Tor独立张量积的同伦李代数
The homotopy Lie algebra of a Tor-independent tensor product
摘要：
本文研究了一对满射局部环映射$S_1\leftarrow R\to S_2$及其与正则投影$R\to S_1\otimes_R S_2$的关系，其中$S_1,S_2$在$R$上是TOR-独立的.我们的主要结果断言了$S:=S_1\otimes_R S_2$的同伦李代数（记作$π(S)$）与$R,S_1$和$S_2$的同伦李代数之间的结构联系。也就是说，$π(S)$是在各种情况下沿着映射$π(S_i)\to π(R)$的（调整的）李代数的拉回，包括当上面的映射具有残差特征零时。结果的主要定理包括结构的结果安德烈-奎伦上同调，稳定上同调，和Tor代数，以及平等有关庞加莱系列的公共剩余领域的$R,S_1,S_2$和$S$。
英文摘要：
In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1\otimes_R S_2$, denoted $π(S)$, in terms of those of $R,S_1$ and $S_2$. Namely, $π(S)$ is the pullback of (adjusted) Lie algebras along the maps $π(S_i)\to π(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on André-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincaré series of the common residue field of $R,S_1,S_2$ and $S$.
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