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Raphaël Ruimy

Publications and source records attributed to Raphaël Ruimy.

8 recordsLinked to original sources
arXiv2026

Integral Artin motives II: Perverse motives and Artin Vanishing Theorem

In this text, we are mainly interested in the existence of the perverse motivic t-structures on the category of Artin étale motives with integral coefficients. We construct the perverse homotopy t-structure which is the best possible approximation to a perverse t-structure on Artin motives with rational coefficients. The heart of this t-structure has properties similar to those of the category of perverse sheaves and contains the Ayoub-Zucker motive. With integral coefficients, we construct the perverse motivic t-structure on Artin motives when the base scheme is of dimension at most $2$ and show that it cannot exist in dimension $4$. This construction relies notably on a an analogue for Artin motives of the Artin Vanishing Theorem.

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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.

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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.

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arXiv2025

The weight filtration on real singular homology is motivic

We give an alternative construction of Totaro's weight filtration on singular homology of the real points of a real algebraic variety. Our construction shows that this filtration comes from Bondarko's weight filtration on Voevodsky motives.

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arXiv2025

Deligne 1-motives with torsion and étale motives

We construct the motivic t-structure on 1-motives with integral coefficients over a scheme of characteristic zero or a Dedekind scheme. When we invert the residue characteristic exponents of the base, this t-structure induces a t-structure on the category of smooth 1-motives whose heart is the category of Deligne 1-motives with torsion which we prove to be abelian. This relies on the fact that over a normal scheme, Deligne 1-motives with torsion are determined by their fiber on the generic point and on an explicit description of good reduction Deligne 1-motives with torsion.

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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.

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arXiv2023

Artin perverse sheaves

We show that the perverse t-structure induces a t-structure on the category $\mathcal{D}^A(S,\mathbb{Z}_\ell)$ of Artin $\ell$-adic complexes when $S$ is an excellent scheme of dimension less than $2$ and provide a counter-example in dimension $3$. The heart $\mathrm{Perv}^A(S,\mathbb{Z}_\ell)$ of this t-structure can be described explicitly in terms of representations in the case of $1$-dimensional schemes. When $S$ is of finite type over a finite field, we also construct a perverse homotopy t-structure over $\mathcal{D}^A(S,\mathbb{Q}_\ell)$ and show that it is the best possible approximation of the perverse t-structure. We describe the simple objects of its heart $\mathrm{Perv}^A(S,\mathbb{Q}_\ell)^\#$ and show that the weightless truncation functor $ω^0$ is t-exact. We also show that the weightless intersection complex $EC_S=ω^0 IC_S$ is a simple Artin homotopy perverse sheaf. If $S$ is a surface, it is also a perverse sheaf but it need not be simple in the category of perverse sheaves.

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