arXiv:2611.13248v1 Announce Type: new Abstract: In analogy with the passage from connective to non-connective algebraic K-theory, Schlichting put forward the Karoubi-Grothendieck-Witt spectrum as an approximation to Grothendieck-Witt theory by a functor invariant under idempotent completion. In this paper, we extend his construction to the setting of Poincar\'e $\infty$-categories and characterise the resulting functor by the expected universal property. Our main result places these Karoubi-Grothendieck-Witt-spectra in an exact sequence involving the ultimately decorated L-spectrum and homotopy orbits of non-connective K-theory. This confirms a conjecture of Williams which had remained open even for rings in which $2$ is invertible, and allows us to deduce universal characterisations for ultimately decorated L-theory. We obtain these results by unifying the Bass-Heller-Swan and Ranicki-Shaneson splittings into a general decomposition theorem that yields, in particular, formulas for Grothendieck-Witt groups of Laurent polynomial rings. Finally, we construct multiplicative structures on both versions of our Grothendieck-Witt and L-functors and refine their universal properties to multiplicative ones. Along the way, we show that the original versions are even idempotent with respect to Day convolution.
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