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Framed transfers and motivic fundamental classes. (English) Zbl 1444.14050
The main goal of this paper is to relate Voevodsky’s theory of framed transfers ([G. Garkusha and I. Panin, “Framed motives of algebraic varieties (after V. Voevodsky)”, J. Am. Math. Soc. (to appear)]) and the theory of fundamental classes ([F. Déglise et al., “Fundamental classes in motivic homotopy theory”, Preprint, arXiv:1805.05920]) in motivic homotopy theory by comparing the associated Gysin transfers.
If $$E$$ is a motivic spectrum over a scheme $$S$$, for every smooth $$S$$-scheme $$X$$ let $E(X)=Maps(\Sigma^\infty X_+,E)$ be the associated cohomology theory. A framed correspondence between smooth $$S$$-schemes $$X$$ and $$Y$$ is a span of the form $$\alpha:X\xleftarrow{f}Z\xrightarrow{g}Y$$ where $$f$$ is finite syntomic, together with a trivialization of the cotangent complex of $$f$$ in the $$K$$-theory space of $$Z$$. To such an $$\alpha$$ there are several ways to associate a Gysin morphism $\alpha^*:E(Y)\to E(X)$ as follows: (1) one can use the fundamental class, constructed using deformation to the normal cone; (2) one can define a Gysin morphism via Voevodsky’s theory of framed transfers; (3) one can use the theory of framed motivic spectra ([E. Elmanto et al., “Motivic infinite loop spaces”, Preprint, arXiv:1711.05248]). The authors show that all these constructions are equivalent (Theorems 3.2.11 and 3.3.10).
In Section 4, the authors construct, for a motivic ring spectrum $$R$$ over $$S$$, an enriched homotopy category $$hCorr^R(Sm_S)$$ of finite $$R$$-correspondences over $$S$$, which generalizes several previous constructions such as ([B. Calmès and J. Fasel, “Finite Chow-Witt correspondences”, Preprint, arXiv:1412.2989]) and [A. Druzhinin and H. Kolderup, Algebr. Geom. Topol. 20, No. 3, 1487–1541 (2020; Zbl 1442.14079)]), and which conjecturally can be enhanced to an $$\infty$$-category of correspondences.

MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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