Martin Gallauer

papers and preprints

all - none - abstract homotopy theory Mackey functors modular representation theory motives motivic homotopy theory non-archimedean analytic geometry periods permutation modules six operations tt-geometry

Exponentiation of coefficient systems and exponential motives
with Simon Pepin Lehalleur
- arxiv
- Description: 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.
- Comments: This is a preliminary version, to be updated soon.
- motives motivic homotopy theory six operations
The tt-geometry of permutation modules. Part I: Stratification
with Paul Balmer
- arxiv
- Description: This is the first in a series of three 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). In this first part we focus on finite groups. (In part 3 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 2.
- modular representation theory permutation modules tt-geometry
An introduction to six-functor formalisms
- arxiv, most recent (28 December 2022)
- Description: 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.
- motivic homotopy theory motives six operations
Permutation modules, Mackey functors, and Artin motives
with Paul Balmer
- Proceedings of ICRA2020 (accepted)
- arxiv
- Description: This is a companion to the papers 08-09. 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.
- permutation modules Mackey functors motives
Supports for constructible systems
- Doc. Math. (2022)
- arxiv
- Description: 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.
- tt-geometry motives
The six-functor formalism for rigid analytic motives
with Joseph Ayoub and Alberto Vezzani
- Forum Sigma (2022)
- journal, arxiv
- Description: 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.
- Comments: An informal account of the main construction in the last chapter of this paper is given in paper 14.
- non-archimedean analytic geometry motives motivic homotopy theory six operations
The universal six-functor formalism
with Brad Drew
- Ann. K-theory (2022)
- arxiv
- Description: We prove that Morel-Voevodsky’s stable 𝔸¹-homotopy theory affords the universal six-functor formalism.
- motivic homotopy theory abstract homotopy theory six operations
Permutation modules and cohomological singularity
with Paul Balmer
- Comment. Math. Helv. (2022)
- journal, arxiv
- Description: This is the sequel to paper 08. 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.
- modular representation theory permutation modules
Finite permutation resolutions
with Paul Balmer
- Duke Math. J. (2023)
- journal, arxiv
- Description: 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 09 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.
- modular representation theory permutation modules
A note on Tannakian categories and mixed motives
- Bull. Lond. Math. Soc. (2021)
- journal, arxiv
- Description: 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.
- motives tt-geometry
Three real Artin-Tate motives
with Paul Balmer
- Adv. Math. (2022)
- journal, arxiv
- Description: We classify mixed Artin-Tate motives over real closed fields up to the tensor-triangular structure. Compared to paper 05, the additional difficulty lies at the prime 2 where we are required to solve some problems in "filtered modular representation theory".
- motives tt-geometry modular representation theory
tt-geometry of Tate motives over algebraically closed fields
- Compositio Math. (2019)
- journal, arxiv
- Description: I classify mixed Tate motives over algebraically closed fields up to the tensor-triangular structure.
- Comments: The description of the spectrum of étale motives with finite coefficients was completed in paper 07.
- motives tt-geometry
Tensor triangular geometry of filtered modules
- Algebra Number Theory (2018)
- journal, arxiv
- 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 paper 05. In the meantime it has been generalized in this paper.
- tt-geometry
Homotopy theory of dg sheaves
with Utsav Choudhury
- Comm. Algebra (2019)
- journal, arxiv
- 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 paper 02.
  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.
- abstract homotopy theory
An isomorphism of motivic Galois groups
with Utsav Choudhury
- Adv. Math. (2017)
- journal, arxiv
- 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.
- motives periods
Traces in monoidal derivators, and homotopy colimits
- Adv. Math. (2014)
- journal, arxiv
- 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 proved independently in this paper.
- abstract homotopy theory