Motivic cohomology spectral sequence and Steenrod operations

Abstract

For a prime number$p$, we show that differentials$d_{n}$in the motivic cohomology spectral sequence with$p$-local coefficients vanish unless$p-1$divides$n-1$. We obtain an explicit formula for the first non-trivial differential$d_{p}$, expressing it in terms of motivic Steenrod$p$-power operations and Bockstein maps. To this end, we compute the algebra of operations of weight$p-1$with$p$-local coefficients. Finally, we construct examples of varieties having non-trivial differentials$d_{p}$in their motivic cohomology spectral sequences.

Second differentials in the Quillen spectral sequence

“…S. Yagunov carried out a similar program for the motivic spectral sequence in his recent paper [Yag16]. For a prime l ≥ 3, he proved that the l-local part of d r vanishes for r < l and obtained an expression for the l-local part of d l in terms of stable motivic operations of Voevodsky.…”