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Modules over algebraic cobordism. (English) Zbl 1458.14027
The article under review is a continuation of [E. Elmanto et. al. “Motivic infinite loop spaces”, Preprint, arXiv:1711.05248]. The main result is a concrete description of the $$\infty$$-category of modules over Voevodsky’s algebraic cobordism spectrum $$\operatorname{MGL}$$ as motivic spectra with finite syntomic transfers $\mathbf{Mod}_{\operatorname{MGL}}(\mathbf{SH}(S)) \simeq \mathbf{SH}^{\operatorname{fsyn}}(S)$ for an arbitrary base scheme $$S$$ (Theorem 4.1.3). The proof of this result strongly relies on the flexible theory of framed correspondences developed in [loc. cit.]. One important result about these is the reconstruction theorem (Theorem 18 in [M. Hoyois, Compos. Math. 157, No. 1, 1–11 (2021; Zbl 1455.14042)]), which says that there is an equivalence of $$\infty$$-categories between motivic spectra and framed motivic spectra $\mathbf{SH}(S) \simeq \mathbf{SH}^\mathrm{fr}(S)$ over any scheme $$S$$. This may be viewed as the “sphere spectrum version” of the above equivalence. The rough idea is that certain motivic spectra of interest may have a more geometric description as motivic spectra with framed correspondences. The second main result of this paper is an example of such a description. Namely for a so called smooth stable tangential structure (Definitions 3.3.1 and 3.3.8) $$\beta$$ there is an equivalence of motivic spectra $M\beta \simeq \Sigma^\infty_{\mathbf{T},\operatorname{fr}} \mathcal{FQ}\operatorname{Sm}_S^\beta$ where the left hand side is the motivic Thom spectrum associated to $$\beta$$ and the right hand side is the framed suspension spectrum of the moduli stack of finite quasi-smooth derived $$S$$-schemes with $$\beta$$-structure (Theorem 3.3.10). This general result has many concrete incarnations, a special case for example is the equivalence of motivic spectra $\operatorname{MGL}_S \simeq \Sigma^\infty_{T,\operatorname{fr}}\mathcal{FS}\mathrm{yn}_S,$ where $$\mathcal{FS}\mathrm{yn}_S$$ is the moduli stack of finite syntomic $$S$$-schemes (Theorem 3.4.1). From this equivalence the authors then deduce the above description of the $$\infty$$-category of $$\operatorname{MGL}_S$$-modules. Furthermore the motivic recognition principle (Theorem 3.5.14 in [E. Elmanto et. al., “Motivic infinite loop spaces”, Preprint, arXiv:1711.05248]) ensures that all of the results mentioned above have a more concrete form when $$S$$ is a perfect field $$k$$. For example one gets an equivalence of $$\infty$$-categories $\mathbf{Mod}_{\operatorname{MGL}}(\mathbf{SH}^\mathrm{veff}(k)) \simeq \mathbf{H}^\mathrm{fsyn}(k)^\mathrm{gp}$ between very effective $$\operatorname{MGL}$$-modules and grouplike motivic spaces with finite syntomic transfers (Theorem 4.1.4). It is also noteworthy that for the proof of Theorem 4.1.3 the language of derived schemes, which is used throughout the paper, is essential even though the statement itself does not involve derived schemes.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14D23 Stacks and moduli problems
##### Keywords:
algebraic cobordism; framed correspondences
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##### References:
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