Strictly commutative algebras in positive characteristic

par M. Oisin Flynn-Connolly (Paris 13)

mardi 17 sept. 2024, 14:00 → 15:00 Europe/Paris

IMT 1R2 207 (Salle Pellos)

In this talk, which consists of two parts, we study the relationship between strictly commutative dg-algebras, topological spaces and $E_\infty$-algebras in positive and mixed characteristics.

 

In the first part, we work over $F_p$. We define a general theory of higher cohomology operations called cotriple products and compute the secondary cohomology operations for a strictly commutative dg-algebra. We study the induced obstruction theory, producing several counterexamples. We conclude by showing a  $E_\infty$ -algebra over  $F_p$  admits a commutative model if and only if its higher Steenrod operations vanish coherently. 

 

In the second part, we generalise the de Rham forms to the p-adic numbers, thus providing a best strictly commutative approximation to the singular cochains complex in this context. We study the question of its formality and discuss Massey products. We remark on some connections to crystalline cohomology for schemes.