Abstract
AbstractWe prove that the$\infty $-category of$\mathrm{MGL} $-modules over any scheme is equivalent to the$\infty $-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite$\mathbf{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 _{\mathbf{P} ^1}\mathrm{MGL} $is the$\mathbf{A} ^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for$n>0$,$\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $is the$\mathbf{A} ^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension$-n$.
Highlights
This article contains two main results:◦ a concrete description of the ∞-category of modules over Voevodsky’s algebraic cobordism spectrum MGL;◦ a computation of the infinite P1-loop spaces of effective motivic Thom spectra in terms of finite quasi-smooth derived schemes with tangential structure and cobordisms.Both results have an incarnation over arbitrary base schemes and take a more concrete form over perfect fields
We prove that the ∞-category of MGL-modules over any scheme is equivalent to the ∞-category of motivic spectra with finite syntomic transfers
Using the recognition principle for infinite P1-loop spaces, we deduce that very effective MGL-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers
Summary
◦ a concrete description of the ∞-category of modules over Voevodsky’s algebraic cobordism spectrum MGL (and its variants such as MSL);. ◦ a computation of the infinite P1-loop spaces of effective motivic Thom spectra in terms of finite quasi-smooth derived schemes with tangential structure and cobordisms. Both results have an incarnation over arbitrary base schemes and take a more concrete form over perfect fields.
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