Smooth affine model for the framed correspondences spectrum

A. E. Druzhinin

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 484 (2019), pp. 59-71

Résumé

The framed correspondences $T$-spectrum of a smooth affine scheme is a $T$-spectrum of Nisnevich sheaves. We construct the motivically equivalent model of the $T$-spectrum representable in the category of pairs of smooth affine ind-schemes in the case of a base scheme of a finite Krull dimension. In other words, the motivic spaces of $(\mathbb{P},\infty)^{\wedge \infty}$-loops in the relative motivic sphere $\mathbb{A}_Y^{\infty+l}/(\mathbb{A}_Y^{\infty+l}-0)$ are represented in the category of pairs of smooth affine ind-schemes. The construction in not functorial on the category of smooth affine schemes, but it is so on the category of smooth affine framed schemes, that is defined in the text.