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An obstruction theory for strictly commutative algebras in positive characteristic

Oisín Flynn-Connolly

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.

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 -algebras in characteristic and mixed characteristic. In this paper we lay the groundwork by defining a new class of cohomology operations over 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 -algebras can be rectified if and only if the higher Steenrod operations vanish coherently.
Paper Structure (20 sections, 24 theorems, 104 equations)

This paper contains 20 sections, 24 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Three flavours of algebra over an operad
  4. E-infinity-algebras and Steenrod operations
  5. Rectification
  6. The homotopy theory of $E_\infty$-algebras and commutative dg-algebras
  7. Cotriple products
  8. Sullivan algebras
  9. Cotriple products in positive characteristic
  10. Cotriple products for strictly commutative dg-algebras
  11. Secondary cotriple products
  12. The relationship between associative and commutative algebras
  13. Cup-1-algebras
  14. Frobenius products for cup-1-algebras
  15. The commutative algebras
  16. ...and 5 more sections

Key Result

Theorem 1.1

campos23 Let $A$ and $B$ be two commutative dg algebras over a field of characteristic zero. Then, $A$ and $B$ are quasi-isomorphic as associative dg algebras if and only if they are also quasi-isomorphic as commutative dg algebras.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem A
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • ...and 68 more