Universal deformation rings and semidihedral 2-groups
Universal deformation rings and semidihedral 2-groups
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万能变形环和半二面体 2 组
DOI：
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发表时间：
2015
期刊：
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通讯作者：
Roberto C. Soto
中科院分区：
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作者：
Roberto C. Soto
变形理论的主要目的是研究数学对象，如模或群表示，在扰动下的行为。这一理论在纯数学和应用数学中都很有用，并导致了许多长期存在的问题的解决。例如，在数论中，伽罗瓦表示的通用变形环在怀尔斯和泰勒证明费马大定理中发挥了重要作用。在这篇论文中，我们考虑了SDn是2阶半二面体2-群的情形，其中n ≥ 3，k是特征为2的代数闭域。不可分解的kSDn-模已经由Bondarenko和Drozd以及Crawley-Boevey完全描述。我们专注于所谓的endo-trivial kSDn-模，它拥有一个定义良好的通用变形环的工作Bleher和Chinburg。利用Carlson和Thevenaz对所有endo平凡kSDn-模的分类，证明了每个endo平凡kSDn-模的泛变形环同构于群环W [Z/2× Z/2]，其中W = W（k）是k上无限Witt向量环.
The main objective of deformation theory is to study the behavior of mathematical objects, such as modules or group representations, under perturbations. This theory is useful in both pure and applied mathematics and has led to the solution of many long-standing problems. For example, in number theory, universal deformation rings of Galois representations played an important role in the proof of Fermat’s Last Theorem by Wiles and Taylor. In this thesis, we consider the case when SDn is a semidihedral 2-group of order 2 for n ≥ 3 and k is an algebraically closed field of characteristic 2. The indecomposable kSDn-modules have been completely described by Bondarenko and Drozd, and Crawley-Boevey. We concentrate on so-called endo-trivial kSDn-modules, which possess a well-defined universal deformation ring by work of Bleher and Chinburg. Using the classification of Carlson and Thevenaz of all endo-trivial kSDn-modules, we show that the universal deformation ring of every endo-trivial kSDn-module is isomorphic to the group ring W [Z/2× Z/2], where W = W (k) is the ring of infinite Witt vectors over k.