arXiv subjects
Publications and source records attributed to Martin Gallauer.
Informed by our understanding of the tt-geometry of permutation modules, we investigate the proper definition of the `stable permutation category' of a finite group. Then we prove that this category decomposes over cyclic and generalized quaternion groups and only in those cases.
We define the period as a multiplicative characteristic of stably symmetric monoidal $\infty$-categories, develop its basic properties, and study many examples, with a focus on `ordinary' equivariant and motivic homotopy theory. We apply the findings to isotropic points in motivic tt-geometry. (Includes an appendix by Ivo Dell'Ambrogio on generalized comparison maps in tt-geometry.)
The spectrum of a tensor-triangulated category carries a compact Hausdorff topology, called the constructible topology, also known as the patch topology. We prove that patch-dense subsets detect tt-ideals and we prove that any infinite family of tt-functors that detects nilpotence provides such a patch-dense subset. We review several applications and examples in algebra, in topology and in the representation theory of profinite groups.
We analyze the tt-geometry of derived Artin motives, via modular representation theory of profinite groups. To illustrate our methods, we discuss Artin motives over a finite field, in which case we also prove stratification.
We consider the derived category of permutation modules for a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the spectrum of its compact objects, by reducing the problem to elementary abelian groups and then by using a twisted form of cohomology to express the spectrum locally in terms of the graded endomorphism ring of the unit. Together, these results yield a classification of thick and of localizing ideals.
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 $C_{\text{exp}}$. This requires in particular constructing the convolution product symmetric monoidal structure at the $\infty$-categorical level. We study $C_{\text{exp}}$ and how it relates to $C$. We also define motives in $C_{\text{exp}}$ attached to varieties with potential and study their properties.
We consider the derived category of permutation modules over a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the set underlying the tt-spectrum of compact objects, and discuss several examples.
These are notes for a mini-course given at the summer school and conference "The Six-Functor Formalism and Motivic Homotopy Theory" in Milan 9/2021. They provide an introduction to the formalism of Grothendieck's six operations in algebraic geometry 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 account for the benefit of the audience and others. No originality is claimed.
We explain in detail the connections between the three concepts in the title, and we discuss how the `big' derived category of permutation modules introduced in arXiv:2009.14093 fits into the picture.
We develop a `universal' support theory for derived categories of constructible (analytic or \'etale) sheaves, holonomic D-modules, mixed Hodge modules and others. As applications we classify such objects up to the tensor triangulated structure and discuss the question of monoidal topological reconstruction of algebraic varieties.
We offer a systematic study of rigid analytic motives over general rigid analytic spaces, and we develop their six-functor formalism. A key ingredient is an extended proper base change theorem that we are able to justify by reducing to the case of algebraic motives. In fact, more generally, we develop a powerful technique for reducing questions about rigid analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well. We pay special attention to establishing our results without noetherianity assumptions on rigid analytic spaces. This is indeed possible using Raynaud's approach to rigid analytic geometry.
We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.
We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.
We prove that Morel-Voevodsky's stable $\mathbb{A}^1$-homotopy theory affords the universal coefficient system, giving rise to Grothendieck's six operations.
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 \'etale motives over F with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of F.
We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by the second-named author. So the novelty is concentrated at the prime 2, where modular representation theory enters the picture via work of Positselski, based on Voevodsky's resolution of the Milnor Conjecture. With coefficients in k=Z/2, our spectrum becomes homeomorphic to the spectrum of the derived category of filtered kC_2-modules with a peculiar exact structure, for the cyclic group C_2=Gal(C/R). This spectrum consists of six points organized in an interesting way. As an application, we find exactly fourteen classes of mod-2 real Artin-Tate motives, up to the tensor-triangular structure. Among those, three special motives stand out, from which we can construct all others. We also discuss the spectrum of Artin motives and of Tate motives.
We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including the field of algebraic numbers and the algebraic closure of a finite field, we arrive at a complete description of the tensor triangular spectrum and a classification of thick tensor ideals.