An obstruction theory for strictly commutative algebras in positive characteristic
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915814313033728 |
|---|---|
| author | Flynn-Connolly, Oisín |
| author_facet | Flynn-Connolly, Oisín |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An obstruction theory for strictly commutative algebras in positive characteristic Flynn-Connolly, Oisín Algebraic Topology K-Theory and Homology 18N40, 18M70, 13D03, 18M60 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. |
| title | An obstruction theory for strictly commutative algebras in positive characteristic |
| topic | Algebraic Topology K-Theory and Homology 18N40, 18M70, 13D03, 18M60 |
| url | https://arxiv.org/abs/2404.16681 |