$E_{\infty}$-coalgebras and $p$-adic homotopy theory

We prove that taking "chains with coefficients in an separably closed field k of characteristic p" induces a fully faithful functor from p-complete, nilpotent spaces to $E_{\infty}$ -coalgebras over k. This removes finiteness assumptions from a theorem of Mandell. (jt. with Robert Burklund).