On the deformation theory of $\mathbb{E}_\infty$-coalgebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929343596331008 |
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| author | Riedel, Florian |
| author_facet | Riedel, Florian |
| contents | We introduce a notion of formally étale $\mathbb{E}_{\infty}$-coalgebras and show that they admit essentially unique, functorial lifts along square zero extensions of $\mathbb{E}_{\infty}$-rings. Using this, we show that for a perfect $\mathbb{F}_p$-algebra $k$, Weil restriction along the augmentation $\mathbb{W}(k)\to k$ induces a fully faithful functor from formally étale, connective $\mathbb{E}_{\infty}$-coalgebras in $k$-modules to connective $\mathbb{E}_{\infty}$-coalgebras in $p$-complete modules over the spherical Witt vectors $\mathbb{W}(k)$. Finally, we prove that for any connected space $X$, the $k$-homology $k[X]$ is a formally étale $\mathbb{E}_\infty$-coalgebra in $k$-modules. This shows that $\mathbb{W}(k)[X]^{\wedge}_p$ can be recovered as the essentially unique lift of $k[X]$ to a connective coalgebra in $p$-complete $\mathbb{W}(k)$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12958 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the deformation theory of $\mathbb{E}_\infty$-coalgebras Riedel, Florian Algebraic Topology We introduce a notion of formally étale $\mathbb{E}_{\infty}$-coalgebras and show that they admit essentially unique, functorial lifts along square zero extensions of $\mathbb{E}_{\infty}$-rings. Using this, we show that for a perfect $\mathbb{F}_p$-algebra $k$, Weil restriction along the augmentation $\mathbb{W}(k)\to k$ induces a fully faithful functor from formally étale, connective $\mathbb{E}_{\infty}$-coalgebras in $k$-modules to connective $\mathbb{E}_{\infty}$-coalgebras in $p$-complete modules over the spherical Witt vectors $\mathbb{W}(k)$. Finally, we prove that for any connected space $X$, the $k$-homology $k[X]$ is a formally étale $\mathbb{E}_\infty$-coalgebra in $k$-modules. This shows that $\mathbb{W}(k)[X]^{\wedge}_p$ can be recovered as the essentially unique lift of $k[X]$ to a connective coalgebra in $p$-complete $\mathbb{W}(k)$-modules. |
| title | On the deformation theory of $\mathbb{E}_\infty$-coalgebras |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2303.12958 |