On the deformation theory of 𝔼_∞-coalgebras

2023/03/22 by Florian Riedel, Riedel, Florian
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2303.12958

openalex publication_date 2023/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a notion of formally étale 𝔼_∞-coalgebras and show that they admit essentially unique, functorial lifts along square zero extensions of 𝔼_∞-rings. Using this, we show that for a perfect \mathbbF_p-algebra k, Weil restriction along the augmentation \mathbbW(k)→ k induces a fully faithful functor from formally étale, connective 𝔼_∞-coalgebras in k-modules to connective 𝔼_∞-coalgebras in p-complete modules over the spherical Witt vectors \mathbbW(k). Finally, we prove that for any connected space X, the k-homology k[X] is a formally étale 𝔼_∞-coalgebra in k-modules. This shows that \mathbbW(k)[X]^\wedge_p can be recovered as the essentially unique lift of k[X] to a connective coalgebra in p-complete \mathbbW(k)-modules.

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