10
$\begingroup$

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.

$\endgroup$

1 Answer 1

4
$\begingroup$

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.)

$\endgroup$
1
  • $\begingroup$ Beautiful! I did not know Jacobson and Karoubi's works. $\endgroup$ Jul 14, 2019 at 7:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.