# Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

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.

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.)

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