Rigidity properties of the cotangent complex
Rigidity properties of the cotangent complex
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DOI：
10.1090/jams/1000
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发表时间：
2020-10
期刊：
影响因子：
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通讯作者：
Benjamin Briggs;S. Iyengar
中科院分区：
文献类型：
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作者：
Benjamin Briggs;S. Iyengar
这项工作涉及交换诺特环的映射 $\varphi \colon R\to S$，局部有限平坦维度。证明如果Andre-Quillen同调函子$\mathrm{D}_n(S/R,-)$对于$n\ge 1$消失，则$\varphi$是局部完全交集。这扩展了阿夫拉莫夫定理，假设 $\mathrm{D}_n(S/R,-)$ 对于所有 $n\gg 0$ 得出相同的结论，证实了 Quillen 的猜想。这也暗示了 Avramov 和 Herzog 关于更高余切模的猜想，以及第一作者最近证明的 Vasconcelos 猜想。导致这些结果的新观察涉及从 Andre-Quillen 上同调到使用通用 Atiyah 类 $\varphi$ 定义的 Hochschild 上同调映射的等方差性。
This work concerns a map $\varphi \colon R\to S$ of commutative noetherian rings, locally of finite flat dimension. It is proved that if the Andre-Quillen homology functor $\mathrm{D}_n(S/R,-)$ vanishes for some $n\ge 1$, then $\varphi$ is locally complete intersection. This extends Avramov's theorem that draws the same conclusion assuming $\mathrm{D}_n(S/R,-)$ for all $n\gg 0$, confirming a conjecture of Quillen. This implies also a conjecture of Avramov and Herzog about the higher cotangent modules, and a conjecture of Vasconcelos that was proved recently by the first author. The new observation leading to these results deals with the equivariance of a map from Andre-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of $\varphi$.