The Steenrod-span conjecture for motivic cohomology operations

The conjecture asks whether every bistable motivic mod-$p$ cohomology operation is generated by motivic Steenrod operations for every field $F$. The all-fields equality remains unproved; existing results separate the case $p\ne\operatorname{char}F$ from the difficult equal-characteristic case.

Known results

• For $p\ne\operatorname{char}F$, admissible monomials in Bocksteins and power operations form a basis of all bistable operations (2013).
• In characteristic $p$, the relevant map is a split monomorphism, not known to be an equivalence (2017).
• Characteristic-$p$ Steenrod power operations satisfy instability, Adem, and Cartan relations, but generation is not established (2019).
• Hoyois proved the related Hopkins–Morel statement after inverting the exponential characteristic; unpublished work of Hopkins and Morel covers characteristic zero.

June 2025 development

The paper “Motivic Steenrod operations at the characteristic via infinite…” constructs characteristic-$p$ power operations and gives a basis for the image of the Steenrod algebra, but still states the all-fields equality as Conjecture 1.7; it reports no proof or counterexample.

Current status (as of September 2026): The conjecture is settled for $p\ne\operatorname{char}F$ and in characteristic zero only in the cited qualified senses, while the full equal-characteristic equality remains open.