Universal deformation rings and semidihedral 2-groups

• Main Author: Soto, Roberto Carlos
• Other Authors: Bleher, Frauke, 1968-
• Format: Others
• Language: English
• Published: University of Iowa 2015
• Subjects: publicabstract; Mathematics
• Online Access: https://ir.uiowa.edu/etd/1908; https://ir.uiowa.edu/cgi/viewcontent.cgi?article=5964&context=etd

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 2n+1 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 x ℤ/2], where W = W (k) is the ring of infinite Witt vectors over k.