Fibrant resolutions for motivic Thom spectra

TL;DR

This paper constructs explicit fibrant resolutions for motivic Thom spectra using framed correspondences and motives, demonstrating their representation in bispectra and applications to algebraic cobordism and topological spectra.

Contribution

It introduces explicit fibrant resolutions for motivic Thom spectra via framed motives, connecting algebraic and topological spectra, and computes the algebraic cobordism spectrum MGL in this framework.

Findings

Constructed explicit fibrant resolutions for motivic Thom spectra.

Proved that the bispectrum of twisted E-framed motives represents E in the bispectrum category.

Computed the algebraic cobordism spectrum MGL in terms of Ω-correspondences.

Abstract

Using the theory of framed correspondences developed by Voevodsky [24] and the machinery of framed motives introduced and developed in [6], various explicit fibrant resolutions for a motivic Thom spectrum $E$ are constructed in this paper. It is shown that the bispectrum $$M_E^{\mathbb G}(X)=(M_{E}(X),M_{E}(X)(1),M_{E}(X)(2),\ldots),$$ each term of which is a twisted $E$-framed motive of $X$, introduced in the paper, represents $X_+\wedge E$ in the category of bispectra. As a topological application, it is proved that the $E$-framed motive with finite coefficients $M_E(pt)(pt)/N$, $N>0$, of the point $pt=Spec (k)$ evaluated at $pt$ is a quasi-fibrant model of the topological $S^2$-spectrum $Re^\epsilon(E)/N$ whenever the base field $k$ is algebraically closed of characteristic zero with an embedding $\epsilon:k\hookrightarrow\mathbb C$. Furthermore, the algebraic cobordism spectrum $MGL$ is computed in terms of $\Omega$-correspondences in the sense of [15]. It is also proved that $MGL$ is represented by a bispectrum each term of which is a sequential colimit of simplicial smooth quasi-projective varieties.