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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.

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.

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 C_{exp}. This requires in particular constructing the convolution product symmetric monoidal structure at the ∞-categorical level. We study C_{exp} and how it relates to C. We also define motives in C_{exp} attached to varieties with potential and study their properties.

This is a preliminary version, to be updated soon (hopefully).

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.

Assigning to a constructible sheaf (or holonomic D-module, mixed Hodge module, ...) its support is shown to have a convenient universal property which leads to their classification up to the tensor triangulated structure at the level of derived categories. A refinement of this result allows the systematic reconstruction of the Zariski topological space underlying an algebraic variety from these derived categories. Together with the reconstruction theorems from this paper one reconstructs even a large class of schemes.

This refinement uses what I called the *smashing spectrum*. The latter has subsequently been shown effective at the level of big tt-categories as well, see here.

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.

An informal account of the main construction in the last chapter of this paper is given in this exposition.

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.

In the first paper we settled the

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.

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.

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".

I classify mixed Tate motives over algebraically closed fields up to the tensor-triangular structure.

The description of the spectrum of étale motives with finite coefficients was completed in this paper.

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.

This result was used in the paper on Tate motives. In the meantime it has been generalized in this paper.

This is a careful study of the homotopy theory of sheaves of complexes on a site, in the language of model categories.

This corresponds to the third chapter of my PhD thesis. Several of the results here were used in our earlier paper.

After completing this note we learned that our description of the fibrant objects had appeared in the literature before. In the meantime, this has been generalized to non-dg contexts.

After completing this note we learned that our description of the fibrant objects had appeared in the literature before. In the meantime, this has been generalized to non-dg contexts.

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.

This corresponds to the fourth chapter of my PhD thesis.

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.

This corresponds to the second chapter of my PhD thesis. In the meantime the same result has been proved independently in this paper.

These are notes for a mini-course given at the summer school and conference *Six-Functor Formalism and Motivic Homotopy Theory* in Milan 9/2021. They provide an introduction to the formalism of Grothendieck’s six operations and end with an excursion to rigid-analytic motives.
The notes do not correspond precisely to the lectures delivered but provide a more self-contained accompaniment for the benefit of the audience. No originality is claimed.

This is a companion to our papers on permutation modules and their tt-geometry. It discusses the `big' derived category of permutation modules, and describes the beautiful connections with cohomological Mackey functors and Artin motives. The note is more expository than those papers.

In this short announcement we describe the spectrum of Artin motives over a finite field, and thereby classify them up to the tensor triangulated structure. Proofs will appear as part of forthcoming work on the tensor-triangular geometry of Artin-Tate motives.

local
most recent version from 5 May 2022

This consists essentially of my (chronologically) first three papers, bundled together and prefaced with an introduction.

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.

In comparison to
the original
article by Varshavsky, this document is mainly more
detailed.

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.

reciprocal plane.