Grothendieck-Witt theory of derived schemes

Abstract

We construct a non-$\mathbb{A}^1$-invariant motivic ring spectrum $\mathrm{KO}$ over $\mathrm{Spec}(\mathbb{Z})$, whose associated cohomology theory on qcqs derived schemes is the Grothendieck-Witt theory of classical symmetric forms (as opposed to homotopy symmetric forms). In particular, we show that this theory satisfies Nisnevich descent, smooth blowup excision, a projective bundle formula, and is locally left Kan extended from smooth $\mathbb{Z}$-schemes up to Bass delooping. More generally, our construction produces $\mathrm{KO}$-modules representing localizing invariants of two different families of Poincar\'e structures on derived schemes, which we call "classical" and "genuine"; the latter Poincar\'e structures are defined for spectral schemes with involution, but the former only for derived schemes. We then establish basic properties of these motivic spectra. As in $\mathbb{A}^1$-homotopy theory, the fracture square of $\mathrm{KO}$ with respect to the Hopf element recovers the fundamental cartesian square relating GW-theory, L-theory, and K-theory. A new phenomenon when $2$ is not a unit is that $\mathrm{KO}$ is not Bott-periodic, and the left and right Bott periodizations of $\mathrm{KO}$ represent the Grothendieck-Witt theories of homotopy symmetric and homotopy quadratic forms, respectively. We also construct the expected metalinear $\mathrm{E}_\infty$-orientation of $\mathrm{KO}$. Finally, we show that the $\mathbb{A}^1$-localization of $\mathrm{KO}$ recovers the motivic spectrum recently constructed by Calm\`es, Harpaz, and Nardin.