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.

arXiv commentTo appear in Algebraic and Geometric Topology; 24 pages; minor revision

References in corpus (6)

• Combinatorial operad actions on cochains
• Model Structures on Commutative Monoids in General Model Categories
• The cotriple resolution of differential graded algebras
• Lie, associative and commutative quasi-isomorphism
• Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products
• Cup-one algebras and 1-minimal models

Cited by in corpus (1)

• E2 formality via obstruction theory