Seminar of Algebra

In this talk, we study the relationship between strictly commutative dg-algebras and $E_\infty$-algebras in positive characteristic. In the first part, we work over $\mathbb F_p$. We construct the secondary cohomology operations for a strictly commutative dg-algebra. We study the obstruction theories these induce, showcasing several counterexamples. We show a $E_\infty$-algebra over $\mathbb F_p$ admits a commutative model if and only if its higher Steenrod operations vanish coherently. In the second part, we construct a de Rham complex over $\padic$ which provides a "best strictly commutative approximation" to the singular cochains complex, and we study the question of its formality and discuss Massey products in this context. This generalises work of Cartan.