haenel@math.uni-wuppertal.de
We consider Chow-Witt groups of a scheme and try to generalise the definition to the tensor triangular (=tt) categories with dualities, i.e. triangulated categories equipped with a "tensor product" and a duality, all behaving sufficiently well.
The invariant of Chow-Witt groups is defined as the cohomology of the so-called Rost-Schmidt complex. It is not possible to translate all the complicated differentials to the tt-setting, so we follow the idea Klein already used to construct tensor triangular Chow groups: We aim to rewrite the respective image and kernel of the differentials in terms of easier maps such as localisation, quotient, idempotent completion or inclusion maps, which we can then translate into the tt-setting. To justify its name, we work on an agreement theorem showing that for the derived category of perfect complexes $D^{perf}(X)$, the tt-definition recovers the usual Chow-Witt groups.