Séminaire de Mathématique
# Framed motives of algebraic varieties (after V. Voevodsky)

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

This is a joint work with Ivan Panin (St. Petersburg). Using the machinery of framed correspondences and framed sheaves developed by Voevodsky in the early 2000-s, a triangulated category of framed motives of smooth algebraic varieties is introduced and studied. To any smooth algebraic variety *X* we associate the framed motive *M _{fr}* (

(*M _{fr}*(

each term of which is a twisted framed motive of *X*, has motivic homotopy type of the suspension bispectrum of *X* (this result is an *A*^{1}-homotopy analog of a theorem of G. Segal). We also construct a triangulated category of framed bispectra and show that it reconstructs the motivic stable homotopy theory *SH*(*k*) in the sense of Morel-Voevodsky. As a topological application, it is shown that the framed motive of the point evaluated at the point yields an explicit model for the classical sphere spectrum whenever the base field is algebraically closed of characteristic zero. Over such a field an explicit model for the space Ω^{∞}Σ^{∞}*S ^{n}* with

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