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Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.

Baeza [B, V.6.3] extended this result to Witt groups of semilocal (commutative) rings $A$ (assume $2\in A^\times$ for simplicity).

Question: Is it known whether the same result holds for arbitrary rings, or more generally, for schemes? Alternatively, are there examples of schemes $X$ such that $W(X)$ has non-trivial odd torsion?

I am particularly interested in the case where $X$ is a real algebraic variety.

The definition of the Witt group of rings and schemes can be found, for instance, in section 1.2 here.

[B] Baeza, Ricardo, Quadratic forms over semilocal rings, Lecture Notes in Mathematics. 655. Berlin-Heidelberg-New York: Springer-Verlag. VI, 199 p. (1978). ZBL0382.10014.

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Examples of smooth real varieties whose Witt group has odd torsion can be found in a paper of Jacobson:

The idea is the following: from the Gersten conjecture for Witt groups, there is a spectral sequence $E^{p,q}_2:H^p_{\rm Zar}(X,\mathbf{W})\Rightarrow W^{p+q}(X)$ which was discussed by Balmer and Walter

  • P. Balmer and C. Walter: A Gersten-Witt spectral sequence for regular schemes. Ann. Sci. École Norm. Sup. (4)35(1), 127–152 (2002).

For schemes of dimension $\leq 7$, this provides an exact sequence $0\to H^4_{\rm Zar}(X,\mathbf{W})\to W^0(X)\to H^0_{\rm Zar}(X,\mathbf{W})$ which describes the kernel of the map from the Witt group to the unramified Witt group (which by the Pfister result over fields has only 2-primary torsion). For a real variety $X$, Jacobson's work on the signature allows to identify $H^4_{\rm Zar}(X,\mathbf{W}[1/2])$ with singular cohomology $H^4_{\rm sing}(X,\mathbb{Z}[1/2])$. This way, odd torsion in $H^4_{\rm sing}(X,\mathbb{Z})$ yields odd torsion in $W(X)$, cf. Corollary 5.6 of Jacobson's paper.

To get an explicit example, take a 5-dimensional lens space $L^5(p)=S^5/\mu_p$ for an odd prime $p$; this has $H^4(L^5(p),\mathbb{Z})=\mathbb{Z}/p$. Use Nash-Tognioli to write $L^5(p)$ as real points of a smooth real variety $X$. This $X$ will have a $\mathbb{Z}/p$ summand in $W(X)$. (See the discussion on p. 21 of Jacobson's paper.)

On p. 3 of the paper Jacobson says that such examples were already discussed by Karoubi in 1976 in the following paper. (Karoubi used comparison to complex K-theory to see the odd torsion.)

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  • $\begingroup$ Beautiful! I did not know Jacobson and Karoubi's works. $\endgroup$ – Uriya First Jul 14 '19 at 7:30

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