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Essential dimension and classifying spaces of algebras

University of British Columbia

The overarching theme of this thesis is to assign, and sometimes find, numerical values which reflect complexity of algebraic objects. The main objects of interest are field extensions of finite degree, and more generally, etale algebras of finite degree over a ring. Of particular interest to us is the invariant known as essential dimension. The essential dimension of separable field extensions was introduced by J. Buhler and Z. Reichstein in their landmark paper. A major (still) open problem arising from that work is to determine the essential dimension of a general separable field extension of degree n (or equivalently, the essential dimension of the symmetric group). Loosening the separability assumption we arrive at the case of inseparable field extensions. In the first part of this thesis we study the problem of determining the essential dimension of inseparable field extensions. In the second part of this thesis, we study the essential dimension of the double covers of symmetric groups and alternating groups, respectively. These groups were first studied by I. Schur and their representations are closely related to projective representations of symmetric and alternating groups. In the third part, we study the problem of determining the minimum number of generators of an etale algebra over a ring. The minimum of number of generators of an etale algebra is a natural measure of its complexity.

Text

2020-04-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74097

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/