On the abelianization of derived categories and a negative solution to Rosicky's problem

We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each infinite regular cardinal lambda. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams lambda-representability for morphisms for any lambda. Equivalently, they are examples of well generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky functor among the lambda-abelianization functors.Comment: 24 page

The Chabauty-Kim Method for Relative Completions

In this thesis we develop a Chabauty-Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one hand, this work generalizes results of Kim (and therefore Chabauty) in the unipotent case by adding a reductive quotient of the fundamental group. From this perspective, the addition of a reductive part allows one to apply Chabauty-type methods to fundamental groups with trivial unipotent completion, such as $SL_2(\mathbb{Z})$. On the other hand, the unipotent part provides a natural extension of the recent work of Lawrence and Venkatesh. We show that their concern with the centralizer of Frobenius goes away as one moves up the unipotent tower and away from the reductive world of flag varieties and the Gauss-Manin connection. One is tempted to hope that the relative completion will provide a unified proof of Mordell's conjecture that takes advantage of the two methods. Toward this end, we apply our work to the Legendre family on the projective line minus three points, a particular example where the method of Lawrence and Venkatesh fails.Comment: 149 pages. This work is the author's doctoral thesis. Comments are very welcome