BIRS Workshop Lecture Videos
Indistinguishability and simple algebras â Duncan, Alexander
The theory of flasque resolutions gives a complete solution to the problem of determining whether an algebraic torus is stably- or retract-rational. More generally, flasque resolutions of more general connected linear algebraic groups are a powerful tool for studying R-equivalence, weak approximation, and the Hasse principle of such groups. There is a dual notion of coflasque for tori, which can also be extended to more general linear algebraic groups. â A common technique for classifying objects over non-closed fields is to associate a set of simple algebras to each object. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. However, it may be that there is no way to distinguish non-isomorphic objects through any such association if it is functorial in the base field. Using coflasque resolutions of general algebraic groups, we can precisely describe the set of such indistinguishable objects. â I will recall flasque and coflasque resolutions, discuss how they can be used to describe these indistinguishable sets, and comment on the implications for detecting arithmetic properties using Brauer groups, K-theory, and derived categories. This is based on joint work with Matthew Ballard, Alicia Lamarche, and Patrick McFaddin.
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