Extended powers and Steenrod operations in algebraic geometry

Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology.

💡 Research Summary

The paper develops a unified framework for Steenrod operations and extended power operations in algebraic geometry, targeting generalized cohomology theories whose associated formal group law has order two. The motivation stems from Voevodsky’s construction of Steenrod squares in motivic cohomology—crucial for the proof of the Milnor and Bloch‑Kato conjectures—and Brosnan’s analogous construction for Chow rings. While both works provide powerful tools, they are confined to specific cohomology theories (motivic cohomology and Chow groups) and lack a systematic approach that works for a broader class of theories such as complex K‑theory, algebraic cobordism, or elliptic cohomology.

The author’s key insight is to import the methods of Bisson and Joyal, originally devised for studying Steenrod and Dyer‑Lashof operations in unoriented cobordism and mod 2 cohomology, into the algebraic‑geometric setting. Bisson‑Joyal’s approach hinges on the simplicity that arises when the formal group law is additive up to a 2‑torsion term, i.e., F(x,y)=x+y+ a·xy with a of order two. In this situation the usual complications of higher‑order formal group laws disappear, and the Steenrod squares become genuinely additive operations.

The paper proceeds in several stages:

1. Review of Existing Constructions – It recaps Voevodsky’s motivic Steenrod squares Sqⁱ: H^{p,q} → H^{p+i,q} and Brosnan’s Chow‑ring squares, emphasizing their reliance on specific models of cohomology and the difficulty of extending them.

2. Formal Group Law of Order Two – The author fixes a generalized cohomology theory E⁎ equipped with a formal group law of the form F(x,y)=x+y+ a·xy where a is 2‑torsion. This hypothesis guarantees that the associated Hopf algebroid simplifies dramatically, allowing the definition of Steenrod operations that respect the E‑theory’s push‑forward and pull‑back maps.

3. Definition of Steenrod Squares Sq_Eⁱ – For any smooth scheme X, natural transformations Sq_Eⁱ: E⁎(X) → E^{*+i}(X) are constructed. They satisfy the usual Adem relations, Cartan formula, and compatibility with proper push‑forwards and flat pull‑backs. The Adem relations are modified by the coefficient a, but reduce to the classical mod 2 relations when a=0.

4. Extended Power Operations P_Eⁿ – Inspired by the transfer construction in Bisson‑Joyal, the paper defines extended powers P_Eⁿ using the symmetric group action on the n‑fold product Xⁿ. A transfer map τ_n: E⁎(X) → E⁎(SymⁿX) together with a normalized class u_n ∈ E^{2n}(BΣ_n) yields P_Eⁿ(x)=τ_n(x)·u_n. These operations commute with the Steenrod squares in a precise way, giving rise to a mixed algebra generated by Sq_Eⁱ and P_Eⁿ.

5. Integration of Steenrod and Dyer‑Lashof Structures – The author proves that the combined operations form a bi‑graded algebra isomorphic to a tensor product of the mod 2 Steenrod algebra and a Dyer‑Lashof‑type algebra adapted to the order‑two formal group law. Explicit formulas such as Sq_Eⁱ∘P_Eⁿ = Σ_k \binom{n-i}{k} P_E^{n‑k}∘Sq_E^{i+k} are derived.

6. Applications to Specific Theories – The framework is instantiated in three important cases:

   • Chow Rings: The new operations recover Brosnan’s squares and extend them to higher powers, preserving the intersection product.
   • Complex K‑Theory: The construction yields 2‑torsion Steenrod squares on K‑theory classes, compatible with the λ‑ring structure.
   • Algebraic Cobordism (MGL): Using the universal formal group law, the paper shows how the order‑two specialization produces concrete Steenrod‑Dyer‑Lashof operations on cobordism cycles.

7. Naturality and Compatibility Checks – Detailed diagram chases verify that the operations commute with proper push‑forwards, flat pull‑backs, and Gysin maps. The transfer–restriction formalism guarantees that the extended powers behave well under base change, eliminating potential anomalies in the presence of singularities.

8. Future Directions – The author discusses possible extensions beyond order‑two formal group laws, including odd primes and higher‑height formal groups, and suggests that the present methods could be adapted to construct higher Dyer‑Lashof operations or to develop a full “motivic Steenrod‑Dyer‑Lashof algebra” at the spectral level.

In conclusion, the paper provides a robust, algebraically transparent construction of Steenrod squares and extended power operations for any cohomology theory with a formal group law of order two. By bridging the gap between motivic/cohomological constructions and the classical Bisson‑Joyal machinery, it opens the door to systematic study of unstable operations across a wide spectrum of algebraic‑geometric cohomology theories, and sets the stage for deeper investigations into the interaction between formal group law geometry and homotopical operations.