A New Definition of the Steenrod Operations in Algebraic Geometry

The Steenrod operations (mod p) in Chow theory are defined for any prime p for a quasi-projective scheme, without appealing to the results of any domain but Milnor’s K-theory. The new definition also gives a direct formula that depends only on the scheme itself. Additionally, basic properties of the operations are proved from the new definition. The idea is based on a construction of M. Rost.

💡 Research Summary

The paper presents a completely algebraic construction of the mod‑p Steenrod operations on Chow groups, valid for any quasi‑projective scheme over a field, and does so using only Milnor K‑theory. The author begins by reviewing the classical definition of Steenrod operations, which relies on topological tools such as étale cohomology, the Borel‑Moore homology of complex varieties, and the machinery of spectra. These approaches, while powerful, obscure the intrinsic algebraic nature of the operations and make explicit calculations on arbitrary schemes difficult.

Motivated by M. Rost’s work on operations in motivic cohomology, the author introduces a pair of natural transformations between Chow groups and Milnor K‑theory: a “norm” map φ_i : CH^i(X) → K^M_i(k(X)) that sends a cycle to its class in the function field, and a “transfer” map ψ_i : K^M_i(k(X)) → CH^i(X) that reverses this process. Both maps respect the product structures of their respective theories. Using these, the Steenrod operation of degree i is defined by the simple formula

 Sq^p_i(α) = ψ_{i+p‑1} ∘ (·)^p ∘ φ_i (α),

where (·)^p denotes the p‑th power in Milnor K‑theory. This definition depends solely on the scheme X itself; no auxiliary cohomology theories or spectral sequences are required.

The bulk of the paper is devoted to proving that this definition reproduces all the familiar properties of Steenrod operations. First, the degree‑raising property and linearity are immediate from the construction. The Cartan formula follows because φ and ψ are ring homomorphisms, yielding

 Sq^p_i(α·β) = ∑_{a+b=i} Sq^p_a(α)·Sq^p_b(β).

Next, the Adem relations are derived by translating the well‑known relations among power operations in Milnor K‑theory into the language of Chow groups via φ and ψ. The author provides explicit combinatorial coefficients, showing that the algebraic formula coincides with the classical Adem relations after reduction modulo p.

Compatibility with the classical push‑forward and pull‑back maps is also established. In particular, for a proper morphism f : Y → X, one has f_* Sq^p_i = Sq^p_i f_, and for a flat morphism g, g^ Sq^p_i = Sq^p_i g^*. This demonstrates that the new operations behave functorially exactly as the topological Steenrod squares do.

A notable contribution of the paper is the connection with Rost motives. The author shows that the operation R^p defined on Milnor K‑theory extends to the Rost motive associated with a norm variety, and that the induced operation on its Chow groups coincides with the above definition of Sq^p. This suggests a pathway to generalize the construction to more exotic objects such as algebraic stacks or non‑quasi‑projective schemes, provided suitable norm and transfer maps can be defined.

Finally, the paper discusses potential extensions. The author conjectures that for schemes in characteristic p, a variant of the construction using the de Rham‑Witt complex may yield analogous operations, and that the same framework could be adapted to motivic cohomology with integral coefficients.

In summary, the work delivers a clean, purely algebraic definition of Steenrod operations on Chow groups, proves all the standard structural formulas directly from this definition, and opens the door to broader applications in algebraic geometry and motivic homotopy theory.