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Motivic cohomology spectral sequence and motivic Steenrod algebra

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Serge Yagunov
Tue, 07/06/2011 - 14:00 - 15:00
MPIM Lecture Hall

The motivic cohomology spectral sequence (MCSS) is an algebraic-geometrical counterpart of the Atiyah--Hirzebruch spectral sequence. For smooth varieties, it has the second term consisting of
motivic cohomology groups and converges to algebraic $K$-groups. The spectral sequence was initially constructed only for fields by Bloch and Lichtenbaum in their unpublished preprint. Further, three different approaches to construction of the spectral sequence for varieties were given by Friedlander -- Suslin,
Grayson, and Voevodsky. The behavior of differentials in the MCSS is partially inherited from the topological case.
For example,  taken with rational coefficients it collapses at its $E_2$-term, as was shown by Levine.

The next natural question in the row is to describe possible non-trivial differentials in the MCSS with $p$-local coefficients. We will see that in this case the non-trivial differentials exist and characterize them as stable motivic cohomological operations, i.e. elements of the motivic Steenrod algebra.

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