# Modules over algebraic cobordism

## TL;DR

This paper establishes an equivalence between modules over algebraic cobordism spectra and motivic spectra with finite syntomic transfers, providing new insights into motivic homotopy theory and moduli stacks.

## Contribution

It proves the equivalence of $ ext{MGL}$-modules with motivic spectra with finite syntomic transfers and describes motivic Thom spectra via moduli stacks of derived schemes.

## Key findings

- $ ext{MGL}$-modules are equivalent to motivic spectra with finite syntomic transfers.
- $ ext{MGL}$-modules over a perfect field relate to grouplike motivic spaces.
- Descriptions of motivic Thom spectra via moduli stacks of derived schemes.

## 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 $\Omega^\infty_{\mathbb{P}^1}\mathrm{MGL}$ 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 $-n$.