On the motivic homotopy theory of Stiefel varieties

This dissertation concerns the motivic homotopy theory of the Stiefel varieties, denoted Vᵣ (Aⁿ). A paper by Haynes Miller shows that there is a filtration on the unitary groups (and more generally, the real, complex, and quaternionic Stiefel manifolds) that splits in the stable homotopy category, where the stable summands are certain Thom spaces over Grassmannians. We give an algebraic version of this result in the context of Voevodsky's tensor triangulated category of stable motivic complexes DM (k, R), where k is a field. Specifically, we show that there are algebraic analogs of the Thom spaces appearing in Miller's splitting that give rise to an analogous splitting of the motive M (GLₙ) in DM (k, R), where GLₙ is the general linear group scheme over k. ⁠ In the second part, we investigate a question of Michèle Raynaud. Let R be a commutative ring. Raynaud asked when a general R-module P that satisfies P+R≅Rⁿ has a free summand of a given rank. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: the projection Vᵣ (Aⁿ) →V₁ (Aⁿ) has a section over k if and only if the following holds: any module P over any k-algebra R with the property that P+R≅Rⁿ has a free summand of rank r-1. Using techniques from A¹-homotopy theory, we characterize those n for which the map Vᵣ (Aⁿ) →V₁ (Aⁿ) has a section in the cases r=3, 4 under some assumptions on the base field.