We relate the recognition principle for infinite ${\mathbf{P}}^1$‐loop spaces to the theory of motivic fundamental classes of Déglise, 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 $R$‐correspondences for $R$ a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calmès 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 $R$‐module spectra factors through the category of finite $R$‐correspondences.

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框架转移和动机基础课程

我们将无限的识别原理联系起来 ${\mathbf{P}}^{\mathrm{1个}}$Déglise，Jin和Khan的动机基础分类理论的循环空间。我们首先比较两种由原动力谱表示的同调理论自然定义的转移：识别原理给出的有框转移，这是由Voevodsky对与${\mathbf{一种}}^{ñ}/（{\mathbf{一种}}^{ñ}-0）$，并且通过Verdier变形将Gysin转移到法线圆锥。然后我们介绍有限类别$\mathrm{[R}$-对应 $\mathrm{[R}$动力环谱，概括了Voevodsky的有限对应类别以及Calmès和Fasel的有限Milnor-Witt对应类别。使用基本类的形式主义，我们证明了自然函子从构架对应的类别到$\mathrm{[R}$通过有限类别的模块频谱因子 $\mathrm{[R}$对应。