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arXiv · 2405.04709

Quillen's Fundamental Spectral Sequences Revisited

Hossein Faridian
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Abstract

Quillen's fundamental spectral sequences relate Andr\'{e}-Quillen homology and cohomology to Tor and Ext functors. The five-term exact sequences arising from these spectral sequences are leveraged to characterize regular and complete intersection local rings. Despite their immense importance in the theory of Andr\'{e}-Quillen homology and cohomology, these spectral sequences are not treated in the literature as they merit. A sketchy argument with gaps for a special case of the first spectral sequence appears in an unpublished manuscript of Quillen, while the second spectral sequence is only stated without proof in another paper of Quillen. A rigorous proof of the spectral sequences requires a delicate investigation of the subtle structures involved. The goal of this expository article is to present a comprehensive and detailed proof of Quillen's fundamental spectral sequences in a readable and well-structured manner.

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Hossein Faridian. 2024-05-07. Quillen's Fundamental Spectral Sequences Revisited. https://arxiv.org/abs/2405.04709

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