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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 $-n$.
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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.
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
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.
We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendiecks six operations. Objects in this category represent generalized cohomology theories for stacks like algebraic K-theory, as well as new examples like genuine motivic cohomology and algebraic cobordism. These cohomology theories admit Gysin maps and satisfy homotopy invariance, localization, and Mayer-Vietoris. We also prove a fixed point localization formula for torus actions. Finally, the construction is contrasted with a limit-extended stable motivic homotopy category: we show for example that limit-extended motivic cohomology of quotient stacks is computed by the equivariant higher Chow groups of Edidin-Graham, and we also get a good new theory of Borel-equivariant algebraic cobordism.
Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).
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