Graded $p$-polar rings and the homology of $Ω^nΣ^nX$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915271500890112 |
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| author | Bauer, Tilman |
| author_facet | Bauer, Tilman |
| contents | As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we show that the homology of the free $E_n$-algebra $H^*(Ω^nΣ^n X;\mathbf F_p)$, as a Hopf algebra, only depends on the $p$-polar structure of $H^*(X;\mathbf F_p)$ in a functorial way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_05286 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Graded $p$-polar rings and the homology of $Ω^nΣ^nX$ Bauer, Tilman Algebraic Topology Algebraic Geometry Rings and Algebras 14L05, 14L15, 14L17, 13A99, 13A35, 16T05 As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we show that the homology of the free $E_n$-algebra $H^*(Ω^nΣ^n X;\mathbf F_p)$, as a Hopf algebra, only depends on the $p$-polar structure of $H^*(X;\mathbf F_p)$ in a functorial way. |
| title | Graded $p$-polar rings and the homology of $Ω^nΣ^nX$ |
| topic | Algebraic Topology Algebraic Geometry Rings and Algebras 14L05, 14L15, 14L17, 13A99, 13A35, 16T05 |
| url | https://arxiv.org/abs/2203.05286 |