Koszul duality for simplicial restricted Lie algebras
Nikolay Konovalov
备注：
84 pages. Comments welcome. v3: more minor changes, Theorem B is fixed and the proof of Theorem 3.2.24 is corrected. v2: minor changes, Theorem C and Proposition 3.2.28 are corrected, Proposition 6.4.1 and Corollary 6.4.4 are improved
令 $\mathsf{s}_0\mathsf{Lie}^r$ 为固定正特征 $p$ 完美域上的 $0$-约化单纯受限李代数范畴。我们证明同伦范畴 $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ 存在一个满子范畴 $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_\xi)$，并且有等价 $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_\xi)\simeq\mathrm{Ho}(\mathsf{s}_1\mathsf{CoAlg}^{tr})$。这里 $\mathsf{s}_1\mathsf{CoAlg}^{tr}$ 是 $1$-约化单纯截断余代数范畴；非正式地说，一个余增广余交换余代数 $C$ 被称为截断的，是指对来自对偶代数 $C^*$ 的增广理想中的任意 $x$ 都有 $x^p=0$。此外，我们用同伦群 $\pi_*(L_\bullet)$ 给出了 $L_\bullet \in \mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ 属于满子范畴 $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_\xi)$ 的充分必要条件。作为上述等价的应用，我们在范畴 $\mathsf{s}\mathsf{Lie}^r$ 中构造并考察了 A. K. Bousfield 与 D. Kan 的非稳定 Adams 谱序列的一个类比。我们利用该谱序列重新计算了自由单纯受限李代数的同伦群。
英文摘要：
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 $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)$ of the homotopy category $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ and an equivalence $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)\simeq\mathrm{Ho}(\mathsf{s}_1\mathsf{CoAlg}^{tr})$. 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 $π_*(L_\bullet)$ for $L_\bullet \in \mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ to lie in the full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)$.
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.
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Manin products, Koszul duality, Loday algebras and Deligne conjecture
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