Modules over algebraic cobordism

Abstract

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 ${\Omega }_{{\mathbb{P}}^1}^{\infty }\mathrm{MGL}$ is the ${\mathbb{A}}^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, ${\Omega }_{{\mathbb{P}}^1}^{\infty }{\Sigma }_{{\mathbb{P}}^1}^n\mathrm{MGL}$ is the ${\mathbb{A}}^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.

arXiv comment41 pages. Final version, to appear in Forum of Mathematics, Pi

References in corpus (15)

• Algebraic cobordism
• Noetherian approximation of algebraic spaces and stacks
• On the algebraic K-theory of higher categories
• Iterated spans and classical topological field theories
• Fundamental classes in motivic homotopy theory
• Modules over algebraic cobordism
• Orientation theory in arithmetic geometry
• Derived algebraic cobordism
• On the effectivity of spectra representing motivic cohomology theories
• On Modules Over Motivic Ring Spectra
• The localization theorem for framed motivic spaces
• Framed transfers and motivic fundamental classes
• Fibrant resolutions for motivic Thom spectra
• On the infinite loop spaces of algebraic cobordism and the motivic sphere
• Framed motives of smooth affine pairs

Cited by in corpus (19)

• Modules over algebraic cobordism
• A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
• Virtual fundamental classes of derived stacks I
• K-theory and G-theory of derived algebraic stacks
• On the rational motivic homotopy category
• Fibrant resolutions for motivic Thom spectra
• Some recent advances in topological Hochschild homology
• Precobordism and cobordism
• On the infinite loop spaces of algebraic cobordism and the motivic sphere
• Cohomology of the moduli stack of algebraic vector bundles
• The unit map of the algebraic special linear cobordism spectrum
• Algebraic Spivak's theorem and applications
• Notes on motivic infinite loop space theory
• Deformation theory of perfect complexes and traces
• Determinant map for the prestack of Tate objects
• Correspondences and stable homotopy theory
• Universality of Barwick's unfurling construction
• Introduction to Framed Correspondences
• The geometric diagonal of the special linear algebraic cobordism