What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory

Question

I would like a reference/argument for the truth/falsity of the following statement:

The etale sheafification of the unramified Milnor-Witt K-theory (Nisnevich) sheaves are the (etale sheafification of?) unramified Milnor K-theory sheaves.

Thanks!

Answer 1

We have a short exact sequence of sheaves of abelian groups (see e.g. Morel's book on $\mathbb{A}^1$-algebraic topology): $$ 0\to\mathbf{I}^{n+1}\to\mathbf{K}^{\rm MW}_n\to \mathbf{K}^{\rm M}_n\to 0, $$ where $\mathbf{I}^{n+1}$ is the sheaf of $n+1$-th powers of the fundamental ideal in the Witt ring. Over a strictly henselian local ring (essentially smooth), this yields a short exact sequence of global sections (by exactness of the Gersten complex). So the question is equivalent to asking if the etale sheafification of $\mathbf{I}^{n+1}$ is trivial. (Alternatively, this can be deduced from the new presentation of Milnor-Witt K-theory of local rings by Gille-Scully-Zhong.)

So let $R$ be an essentially smooth strictly henselian local ring with $1/2\in R$. Its 2-cohomological dimension will be at most the Krull dimension of $R$. This implies that the restriction of $\mathbf{I}^{n+1}$ to the small Nisnevich site of $R$ vanishes whenever $n>\dim R$ (well, actually we only need the global sections). This follows as in Proposition 5.1 of

• A. Asok and J. Fasel. A cohomological classification of vector bundles on smooth affine threefolds. Duke Math. J. 163 (2014), 2561-2601.

Now we want to show that $\mathbf{I}^j(R)=0$ for all $j$. By descending induction, it suffices to show that $I^n(R)/I^{n+1}(R)=0$. By Voevodsky's solution of the Milnor conjecture, we know that we have an identification of Nisnevich sheaves $\mathbf{I}^j/\mathbf{I}^{j+1}\cong \mathbf{K}^{\rm M}_j/2$. Now if $(R,\mathfrak{m})$ is a local henselian ring with residue characteristic $\neq 2$, then $K^{\rm M}_j(R)/2\cong K^{\rm M}_j(R/\mathfrak{m})/2$ by rigidity. Since $(R,\mathfrak{m})$ is in fact a strictly henselian local ring, the residue field $R/\mathfrak{m}$ is algebraically closed and therefore has trivial mod 2 Milnor K-theory. This shows that $I^j(R)=0$ for all $j$, proving that the natural projection $\mathbf{K}^{\rm MW}_n\to\mathbf{K}^{\rm M}_n$ becomes an isomorphism after etale sheafification.

Some more remarks on rigidity: one possibility is to apply Hornbostel-Yagunov rigidity for orientable $\mathbb{A}^1$-representable theories which works for Milnor K-theory. Another possibility is to use Kerz's identification of Milnor K-theory of local rings with motivic cohomology. Yet another possibility is to use the other part of the Milnor conjecture, the identification of $\mathbf{I}^n/\mathbf{I}^{n+1}$ with the Nisnevich sheafification of ${\rm H}^n_{\rm et}(-,\mu_2^{\otimes n})$. This would also satisfy rigidity by results of Gabber. Essentially, the rigidity argument requires $\mathbb{A}^1$-invariance, existence of suitable transfers and mod $p$ coefficients (prime to the characteristic). These things are satisfied for either Milnor K-theory or etale cohomology.