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Register a new userMilnor excision for motivic spectra
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We prove that the $infty$-category of motivic spectra satisfies Milnor excision: if $Ato B$ is a morphism of commutative rings sending an ideal $Isubset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is equivalent to a pair of motivic spectra over $B$ and $A/I$ that are identified over $B/IB$. Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoubs etale motives over schemes of finite virtual cohomological dimension.
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We give necessary and sufficient conditions for a cdh sheaf to satisfy Milnor excision, following ideas of Bhatt and Mathew. Along the way, we show that the cdh infinity-topos of a quasi-compact quasi-separated scheme of finite valuative dimension is hypercomplete, extending a theorem of Voevodsky to nonnoetherian schemes. As an application, we show that if E is a motivic spectrum over a field k which is n-torsion for some n invertible in k, then the cohomology theory on k-schemes defined by E satisfies Milnor excision.
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 finite etale, we show that it stabilizes to a functor $f_otimes: mathcal{SH}(S) to mathcal{SH}(S)$, where $mathcal{SH}(S)$ is the $mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendiecks Galois theory, with Betti realization, and with Voevodskys slice filtration; we prove that the norm functors categorify Rosts multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $Hmathbb Z$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $Hmathbb Z$ is a common refinement of Fulton and MacPhersons mutliplicative transfers on Chow groups and of Voevodskys power operations in motivic cohomology.
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 homotopy theory: the (unstable) purity and gluing theorems of Morel and Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoubs and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G-schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology.
We obtain geometric models for the infinite loop spaces of the motivic spectra $mathrm{MGL}$, $mathrm{MSL}$, and $mathbf{1}$ over a field. They are motivically equivalent to $mathbb{Z}times mathrm{Hilb}_infty^mathrm{lci}(mathbb{A}^infty)^+$, $mathbb{Z}times mathrm{Hilb}_infty^mathrm{or}(mathbb{A}^infty)^+$, and $mathbb{Z}times mathrm{Hilb}_infty^mathrm{fr}(mathbb{A}^infty)^+$, respectively, where $mathrm{Hilb}_d^mathrm{lci}(mathbb{A}^n)$ (resp. $mathrm{Hilb}_d^mathrm{or}(mathbb{A}^n)$, $mathrm{Hilb}_d^mathrm{fr}(mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $mathbb{A}^n$, and $+$ is Quillens plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.
We construct a new motivic integration morphism, the so-call bounded integral, that interpolates both the integration morphisms with and without volume forms of Hrushovski and Kazhdan. This is done within the framework of model theory of algebraically closed valued fields of equicharacteristic zero. As an application, we recover and extend some results of Hrushovski and Loeser about the motivic Milnor fiber.
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