Abstract

Using the theory of framed correspondences developed by Voevodsky and the machinery of framed motives introduced and developed in recent work of Garkusha and Panin, various explicit fibrant resolutions for a motivic Thom spectrum $E$ are constructed in this paper. The bispectrum

each term of which is a twisted $E$-framed motive of $X$, is introduced and shown to represent $X_{+}\wedge E$ in the category of bispectra. As a topological application, it is proved that the $E$-framed motive with finite coefficients $M_E(\mathrm{pt}<!--FUNCTION\; APPLICATION-->)(\mathrm{pt}<!--FUNCTION\; APPLICATION-->)∕N$, $N>0$, of the point $\mathrm{pt}<!--FUNCTION\; APPLICATION-->=\mathrm{Spec}<!--FUNCTION\; APPLICATION-->k$ evaluated at $\mathrm{pt}<!--FUNCTION\; APPLICATION-->$ is a quasifibrant model of the topological $S^2$-spectrum ${\mathrm{Re}<!--FUNCTION\; APPLICATION-->}^{𝜖}(E)∕N$ whenever the base field $k$ is algebraically closed of characteristic zero with an embedding $𝜀:k\hookrightarrow ℂ$. Furthermore, the algebraic cobordism spectrum $\mathrm{MGL}<!--FUNCTION\; APPLICATION-->$ is computed in terms of $\mathrm{\Omega }$-correspondences. It is also proved that $\mathrm{MGL}<!--FUNCTION\; APPLICATION-->$ is represented by a bispectrum each term of which is a sequential colimit of simplicial smooth quasiprojective varieties.

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Mathematical Subject Classification

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