MOTIVIC STEENROD OPERATIONS

I. Motivic Steenrod Algebra

Voevodsky constructed motivic cohomology operations by computing the endomorphism algebra of the motivic Eilenberg-MacLane spectrum HZ/p in SH(k):

A^{*,*} = π_{*,*} End_{SH(k)}(HZ/p)

This is the motivic Steenrod algebra, bigraded by (topological degree, weight). It's larger than the classical Steenrod algebra due to the extra weight grading.

II. Motivic Sq^i Operations

For p=2, the motivic Steenrod algebra is generated by operations Sq^{2i} of bidegree (2i, i) and Sq^{2i+1} of bidegree (2i+1, i):

Sq^{2i}: H^{p,q}(X, Z/2) → H^{p+2i,q+i}(X, Z/2)

Sq^{2i+1} = τ · Sq^{2i} where τ ∈ H^{0,1}(k, Z/2)

The element τ = [-1] ∈ k*/k*^2 plays a fundamental role. Over algebraically closed fields, τ = 0 and the odd operations vanish.

III. Milnor Basis

The dual motivic Steenrod algebra has a Milnor-type description:

A_* = Z/2[τ_0, τ_1, ..., ξ_1, ξ_2, ...] / (τ_i^2 = τξ_{i+1} + ρτ_{i+1} + ρτ_0ξ_{i+1})

where ρ = [-1] ∈ H^{1,1}. The Milnor basis elements are dual to iterated Sq operations.

IV. Motivic Adams Spectral Sequence

E_2^{s,f,w} = Ext^{s,f,w}_{A^{*,*}}(H^{*,*}(X), H^{*,*}(pt)) ⇒ π_{f-s,w}(X)

Computes the motivic stable homotopy groups. Over C, converges to classical groups after base-change.

◆ MOTIVIC STEENROD OPERATIONS

I. Motivic Steenrod Algebra

II. Motivic Sq^i Operations

III. Milnor Basis

IV. Motivic Adams Spectral Sequence