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For a prime number
, we show that differentials
in the motivic cohomology spectral sequence with
-local coefficients vanish unless
. We obtain an explicit formula for the first non-trivial differential
, expressing it in terms of motivic Steenrod
-power operations and Bockstein maps. To this end, we compute the algebra of operations of weight
-local coefficients. Finally, we construct examples of varieties having non-trivial differentials
in their motivic cohomology spectral sequences.
It is shown that the Grayson tower for
-theory of smooth algebraic varieties is isomorphic to the slice tower of
-spectra. We also extend the Grayson tower to bispectra, and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic
. This solves Suslin’s problem about these two spectral sequences in the affirmative.
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