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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!

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  • $\begingroup$ The quadratic closure (or the separable, if you prefer) is an etale cover. For quadratically closed fields, milnorwitt is iso to milnor. So it's true. Sorry for not writing this as answer, I'm on the mobile.. $\endgroup$ – Konrad Voelkel Oct 5 '15 at 21:43
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    $\begingroup$ @KonradVoelkel: your argument only shows that the sheaves agree on separably closed fields, for the identification of sheafifications you need agreement on strict henselizations of local rings of smooth varieties. One way to prove this would be to show that the sheafification is unramified and strictly $\mathbb{A}^1$-invariant, but I am not sure if this so easy to do with the étale sheafification. You could also compare the Gersten resolutions for Milnor and Milnor-Witt, but they will not be isomorphic because only the closed point has algebraically closed residue field. $\endgroup$ – Matthias Wendt Oct 6 '15 at 8:49
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    $\begingroup$ @MatthiasWendt. Just to clarify: you're saying that we should compare the $E_1$-pages of the Brown-Gersten spectral sequence (I guess for MW this would be in Morel's book and for M this is Kerz), which doesn't seem equivalent but it should give us an equivalence eventually? Please correct me if I am wrong. $\endgroup$ – Elden Elmanto Oct 6 '15 at 14:27
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    $\begingroup$ @EldenElmanto: yes. If you want to prove that the étale sheafifications are the same, then you want to prove that, for strictly henselian local rings, the induced map on $H^0$ of the Gersten complex (which comes from the $E_1$-page of the Brown--Gersten spectral sequence) is an isomorphism. It usually won't be an isomorphism before taking cohomology because the field of fractions of a strictly henselian local ring could have nontrivial quadratic extensions (as happens e.g. for $\mathbb{Q}_p^{\operatorname{nr}}$). $\endgroup$ – Matthias Wendt Oct 6 '15 at 15:03
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    $\begingroup$ @EldenElmanto: sure, no problem, you can contact me via email. $\endgroup$ – Matthias Wendt Oct 6 '15 at 19:11
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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.

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