The motivic Steenrod algebra in positive characteristic

Abstract

Let S be an essentially smooth scheme over a field and ℓ = char S a prime number. We show that the algebra of bistable operations in the mod ℓ motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for S a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by usingétale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.

The first stable homotopy groups of motivic spheres

“…The formalism of this approach is remarkably convenient. It allows us to calculate differentials using the action of the motivic Steenrod algebra on motivic cohomology groups [27], [75]. Our focus in this paper is not just restricted to specific computations, but also the context we lay out to formulate and carry them out.…”

“…Voevodsky [75], Hoyois-Kelly-Østvaer[27], Spitzweck[68]): There exists a weak equivalence of right MZ/2-modulesMZ/2 ∧ MZ/Σ 1,0 MZ/2 ∨ (i,j)∈I Σ i,j MZ/2, (A.1) where I ⊂ N × N consists of pairs (i, j) of integers with i ≥ 2j > 0.With respect to this weak equivalence, the unit and multiplication maps are given by(id, 0, . .…”

On very effective hermitian K-theory

“…The same result holds for η, KQ, KGL, and base schemes over Spec(Z[ 1 2 ]) by [19,Theorem 3.4], but it is plainly false for the effective covers of KQ and KGL by the proof of [21, Corollary 5.1]. By using (1.2) we identify the Betti realization of kq with ko and calculate the mod-2 motivic cohomology MZ/2 ⋆ kq as A ⋆ //A ⋆ (1); the quotient of the mod-2 motivic Steenord algebra A ⋆ by the augmentation ideal of the MZ/2 ⋆ -subalgebra generated by Sq 1 and Sq 2 [8], [28]. By dualizing, the mod-2 motivic homology MZ/2 ⋆ kq identifies with A ⋆ A⋆(1) MZ/2 ⋆ as an A ⋆ -comodule algebra, and by change-of-rings the MZ/2-based Adams spectral sequence for kq takes the form Ext * ,⋆ A⋆ (1)…”

Two-complete stable motivic stems over finite fields

“…Just as for the stable homotopy category SH, it is an interesting and deep problem to compute the stable motivic homotopy groups of spheres π s,w (k) over k, that is, SH k (Σ s,w ½, ½), where ½ denotes the motivic sphere spectrum over k. When k has finite mod 2 cohomological dimension and s ≥ w ≥ 0, the motivic Adams spectral sequence (MASS) converges to the two-completion of the stable motivic stems E f,(s,w) 2 = Ext f,(s+f,w) A * * (H * * , H * * ) =⇒ (π s,w ½) ∧ 2 . This is a trigraded spectral sequence, where A * * is the bigraded mod 2 motivic Steenrod algebra (see the work of Hoyois, Kelly and Østvaer [22] and Voevodsky [48]), and H * * is the bigraded mod 2 motivic cohomology ring of k. A construction of the motivic Adams spectral sequence is given in section 5. The calculational challenges are to: (1) identify the motivic Ext groups, (2) determine the differentials, and (3) reconstruct the abutment from the filtration quotients.…”