Koszul duality for simplicial restricted Lie algebras

Abstract

Let $\mathsf{s}_0\mathsf{Lie}^r$ be the category of $0$-reduced simplicial restricted Lie algebras over a fixed perfect field of positive characteristic $p$. We prove that there is a full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})$ of the homotopy category $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ and an equivalence $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})\simeq\mathrm{Ho}(\mathsf{s}_1\mathsf{CoAlg}^{tr})$. Here $\mathsf{s}_1\mathsf{CoAlg}^{tr}$ is the category of $1$-reduced simplicial truncated coalgebras; informally, a coaugmented cocommutative coalgebra $C$ is truncated if $x^p=0$ for any $x$ from the augmentation ideal of the dual algebra $C^*$. Moreover, we provide a sufficient and necessary condition in terms of the homotopy groups $\pi_*(L_\bullet)$ for $L_\bullet \in \mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ to lie in the full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})$. As an application of the equivalence above, we construct and examine an analog of the unstable Adams spectral sequence of A. K. Bousfield and D. Kan in the category $\mathsf{s}\mathsf{Lie}^r$. We use this spectral sequence to recompute the homotopy groups of a free simplicial restricted Lie algebra.