Séminaire de Mathématique

Framed motives of algebraic varieties (after V. Voevodsky)

by Prof. Grigory GARKUSHA (Swansea University, UK & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


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

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 Mfr (X), which is an object of this category. One of the main results states that the bispectrum

(Mfr(X), Mfr(X)(1), Mfr(X)(2), …)

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 A1-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 ΩΣSn with Sn a sphere is given in terms of framed correspondences. This machinery also allows to recover in characteristic zero the celebrated theorem of Morel stating that the stable π 0,0(k) equals the Grothendiek-Witt ring of the field k.