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We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck-Verdier duality for SH_Q. Next, we prove that SH_Q is canonically SL-oriented; we compare SH_Q with the category of rational Milnor-Witt motives; and we relate the rational bivariant A^1-theory to Chow-Witt groups. These results are derived from analogous statements for the minus part of SH[1/2].
We prove a topological invariance statement for the Morel-Voevodsky motivic homotopy category, up to inverting exponential characteristics of residue fields. This implies in particular that SH[1/p] of characteristic p>0 schemes is invariant under pas
If $f:S to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal norm functor $f_otimes: mathcal H_*(S) tomathcal H_*(S)$, where $mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finit
We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fultons intersection t
We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic hom
For each prime $p$, we define a $t$-structure on the category $widehat{S^{0,0}}/tautext{-}mathbf{Mod}_{harm}^b$ of harmonic $mathbb{C}$-motivic left module spectra over $widehat{S^{0,0}}/tau$, whose MGL-homology has bounded Chow-Novikov degree, such