Fibrant resolutions for motivic Thom spectra

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 $$M_E^{\mathbb{G}}(X)=(M_E(X),M_E(X)(1),M_E(X)(2),\dots ),$$ 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 $R{e}^{\epsilon }(E)/N$ 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.

the final accepted version

References in corpus (1)

• Modules over algebraic cobordism

Cited by in corpus (2)

• Modules over algebraic cobordism
• Correspondences and stable homotopy theory