Abstract
AbstractFor a primepand a fieldkof characteristic$p,$we define Steenrod operations$P^{n}_{k}$on motivic cohomology with$\mathbb {F}_{p}$-coefficients of smooth varieties defined over the base field$k.$We show that$P^{n}_{k}$is thepth power on$H^{2n,n}(-,\mathbb {F}_{p}) \cong CH^{n}(-)/p$and prove an instability result for the operations. Restricted to modpChow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. Using these new operations, we remove previous restrictions on the characteristic of the base field for Rost’s degree formula. Over a base field of characteristic$2,$we obtain new results on quadratic forms.
Highlights
For a prime, Voevodsky’s construction of Steenrod operations for the coefficient field F uses the calculation of the motivic cohomology of
Haution made progress on this problem by constructing the first − 1 homological Steenrod operations on Chow groups mod and -primary torsion over any base field [12], defining the first Steenrod square on mod 2 Chow groups over any base field [13] and constructing weak forms of the second and third Steenrod squares over a field of characteristic 2 [15]
Restricted to mod Chow groups, I prove that the satisfy expected properties such as Adem relations and the Cartan formula. we show that the operations agree with the operations, constructed by Voevodsky for char( ) = 0, on the mod Chow rings of flag varieties in characteristic 0
Summary
Let be a field of characteristic > 0. Note that ∗,∗ ( , F ) is defined in [35] as a limit of motivic cohomology rings of smooth schemes over the base field. This explains why power series appear in this theorem. For a morphism : 1 → 2 of base schemes, let ∗ ≔ ∗ : ( 1) → ( 2) and ∗ ≔ ∗ : ( 2) → ( 1) denote the right derived push-forward and left derived pullback functors, respectively. The isomorphism between motivic cohomology and Bloch’s higher Chow groups is compatible with pullback maps and product structures [30, Theorem 6.7].
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