Abstract
We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to the theory of motivic fundamental classes of D\'eglise, Jin, and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite E-correspondences for E a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calm\`es and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of E-module spectra factors through the category of finite E-correspondences.
Highlights
We relate the recognition principle for infinite P1-loop spaces to the theory of motivic fundamental classes of Deglise, Jin and Khan
Both the theory of fundamental classes and that of framed motives imply the existence of certain transfers, called framed transfers, in such a cohomology theory
These transfers can be encoded by an extension of E∗,∗(−) to the category hCorrfr(SmS) of framed correspondences: In the first part of this paper, we show that the framed transfers produced by both theories agree
Summary
We introduce finite R-correspondences for a motivic ring spectrum R, generalizing the finite correspondences of Voevodsky and the finite Milnor–Witt correspondences of Calmes and Fasel. Throughout this section, S is a fixed base scheme. All S-schemes are assumed to be separated
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