余切配合物和Thom光谱

IF 0.4 4区 数学 Q4 MATHEMATICS
Nima Rasekh, Bruno Stonek
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引用次数: 1

摘要

交换环映射的余切复形是变形理论的中心对象。自20世纪90年代以来,它以各种方式被推广到\(E_infty\)-环谱的同位设置。在这项工作中,我们首先在\(\infty\)-范畴的背景下,并使用Goodwillie的函子演算,建立了文献中存在的\(E_\infty\)-环谱映射的余切复形的各种定义是等价的。然后,我们将注意力转向一个具体的例子。设R是一个\(E_infty\)-环谱,\(\mathrm{Pic}(R)\)表示它的Picard\(E_infty\)群。设Mf表示\(E_\infty\)-群\(f:G\rightarrow\mathrm{Pic}(R)\)的映射的Thom\(E_\infty\)-R-代数;Mf的例子由各种风格的共基光谱给出。我们证明了\(R\rightarrow-Mf\)的余切复合物等价于Mf和G的连接谱的砸积。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The cotangent complex and Thom spectra

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of \(E_\infty \)-ring spectra in various ways. In this work we first establish, in the context of \(\infty \)-categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of \(E_\infty \)-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an \(E_\infty \)-ring spectrum and \(\mathrm {Pic}(R)\) denote its Picard \(E_\infty \)-group. Let Mf denote the Thom \(E_\infty \)-R-algebra of a map of \(E_\infty \)-groups \(f:G\rightarrow \mathrm {Pic}(R)\); examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of \(R\rightarrow Mf\) is equivalent to the smash product of Mf and the connective spectrum associated to G.

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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
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