Motivic monodromy and p-adic cohomology theories
We build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo-Kato cohomology without recourse to log-geometry, as predicted by Fontaine, and we produce an induced Clemens-Schmid chain complex.
Exponentiation of coefficient systems and exponential motives
We construct new six-functor formalisms capturing cohomological invariants of varieties with potentials. Starting from any six-functor formalism C, encoded as a coefficient system, we associate a new six-functor formalism Cexp. This requires in particular constructing the convolution product symmetric monoidal structure at the ∞-categorical level. We study Cexp and how it relates to C. We also define motives in Cexp attached to varieties with potential and study their properties.
The six-functor formalism for rigid analytic motives
The original goal of this project was to establish a full six-functor formalism for rigid-analytic motives over general rigid-analytic spaces. We achieve this as an application of a powerful technique that we also develop in this (long) paper. It allows reducing certain questions about rigid-analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well.
The universal six-functor formalism
We prove that Morel-Voevodsky’s stable 𝔸1-homotopy theory affords the universal six-functor
A note on Tannakian categories and mixed motives
Assuming "all" motivic conjectures, the triangulated category of mixed
motives over a field F is the derived category of a Tannakian category.
I explain why one should therefore expect this category to be simple in
the tensor-triangular sense.
In other words, why every non-zero motive generates the whole category
up to the tensor-triangulated structure.
Under the same assumptions, I also completely classify triangulated
étale motives over F with integral coefficients, up to the
tensor-triangulated structure, in terms of the characteristic and the
orderings of F.
tt-geometry of Tate motives over algebraically closed fields
I classify mixed Tate motives over algebraically closed fields up to the tensor-triangular structure.
Tensor triangular geometry of filtered modules
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.
Homotopy theory of dg sheaves
This is a careful study of the homotopy theory of sheaves of complexes on a site, in the language of model categories.
An isomorphism of motivic Galois groups
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.
Traces in monoidal derivators, and homotopy colimits
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