An obstruction theory for strictly commutative algebras in positive characteristic

Abstract: This is the first in a sequence of articles exploring the relationship between commutative algebras and $E_\infty$-algebras in characteristic $p$ and mixed characteristic. In this paper we lay the groundwork by defining a new class of cohomology operations over $\mathbb F_p$ called cotriple products, generalising Massey products. We compute the secondary cohomology operations for a strictly commutative dg-algebra and the obstruction theories these induce, constructing several counterexamples to characteristic 0 behaviour, one of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We construct some families of higher cotriple products and comment on their behaviour. Finally, we distingush a subclass of cotriple products that we call higher Steenrod operations and conclude with our main theorem, which says that $E_\infty$-algebras can be rectified if and only if the higher Steenrod operations vanish coherently.