arXiv subjects

Fabio Tanania

Publications and source records attributed to Fabio Tanania.

8 recordsLinked to original sources

arXiv2025

Real isotropic motivic spectra

In this paper, we introduce the category of real isotropic motivic spectra, and show that the real realization functor from motivic spectra over $\mathbb{R}$ to classical spectra factors through it. We then describe its cellular subcategory as a one-parameter deformation of the category of spectra, with parameter $\rho$ corresponding to $-1 \in \mathbb{R}^{\times}$, whose special fiber is the derived category of comodules over the dual Steenrod algebra. This leads to an identification of real isotropic cellular spectra with $\mathbb{F}_2$-synthetic spectra, and sheds light on the relation between motivic homotopy theory over $\mathbb{R}$ and $\mathbb{F}_2$-synthetic homotopy theory.

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arXiv2024

Isotropic motivic fundamental groups

The main goal of this paper is to study relative versions of the category of modules over the isotropic motivic Brown-Peterson spectrum, with a particular emphasis on their cellular subcategories. Using techniques developed by Levine, we equip these categories with motivic $t$-structures, whose hearts are Tannakian categories over ${\mathbb F}_2$. This allows to define isotropic motivic fundamental groups, and to interpret relative isotropic Tate motives in the heart as their representations. Moreover, we compute these groups in the cases of the punctured projective line and split tori. Finally, we also apply Spitzweck's derived approach to establish an identification between relative isotropic Tate motives and representations of certain affine derived group schemes, whose 0-truncations coincide with the aforementioned isotropic motivic fundamental groups.

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arXiv2023

Cellular objects in isotropic motivic categories

Our main purpose is to describe the category of isotropic cellular spectra over flexible fields. Guided by [6], we show that it is equivalent, as a stable $\infty$-category equipped with a $t$-structure, to the derived category of left comodules over the dual of the classical topological Steenrod algebra. In order to obtain this result, the category of isotropic cellular modules over the motivic Brown-Peterson spectrum is also studied, and isotropic Adams and Adams-Novikov spectral sequences are developed. As a consequence, we also compute hom sets in the category of isotropic Tate motives between motives of isotropic cellular spectra.

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arXiv2023

Motivic cohomology of the Nisnevich classifying space of even Clifford groups

In this paper, we consider the split even Clifford group $\Gamma^+_n$ and compute the mod 2 motivic cohomology ring of its Nisnevich classifying space. The description we obtain is quite similar to the one provided for spin groups in [11]. The fundamental difference resides in the behaviour of the second subtle Stiefel-Whitney class that is non-trivial for even Clifford groups, while it vanished in the spin-case.

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arXiv2022

Subtle characteristic classes and Hermitian forms

Following [14], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfister form. Moreover, we present a description of the motive of the torsor corresponding to a hermitian form in terms of its subtle characteristic classes.

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arXiv2022

Subtle characteristic classes for $Spin$-torsors

Extending [14], we obtain a complete description of the motivic cohomology with ${\mathbb Z}/2$-coefficients of the Nisnevich classifying space of the spin group $Spin_n$ associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel-Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from $I^3$, which are closely related to $Spin_n$-torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel-Whitney class. Moreover, exploiting the relation between $Spin_7$ and $G_2$, we describe completely the motivic cohomology ring of the Nisnevich classifying space of $G_2$. The result in topology was obtained by Quillen in [13].

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arXiv2022

Isotropic stable motivic homotopy groups of spheres

In this paper we explore the isotropic stable motivic homotopy category constructed from the usual stable motivic homotopy category, following the work of Vishik on isotropic motives (see [29]), by killing anisotropic varieties. In particular, we focus on cohomology operations in the isotropic realm and study the structure of the isotropic Steenrod algebra. Then, we construct an isotropic version of the motivic Adams spectral sequence and apply it to find a complete description of the isotropic stable homotopy groups of the sphere spectrum, which happen to be isomorphic to the $Ext$-groups of the topological Steenrod algebra. At the end, we will see that this isomorphism is not only additive but respects higher products, completely identifying, as rings with higher structure, the cohomology of the classical Steenrod algebra with isotropic stable homotopy groups of spheres.

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arXiv2022

A Serre-type spectral sequence for motivic cohomology

In this paper, we construct and study a Serre-type spectral sequence for motivic cohomology associated to a map of bisimplicial schemes with motivically cellular fiber. Then, we show how to apply it in order to approach the computation of the motivic cohomology of the Nisnevich classifying space of projective general linear groups. This naturally yields an explicit description of the motive of a Severi-Brauer variety in terms of twisted motives of its \v{C}ech simplicial scheme.

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