We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures.
In other words, every non-zero motive generates the whole category up to the tensor-triangulated structure.
Under the same assumptions, we also give a complete classification of triangulated étale motives over F with integral coefficients, up to the tensor-triangulated structure, in terms of the characteristic and the orderings of F.
We classify mixed Artin-Tate motives over real closed fields up to the tensor-triangular structure. Compared to the paper below on Tate motives, the additional difficulty lies at the prime 2 where we are required to solve some problems in "filtered modular representation theory".
tt-geometry of Tate motives over algebraically closed fields
Description. A classical result of Hopkins, Neeman, and Thomason classifies the thick subcategories of the category of perfect complexes over a (commutative) ring. Here I prove an analogous result for perfect filtered complexes, taking into account the tensor structure.
Comments. This result was used in the paper above on Tate motives over algebraically closed fields.
Description. This is a careful study of the homotopy theory of sheaves of complexes on a site, in the language of model categories.
Comments. This corresponds to the third chapter of my PhD thesis. Several of the results here were used in the paper below on motivic Galois groups. After completing this note we learned that our description of the fibrant objects also appeared in doi:10.1016/j.aim.2004.07.007. In the meantime, this has been generalized to non-dg contexts in arXiv:1801.10129.
Description. In characteristic 0 there are two approaches to the conjectural theory of mixed motives: Nori motives and Voevodsky motives. Here we prove that their associated motivic Galois groups are canonically isomorphic, thereby providing some evidence that the two approaches are essentially equivalent.
Comments. This corresponds to the fourth chapter of my PhD thesis.
Traces in monoidal derivators, and homotopy colimits
Description. I define and study traces and Euler characteristics in abstract homotopy theory (using the language of derivators). As an application I prove a formula for the trace of the
homotopy colimit of endomorphisms over finite categories in which
all endomorphisms are invertible. This
generalizes the additivity of traces in triangulated categories proved by May.
Comments. This corresponds to the second chapter of my PhD thesis. In the meantime the same result has
been obtained independently
Description.This is a study of trace maps in algebraic geometry, including their additivity, commutation with many natural operations, and their computation in good local situations. As an application one obtains a proof of Deligne's conjecture regarding the Lefschetz-Verdier trace formula in positive characteristic.
Comments. In comparison to
article by Varshavsky, this document is mainly more
Beweise und mathematisches Wissen (Proofs and mathematical knowledge)
Description.The first part consists of a critique of some conceptions of proofs rather popular in the philosophy of mathematics. Common to these conceptions is that they reduce the role of proofs to justifying theorems. This leads to the second part, a discussion of how proofs convey implicit knowledge: often called "methods", "techniques", "ideas" etc. Finally, some examples are presented in which making such implicit knowledge explicit led to tangible mathematical progress.