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SUMMARY:Framed motives of algebraic varieties (after V. Voevodsky)
DTSTART;VALUE=DATE-TIME:20160915T090000Z
DTEND;VALUE=DATE-TIME:20160915T100000Z
DTSTAMP;VALUE=DATE-TIME:20191024T021534Z
UID:indico-event-1421@indico.math.cnrs.fr
DESCRIPTION:This is a joint work with Ivan Panin (St. Petersburg). Using t
he machinery of framed correspondences and framed sheaves developed by Voe
vodsky in the early 2000-s\, a triangulated category of framed motives of
smooth algebraic varieties is introduced and studied. To any smooth algebr
aic 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\n \n
\n(Mfr(X)\, Mfr(X)(1)\, Mfr(X)(2)\, …)\n\n\neach term of which is a twis
ted framed motive of X\, has motivic homotopy type of the suspension bispe
ctrum 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 t
hat it reconstructs the motivic stable homotopy theory SH(k) in the sense
of Morel-Voevodsky. As a topological application\, it is shown that the fr
amed motive of the point evaluated at the point yields an explicit model f
or 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∞Sn with Sn a sphere is given in terms of framed correspon
dences. This machinery also allows to recover in characteristic zero the c
elebrated theorem of Morel stating that the stable π 0\,0(k) equals the G
rothendiek-Witt ring of the field k.\n\nhttps://indico.math.cnrs.fr/event/
1421/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/1421/
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