My question is whether the construction of higher Witt groups of a scheme in Schlichting's Hermitian K-theory of Exact Categories agrees with the definition in Balmer's chapter in the Handbook of K-theory when 2 is inverted (i.e. the schemes are over $\mathbb{Z}[\frac{1}{2}]$, although I'd also be happy to know whether this is true when we tensor the Witt groups with $\mathbb{Z}[\frac{1}{2}]$). See below for both definitions of higher Witt groups. When 2 is not inverted, they only necessarily agree on the 0-th Witt group.
More background if necessary
Schlichting defines higher Witt groups of an exact category with duality $(\mathcal{E}, *, \eta)$ as the homotopy groups of the geometric realization of a Hermitian Q construction, which is roughly as follows
Given an exact category $\mathcal{E}$ with duality, define the category $Q^h\mathcal{E}$ as the category where
- objects are symmetric spaces in $\mathcal{E}$, i.e. pairs $(X,\varphi)$ of $X\in \text{ob }\mathcal{E}$ and an isomorphism $\varphi: X\mapsto X^{*}$ such that $\varphi^*\eta_X = \varphi$
- morphisms from $(X, \varphi)$ to $(Y, \psi)$ are equivalence classes of diagrams $X \xleftarrow{p}U\xrightarrow{i}Y$ where $p$ is an admissible epimorphism, $i$ an admissible monomorphism, and the restrictions of the symmetric forms on $X$ and $Y$ to $U$ agree.
- composition is pullback
(For a more precise statement, see Definition 4.1 in p.12 of Schlichting)
The $0$-th Witt group $\pi_0(|Q^h\mathcal{E}|)$ turns out to be the usual $$W_0(\mathcal{E}) = \{\text{isoclasses of symmetric spaces}\}/\{\text{metabolics}\}$$ (metabolics are defined in Schlichting, p. 6, Def 2.5, the 0th Witt group is defined in Schlichting p. 7, first paragraph of 2.2 and also in Balmer p. 7 Definition 1.1.27)
We can take Witt groups of a scheme $X$ by using the exact category of vector bundles over $X$ with the usual duality.
On the other hand, Balmer defines higher Witt groups of a triangulated category $\mathcal{K}$ with duality by taking $$W_0(\mathcal{K}) = \{\text{isoclasses of symmetric spaces}\}/\{\text{metabolics}\}$$ as above, and taking $$W_n(\mathcal{K}) = W_0(T^n\mathcal{K})$$ where $T^n$ denotes "applying the shift functor $n$ times" in the triangulated category, which can change the duality (see Balmer Def 1.4.1 and Def 1.4.4 for details). Now we can take Balmer Witt groups of a scheme $X$ by using the category of perfect complexes over $X$.
These definitions certainly agree on $W_0$, but they don't agree in the higher Witt groups (as can be seen in Remark 4.2 of Schlichting), and I want to know whether they do agree when $2$ is inverted as I said above. Thanks!
