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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.

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