arXiv subjects

Swann Tubach

Publications and source records attributed to Swann Tubach.

7 recordsLinked to original sources

arXiv2026

Nori motives (and mixed Hodge modules) with integral coefficients

We construct abelian categories of integral Nori motivic sheaves over a scheme of characteristic zero. The first step is to study the presentable derived category of Nori motives over a field. Next we construct an algebra in étale motives such that modules over it afford a t-structure that restricts to constructible objects. This category of integral Nori motives has the six operations and arc-descent. We finish by providing analogous constructions and results for mixed Hodge modules on schemes over the reals.

math.AG↗

arXiv2026

Pro-étale motives and solid rigidity

We introduce coefficient systems of pro-étale motives and pro-étale motivic spectra with coefficients in any condensed ring spectrum and show that they afford the six operations. Over locally étale bounded schemes, étale motivic spectra embed into pro-étale motivic spectra. We then use the framework of condensed category theory to define a solidification process for any $\widehat{\mathbb{Z}}$-linear condensed category. Pro-étale motives naturally enhance to a condensed category and we show that their solidification is very close to the category of solid sheaves defined by Fargues-Scholze, suitably modified to work on schemes: this is a rigidity result. As a consequence, we obtain that in contrast with the rigid-analytic setting, solid sheaves on schemes afford the six operations, and we obtain a solid realization functor of motives, extending the $\ell$-adic realization functor. The solid realization functor is compatible with change of coefficients, which allows one to recover the $\mathbb{Q}_\ell$-adic realization functor while remaining in a setting of presentable categories.

math.AG↗

arXiv2025

Nilpotence of $η$ in étale motivic spectra

We show that every object of the stable étale motivic homotopy category over any scheme is $η$-complete. In some cases we show that in fact the fourth power of $η$ is null, whereas the third power of $η$ is always nonvanishing, similar to the situation in topology. Moreover, we prove an étale version of May's nilpotence conjecture, that states that $H\mathbb{Z} \in \mathrm{Sp}$ detects the vanishing of $\mathbf{E}_\infty$-rings. We use this to show a version of Nishida's nilpotence theorem in $\mathrm{SH}_{\operatorname{\acute{e}t}}(S)$, i.e. that any positive degree self map of the unit is nilpotent.

math.AG↗

arXiv2025

Mixed Hodge modules on stacks

Using the $\infty$-categorical enhancement of mixed Hodge modules constructed by the author in a previous paper, we explain how mixed Hodge modules canonically extend to algebraic stacks, together with all the $6$ operations and weights. We also prove that Drew's approach to motivic Hodge modules gives an $\infty$-category that embeds fully faithfully in mixed Hodge modules, and we identify the image as mixed Hodge modules of geometric origin.

math.AG↗

arXiv2025

On the Nori and Hodge realisations of Voevodsky motives

We show that the derived category of perverse Nori motives and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations and therefore to obtain realisation functors from the category of Voevodsky étale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We give a proof that the realisation induces an equivalence of categories between Artin motives in the category of étale motives and Artin motives in the derived category of Nori motives. We also prove that if a motivic $t$-structure exists then Voevodsky étale motives and the derived category of perverse Nori motives are equivalent. Finally we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky étale motives that gives a continuity result for perverse Nori motives.

math.AG↗

arXiv2024

Artin motives in relative Nori and Voevodsky motives

Over a scheme of finite type over a field of characteristic zero, we prove that Nori an Voevodsky categories of relative Artin motives, that is the full subcategories generated by the motives of étale morphisms in relative Nori and Voevodsky motives, are canonically equivalent. As an application, we show that over a normal base of characteristic zero an Artin motive is dualisable if and only if it lies in the thick category spanned by the motives of finite étale schemes. We finish with an application to motivic Galois groups and obtain an analogue of the classical exact sequence of étale fundamental groups relating a variety over a field and its base change to the algebraic closure.

math.AG↗

arXiv2024

Étale motives of geometric origin

Over qcqs finite-dimensional schemes, we prove that étale motives of geometric origin can be characterised by a constructibility property which is purely categorical, giving a full answer to the question "Do all constructible étale motives come from geometry?" which dates back to Cisinski and Déglise's work. We also show that they afford the continuity property and satisfy h-descent and Milnor excision.

math.AG↗