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Register a new userOn the infinite loop spaces of algebraic cobordism and the motivic sphere
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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.
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We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $mathbb{P}^infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $mathbb{C}$ and passing to spaces of $mathbb{C}$-valued points, we obtain new proofs of the topologic
We prove that the $infty$-category of $mathrm{MGL}$-modules over any scheme is equivalent to the $infty$-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $mathbb{P}^1$-loop spaces, we deduce that very effective $mathrm{MGL}$-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $Omega^infty_{mathbb{P}^1}mathrm{MGL}$ is the $mathbb{A}^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $Omega^infty_{mathbb{P}^1} Sigma^n_{mathbb{P}^1} mathrm{MGL}$ is the $mathbb{A}^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.
In this paper, we construct a refined, relative version of the etale realization functor of motivic spaces, first studied by Isaksen and Schmidt. Their functor goes from the $infty$-category of motivic spaces over a base scheme $S$ to the $infty$-category of $p$-profinite spaces, where $p$ is a prime which is invertible in all residue fields of $S$. In the first part of this paper, we refine the target of this functor to an $infty$-category where $p$-profinite spaces is a further completion. Roughly speaking, this $infty$-category is generated under cofiltered limits by those spaces whose associated local system on $S$ is $A^1$-invariant. We then construct a new, relative version of their etale realization functor which takes into account the geometry and arithmetic of the base scheme $S$. For example, when $S$ is the spectrum of a field $k$, our functor lands in a certain $infty$-category equivariant for the absolute Galois group. Our construction relies on a relative version of etale homotopy types in the sense of Artin-Mazur-Friedlander, which we also develop in some detail, expanding on previous work of Barnea-Harpaz-Schlank. We then stabilize our functor, in the $S^1$-direction, to produce an etale realization functor for motivic $S^1$-spectra (in other words, Nisnevich sheaves of spectra which are $A^1$-invariant). To this end, we also develop an $infty$-categorical version of the theory of profinite spectra, first explored by Quick. As an application, we refine the construction of the etale $K$-theory of Dwyer and Friedlander, and define its non-commutative extension. This latter invariant should be seen as an $ell$-adic analog of Blancs theory of semi-topological $K$-theory of non-commutative schemes. We then formulate and prove an analog of Blancs conjecture on the torsion part of this theory, generalizing the work of Antieau and Heller.
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.
In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties.
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