The spectrum of Artin motives
This is the culmination of the series of papers on the tt-geometry of permutation modules we have written over the years.
It gives a complete classification of thick tensor ideals in the category of Artin motives over arbitrary fields with coefficients in an arbitrary field.
The tt-geometry of permutation modules. Part II: Twisted cohomology
This is the follow-up to
part I on the tt-geometry of permutation modules / Artin motives / modules over constant Mackey functors. Here, we determine the topology of the spectrum and thereby classify the thick ideals. Our proof shows that the spectrum naturally underlies objects from
algebraic geometry, at least for elementary abelian groups.
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 tt-geometry of permutation modules. Part I: Stratification
This is the first in a series of papers in which we study the derived category of permutation modules in modular representation theory (equivalently, the triangulated category of Artin motives in algebraic geometry or, also equivalently, modules over constant Mackey functors in equivariant homotopy theory). In this first part we focus on finite groups. (In part III we will also discuss profinite groups.) In this setting we completely classify the localizing ideals. And as part of the proof we determine the tt-spectrum of the compact objects
as a set. The topology (and hence the classification of
thick ideals) is the subject of
part II.
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
formalism.
Permutation modules and cohomological singularity
This is the sequel to
the paper just below.
The question we investigate in both of them is
How and to what extent are general representations controlled by permutation ones?
In the first paper we settled the
How?, and in this paper we do the same for the
To what extent?. For this we construct an invariant, using cohomology and singularity categories, that detects which representations are controlled by permutation modules.
Finite permutation resolutions
Modular representation theory is well-known to be "wild" for most groups, whereas permutation representations with their finitely many isomorphism types of indecomposables seem relatively "tame". In this paper and its
sequel we investigate how and to what extent the former is controlled by the latter. For example we prove that, contrary to what one might expect, every finite dimensional representation of a finite
group over a field of characteristic p admits a finite resolution by p-permutation
modules.
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.
Three real Artin-Tate motives
We classify mixed Artin-Tate motives over real closed fields up to the tensor-triangular structure. Compared to the earlier
paper,
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
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
by May.