{\mathrm{cts}}(G_F, \mathbb Z/2)$. Here $\mathbb Z/2$ has the trivial $G_F$-action and the discrete topology.
• It is the degree-$n$ homology of Mathlib's complex of homogeneous cochains. Literally, degree-$n$ cochains are the $G_F$-invariant elements of the iterated space $C(G_F, C(G_F, \dots, C(G_F, \mathbb Z/2)\dots))$, with $n+1$ copies of $G_F$ and compact-open topologies.
• $G_F$ acts on these cochains by $(g\cdot f)(x_0,\dots,x_n) = f(g^{-1}x_0,\dots,g^{-1}x_n)$.
• Mathlib defines the differential inductively.
• The bundle's definitions file identifies these cochains with continuous (i.e. locally constant) functions $G_F^{\,n+1} \to \mathbb Z/2$ by currying. It uses local compactness of $G_F$ to do so, and proves that under this identification the differential is the usual alternating sum of face maps.
• The result is a topological $\mathbb Z/2$-module. In the theorem only its additive group structure is used.
• For $n=0$ this is the group of invariant, i.e. constant, functions $G_F \to \mathbb Z/2$, $\cong \mathbb Z/2$.

The Galois symbol (galoisSymbol a).

• For $a \in F^\times$, let $\sqrt a \in F^{\mathrm{sep}}$ be a fixed but arbitrary (choice-selected) element with $(\sqrt a)^2 = a$. The bundle proves that $\sigma(\sqrt a) \in \{\sqrt a, -\sqrt a\}$ and that these two values are distinct.
• Define $\chi_a : G_F \to \mathbb Z/2$ by $\chi_a(\sigma) = 0$ if $\sigma(\sqrt a) = \sqrt a$, and $\chi_a(\sigma) = 1$ otherwise. The bundle proves that $\chi_a$ is a continuous homomorphism.
• For $a = (a_1,\dots,a_n)$, define the homogeneous $n$-cochain

$$c_a(\sigma_0,\dots,\sigma_n) = \prod_{j=1}^{n} \bigl(\chi_{a_j}(\sigma_j) - \chi_{a_j}(\sigma_{j-1})\bigr) \in \mathbb Z/2 .$$

The bundle proves that $c_a$ is invariant under left translation and is a cocycle.

• $\mathrm{gs}(a) \in H^n(F)$ is its cohomology class.
• For $n = 0$ the product is empty, so $\mathrm{gs}()$ is the class of the constant cochain $1$, i.e. the nonzero element of $H^0 \cong \mathbb Z/2$.
• (Remark, not part of the definitions: under the standard homogeneous/inhomogeneous dictionary, $c_a$ corresponds to the inhomogeneous cocycle $(g_1,\dots,g_n) \mapsto \chi_{a_1}(g_1)\cdots\chi_{a_n}(g_n)$.)

Conclusion. For every such $F$ and every $n$, there exists (existence only; $\exists$, not $\exists!$) a map

$$\varphi : K_n(F) \to H^n(F).$$

It is required only to be an additive group homomorphism. No continuity and no $\mathbb Z/2$-linearity is asserted. It must satisfy all three of the following:

1. $\varphi(\{a_1,\dots,a_n\}) = \mathrm{gs}(a_1,\dots,a_n)$ for every $a \in (F^\times)^n$.
2. $\varphi$ is surjective.
3. For every $x \in K_n(F)$:

$$\varphi(x) = 0 \iff \exists\, y \in K_n(F),\ x = 2y,$$

where $2y$ means $y + y$ (the scalar $2$ is read as a natural number; in any case $2y = y+y$). So $\ker \varphi = 2K_n(F)$ exactly.

(Consequences, not literal parts of the statement:

• Pure tensors span $T_n$, so the symbols generate $K_n(F)$. Condition (1) therefore pins down $\varphi$ on generators, even though only existence is asserted.

• Conditions (2) and (3) together are equivalent to $\varphi$ inducing an isomorphism $K_n(F)/2K_n(F) \cong H^n(F)$.)

• For $n = 0$ the claim is: there is a surjection $\mathbb Z \to H^0 \cong \mathbb Z/2$ sending $1$ to the class of the constant $1$, with kernel $2\mathbb Z$.

• For $n = 1$ the claim is: there is a homomorphism $F^\times \to H^1(F)$ sending $a$ to the class of the cocycle $(\sigma_0,\sigma_1) \mapsto \chi_a(\sigma_1) - \chi_a(\sigma_0)$. It must be surjective with kernel exactly the squares $(F^\times)^2$, since $2y$ in additive notation is $y^2$ in $F^\times$.

The hypothesis $2 \ne 0$ excludes exactly the fields of characteristic $2$. It is satisfiable, e.g. by $\mathbb Q$.