Operations in Milnor K-theory

We show that operations in Milnor K-theory mod $p$ of a field are spanned by divided power operations. After giving an explicit formula for divided power operations and extending them to some new cases, we determine for all fields $k$ and all prime numbers $p$, all the operations $K^M_i/p \to K^M_j/p$ commuting with field extensions over the base field $k$. Moreover, the integral case is discussed and we determine the operations $K^M_i/p \to K^M_j/p$ for smooth schemes over a field.

💡 Research Summary

The paper provides a complete classification of natural operations on Milnor K‑theory modulo a prime p, for any base field k₀. An operation is defined as a family of maps K⁽ᴹ⁾ᵢ(k)/p → K⁽ᴹ⁾*(k)/p, one for each field extension k/k₀, compatible with further extensions. The central construction is the divided‑power operation γₙ, which sends an element represented as a sum of symbols to the sum of all n‑fold products of distinct symbols. Under suitable hypotheses (i even and p odd; or p = 2 with –1 a square; or characteristic 2), the relations {a,a}=0 and {a,–1}=0 guarantee that γₙ is independent of the chosen representation and thus defines a well‑posed map K⁽ᴹ⁾ᵢ(k)/p → K⁽ᴹ⁾{ni}(k)/p.

The paper lists six axioms satisfied by the γₙ’s: γ₀=1, γ₁=id, multiplicativity γₙ(xy)=xⁿγₙ(y), the binomial formula for products γ_m·γ_n = (m+n choose n)γ_{m+n}, the additive formula γₙ(x+y)=∑{i=0}ⁿγ_i(x)γ{n−i}(y), and the iteration formula γ_m(γ_n(x)) = (nm)!/(m! n!) γ_{nm}(x). These axioms make Milnor K‑theory a divided‑power algebra in the sense of Revoy.

The main results are two theorems describing the algebra of all operations. For odd p, the algebra is generated as a free K⁽ᴹ⁾*(k₀)/p‑module by γ₀ and γ₁ when i is odd, and by all γₙ (n≥0) when i≥2 is even; for i=0 it is simply the space of arbitrary functions F_p → K⁽ᴹ⁾(k₀)/p. For p=2 the situation is more subtle: when i=0 or i=1 the algebra is generated by γ₀ and γ₁; for i≥2 it is generated by γ₀, γ₁ and the elements y·γₙ with y in the kernel of the map τ_i:x↦{–1}^{i−1}·x. If –1 is a square in k₀ then τ_i is zero, so the kernel is the whole K⁽ᴹ⁾_(k₀)/2 and all γₙ appear. The product relations among the generators remain the same binomial formulas.

Assuming the Bloch‑Kato conjecture (now a theorem), the same description transfers to motivic cohomology groups H^{n,n}(Spec k,ℤ/p) via the isomorphism K⁽ᴹ⁾ₙ(k)/p ≅ Hⁿ(k,μ_p^{⊗n}). Consequently, the divided‑power operations give explicit Steenrod‑type operations on motivic cohomology and, via the Galois symbol, on Galois cohomology.

The paper also treats integral Milnor K‑theory. By introducing weaker divided‑power operations (which may be defined without the full set of relations), the author shows that any natural operation K⁽ᴹ⁾ᵢ → K⁽ᴹ⁾* is a K⁽ᴹ⁾*(k₀)‑linear combination of these weak operations, under mild hypotheses on the operation (see Propositions 3.19 and 3.21).

Finally, the results are extended from fields to smooth schemes X over k₀. The unramified Milnor K‑theory K⁽ᴹ⁾*(X) is defined as the subgroup of K⁽ᴹ⁾(k(X)) consisting of elements unramified along all codimension‑1 points. The residue and specialization maps behave functorially on X, allowing the divided‑power operations to be defined globally on X. Theorem 3 states that for any smooth X, the algebra of operations K⁽ᴹ⁾ᵢ(X)/p → K⁽ᴹ⁾_(X)/p is again a free module over K⁽ᴹ⁾_*(X)/p generated by the appropriate γₙ’s, exactly as in the field case. By the Bloch‑Kato theorem, this yields a complete description of operations on the unramified motivic cohomology of smooth schemes.

In summary, the paper shows that all natural operations on Milnor K‑theory modulo p are generated by the divided‑power operations γₙ, together with a small correction term when p=2 and –1 is not a square. This provides a unified, explicit, and computationally tractable framework for Steenrod‑type operations across Milnor K‑theory, motivic cohomology, Galois cohomology, and even integral K‑theory, and it extends seamlessly to the setting of smooth algebraic varieties.