On the deformation theory of ${\mathbb{E}}_{\infty }$-coalgebras

Abstract

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 $\mathbb{W}(k)[X{]}_p^{\wedge }$ can be recovered as the essentially unique lift of $k[X]$ to a connective coalgebra in $p$-complete $\mathbb{W}(k)$-modules.