Searcharxiv⌕ Search

arXiv subjects

Antoine Touzé

Publications and source records attributed to Antoine Touzé.

At least 19 recordsLinked to original sources
arXiv2026

The universal property of strict polynomial functors

In characteristic zero, the category of strict polynomial functors is well-known to be the tensor abelian category freely generated by one object. We show that this property fails in positive characteristic, but that it can be repaired by restricting the class of tensor abelian categories considered. The new universal property recovers several known constructions and shows that Ext-algebras of strict polynomial functors act on cohomological computations in many other contexts.

math.RT↗
arXiv2026

Separation and excision in functor homology

We prove separation and excision results in functor homology. These results explain how the global Steinberg decomposition of functors proved by Djament, Touz{é} and Vespa behaves in Ext and Tor computations.

math.KT↗
arXiv2026

On formality of diagrams of Eilenberg-MacLane spaces

In this paper, we establish formality (over $\mathbb{Q}$) for diagrams of Eilenberg-MacLane spaces of any height $n\geq 1$. This implies spectral sequence (over $\mathbb{Q}$) collapse at page $2$ for any diagram of EML spaces over any small category. We prove by functor calculus argument that formality does not hold over any fixed commutative ring $\mathbf{k}$ not containing $\mathbb{Q}$, where the category of diagrams is over the category generated by finite direct sums of a cyclic group.

math.AT↗
arXiv2026

The homology of additive functors in prime characteristic

We compute certain Ext and Tor groups in the category of all functors from an Z/p-linear additive category A to vector spaces in terms of Ext and Tor computed in the full subcategory of additive functors from A to vector spaces. We thus obtain group homology computations for general linear groups.

math.KT↗
arXiv2024

Sur la structure des repr{é}sentations g{é}n{é}riques des groupes lin{é}aires infinis

We study several structure aspects of functor categories from a small additive category to a module category, in particular the category F(A,K) of functors from finitely generated free modules over a commutative ring A to vector spaces over a field K -- such functors are sometimes called \textit{generic representations} of linear groups over A with coefficients in K. We are especially interested with finitely generated functors of F(A,K) taking finite dimensional values. We prove that they can, under a mild extra assumption (always satisfied if the ring A is noetherian), be built from much better understood functors, namely polynomial functors (in the sense of Eilenberg-MacLane), or factorising at the source through reduction modulo a cofinite ideal of A. We deduce that such functors are always noetherian et that, if the ring A is finitely generated, they have finitely generated projective resolutions.Our methods rely mainly on the study of weight decompositions of functors and their cross-effects, our recent previous work with Vespa (Ann. ENS 2023) and elementary commutative algebra.

math.CT↗
arXiv2024

Homology of strict polynomial functors over Fp-linear additive categories

We generalize the strong comparison theorem of Franjou, Friedlander, Scorichenko and Suslin to the setting of Fp-linear additive categories. Our results have a strong impact in terms of explicit computations of functor homology, and they open the way to new applications to stable homology of groups or to K-theory. As an illustration, we prove comparison theorems between cohomologies of classical algebraic groups over infinite perfect fields, in the spirit of a celebrated result of Cline, Parshall, Scott et van der Kallen for finite fields.

math.AT↗
arXiv2023

Homological finiteness of functors on an additive category and applications

We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also a weaker homological finiteness property, which applies to twisted homological stability for matrix monoids. This is inspired by works by Schwartz and Betley-Pirashvili, which are generalised; this also uses decompositions {à} la Steinberg over an additive category that we recently got with Vespa. We show also, as an application, a finiteness property for stable homology of linear groups on suitable rings.

math.KT↗
arXiv2023

Sur la noeth{é}rianit{é} locale des foncteurs polynomiaux

Let A be a finitely-generated commutative ring and k a noetherian commutative ring. We show that, in the category of functors from finitely-generated projective A-modules to k-modules, each finitely-generated polynomial functor is noetherian and has a finitely-generated projective resolution.

math.KT↗
arXiv2022

Functor homology over an additive category

We uncover several general phenomenas governing functor homology over additive categories. In particular, we generalize the strong comparison theorem of Franjou Friedlander Scorichenko and Suslin to the setting of Fp-linear additive categories. Our results have a strong impact in terms of explicit computations of functor homology, and they open the way to new applications to stable homology of groups or to K-theory. As an illustration, we prove comparison theorems between cohomologies of classical algebraic groups over infinite perfect fields, in the spirit of a celebrated result of Cline, Parshall, Scott et van der Kallen for finite fields.

math.AT↗
arXiv2021

Décompositions à la Steinberg sur une catégorie additive

We give a description of simple functors taking finitely generated values, from a small additive category to the category of vector spaces over a field. This result is analogous to Steinberg's tensor product theorems in group representation theory. Our results rest on the notion of polynomial functor introduced by Eilenberg and Mac Lane. We give applications to representations of general linear groups or to finiteness properties of functor categories.

math.RT↗
arXiv2020

On the structure of graded commutative exponential funtors

We investigate the structure of graded commutative exponential functors. We give applications of these structure results, including computations of the homology of the symmetric groups and of extensions in the category of strict polynomial functors.

math.KT↗
arXiv2018

Cohomology of algebraic groups with coefficients in twisted representations

This article is a survey on the cohomology of a reductive algebraic group with coefficients in twisted representations. A large part of the paper is devoted to the advances obtained by the theory of strict polynomial functors initiated by Friedlander and Suslin in the late nineties. The last section explains that the existence of certain `universal classes' used to prove cohomological finite generation is equivalent to some recent `untwisting theorems' in the theory of strict polynomial functors. We actually provide thereby a new proof of these theorems.

math.KT↗
arXiv2016

Connectedness of cup products for polynomial representations of $GL_n$ and applications

We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to prove generalizations and variants of the Steinberg tensor product theorem. Our connectedness bounds for cup product maps depend on numerical invariants which seem also relevant to other problems, for example to study the cohomological behavior of the Schur functor.

math.RT↗
arXiv2015

A functorial control of integral torsion in homology

We study the integral torsion of the values of strict polynomial functors defined over the integers. We interpret some classical homological invariants as values of strict polynomial functors and therefore obtain estimates of the integral torsion of these invariants, including cancellation results.

math.RT↗
arXiv2013

Bar complexes and extensions of classical exponential functors

We compute Ext-groups between classical exponential functors (i.e. symmetric, exterior or divided powers) and their Frobenius twists. Our method relies on bar constructions, and bridges these Ext-groups with the homology of Eilenberg-Mac Lane Spaces. Together with the article "Troesch complexes and the twisting spectral sequence" (arXiv 1005.3133), this article provides an alternative approach to classical Ext-computations in the category of strict polynomial functors over fields. We also obtain significant Ext-computations for strict polynomial functors over the integers.

math.RT↗
arXiv2013

Ringel duality and derivatives of non-additive functors

We prove that Ringel duality in the category of strict polynomial functors can be interpreted as derived functors of non-additive functors (in the sense of Dold and Puppe). We give applications of this fact for both theories.

math.RT↗