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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