Université de Nantes, April 23-27, 2012 — Functor homology and applications
Scientific program

Aurélien Djament
Homologie stable des groupes à coefficients polynomiaux

Résumé :
les systèmes inductifs de groupes tels ceux des groupes symétriques, linéaires ou unitaires, donnent lieu à une riche structure algébrique quand on considère leur homologie comme groupes discrets stablement, c'est-à-dire après passage à la colimite (structures d'algèbres de Hopf, liens avec la topologie pour les groupes symétriques, avec la K-théorie algébrique, éventuellement hermitienne, pour les groupes linéaires ou unitaires...). Après avoir rappelé très brièvement quelques propriétés fondamentales de cette homologie stable à coefficients constants, nous nous pencherons sur le problème du calcul de l'homologie stable à coefficients tordus par un foncteur polynomial - typiquement, une puissance tensorielle, symétrique ou divisée de la représentation tautologique. Nous montrerons comment on peut la calculer, dans les cas favorables, à l'aide de l'homologie stable à coefficients constants et de groupes d'homologie des foncteurs. On commence pour cela par ramener, par des arguments assez formels, cette homologie stable à l'homologie de catégories peu appropriées, sur des foncteurs généraux, avant de montrer, suivant une méthode remarquablement simple, générale et efficace inaugurée par Scorichenko, que l'homologie de ces catégories à coefficients polynomiaux est isomorphe à l'homologie de catégories beaucoup plus maniables et étudiées (des catégories additives).

Plan indicatif :
1) l'homologie des groupes d'automorphismes dans une catégorie monoïdale symétrique : présentation des problèmes essentiels, des cas abordés dans ce cours, de quelques résultats connus en matière de stabilité homologique et de calcul d'homologie stable à coefficients constants ; aperçu des résultats principaux pour les coefficients polynomiaux ; liens avec d'autres problèmes d'algèbre homologique (K-théorie, groupes de découpage, cohomologie des groupes algébriques...).
2) Premier lien entre homologie des foncteurs et homologie stable à coefficients tordus (pour les groupes d'automorphismes dans une catégorie monoïdale symétrique) : rappels sur l'homologie des petites catégories, un cadre axiomatique simple pour obtenir un tel lien sans hypothèse sur les coefficients, application aux groupes symétriques (d'après Betley).
3) Le critère d'annulation homologique de Scorichenko : il assure l'annulation des groupes de torsion entre certains foncteurs et des foncteurs polynomiaux arbitraires, sur une petite catégorie additive. On commencera par quelques rudiments sur les foncteurs polynomiaux et les effets croisés, avant d'énoncer et démontrer ce résultat, qui repose sur une utilisation judicieuse d'adjonctions.
4) Application aux groupes linéaires et unitaires : on montrera comment déduire des considérations des deux séances précédentes le théorème de Scorichenko sur la K-théorie stable à coefficients polynomiaux sur un anneau quelconque et un analogue pour les groupes unitaires.
5) On discutera quelques applications des résultats précédents : calculs de groupes d'homologie stable de groupes linéaires ou orthogonaux (sur un corps ou un sous-anneau de Q, en admettant les calculs homologiques profonds dus à Franjou, Friedlander, Lannes, Pirashvili, Schwartz, Scorichenko, Suslin) et propriétés qualitatives générales (comportement par changement de base, compatibilité aux coproduits...).

The content of this course can be found (in pdf format) on the author's page.

Roman Mikhailov
Derived functors of nonadditive functors and homotopy theory

Abstract:
Here is a tentative plan for the series. I'll give as many examples as possible.
Lecture 1. Introduction to the theory of polynomial functors.
The splitting and non-splitting of certain functors. Simple methods for computing of functorial Hom and Ext-groups. Koszul and de Rham complexes.
Lecture 2. Derived functors of non-additive functors.
Decalage isomorphisms. Description of the derived functors of certain quadratic and cubical functors. Homotopy theory of Lie and super-Lie functors.
Lecture 3. Applications of derived functors.
Homology of Eilenberg-MacLane spaces from the functorial point of view, homotopy groups of the suspensions of classifying spaces, homotopy groups of spheres, K-functors.
Lecture 4. The life of spectral sequences from the functorial point of view I.
Hierarchies of special functors, description of certain differentials in the homotopy spectral sequences.
Lecture 5. The life of spectral sequences from the functorial point of view II.
Predator spectral sequences. Homotopy groups of wedges of Eilenberg-MacLane spaces (if time permits).

The preprint On the splitting of polynomial functors is closely related to this course.

Wilberd van der Kallen
Cohomological finite generation for reductive groups

Abstract:
Let a reductive algebraic group G (e.g. G=SL₃) act on a finitely generated algebra A over the ground field k. If k is C, then classical invariant theory states that the subring A^G of invariants in A is a finitely generated k-algebra. For a general field k this still holds by the theorem of Haboush. Now view A^G as the degree zero component of the cohomology algebra H*(G,A). The speaker conjectured that the full cohomology algebra H*(G,A) is also finitely generated. This conjecture was first reduced to the existence of certain universal cohomology classes, enriching the family of classes employed by Friedlander and Suslin in their proof of finite generation of cohomology of finite group schemes. Touzé proved the conjecture by constructing the requisite classes using the homological algebra of strict polynomial bifunctors introduced by Franjou and Friedlander. (Strict polynomial bifunctors have two variables, while the strict polynomial functors of Friedlander and Suslin have one variable.) Touzé also made a conjecture concerning the effect of Frobenius twist on Ext groups in the category of strict polynomial functors. Proving this conjecture would provide a different way to construct the classes of Touzé.

The content of this course can be found (in pdf format) on the author's page.
Part I of Waterhouse's book Introduction to Affine Group Schemes is useful for preparation.

Antoine Touzé
Prerequisites

The content of this short course in Homological Algebra (and a few more prerequisites) can be found (in pdf format) on the author's page.

Serge Bouc
The Roquette category of finite p-groups

Abstract:
Let p be a prime number. There are many examples of pairs of non-isomorphic finite p-groups P and Q such that F(P) ≅ F(Q) for various functors F coming from representation theory, such as the functors of rational or complex representations, or the group of units of Burnside rings, or the torsion part of the Dade group. These similarities can be explained by introducing the Roquette category ℜp, which is a rigid additive tensor category with the following properties :

• Every finite p-group P can be viewed as an object of  ℜp.  The tensor product of two finite p-groups P and Q in  ℜp is their cartesian product P × Q.
•  In ℜp , every finite finite p-group P has a particular direct summand ∂P, called the edge of P, and P splits as the direct sum of the edges of all its quotients. This edge ∂P should be thought as the faithful part of (and indeed ∂P=0 if the center of P is not cyclic).
• In ℜp, every finite finite p-group P splits as a direct sum of edges of  Roquette p-groups, i.e. p-groups of normal rank 1. Such a decomposition is essentially unique, and follows from the knowledge of a genetic basis of P.
• If P and Q are Roquette p-groups, then ∂P × ∂Q  is isomorphic to the direct sum of (an explicit number of) copies of the edge of another (explicit) Roquette p-group P ⋄ Q.
• The additive functors from ℜp to abelian groups are the rational p-biset functors.
  

Larry Breen
Divided powers and (co)homology

Abstract:
I will discuss a number of homological and homotopical situations in which one encounters the derived functors of the divided power functors and of other related ones.

Marcin Chałupnik
Affine strict poynomial functors

Abstract:
I introduce the category of affine strict polynomial functors P_d^af. This category may be thought of as sort of functorial analogy of the category of representations of the group of loops on GL_n. It encodes an important part of the homological algebra of the category P_d of strict polynomial functors over a field of positive characteristic. For example, the set of simples in P_d^af is indexed by p-tuples of Young diagrams of multiweight d, which explains the role (quite mysterious until now) played by p-quotients of Young diagrams in computations of Ext-groups in P_pd. Also, the main computational tool in P_d, the de Rham complex, can be nicely interpreted within this formalism.

Ivo Dell'Ambrogio
From equivariant K-theory to equivariant KK-theory via restricted Yoneda and Mackey functors

Abstract:
I will explain the general and natural homological techniques that one can use to compute (among other things) the Hom groups in a triangulated category, by way of a restricted Yoneda functor. As a (perhaps exotic) illustration, I will consider the Kasparov category of a finite group G, where the objects are C*-algebras equipped with a G-action by *-isomorphisms and the Hom sets are Kasparov's bivariant K-theory groups KK^G(A,B) . By applying the general techniques, we obtain a new universal coefficient spectral sequence and a new Künneth spectral sequence abutting to KK^G(A,B) and K^G(A ⊗ B) , respectively, which can be computed in a suitable nice abelian category of Mackey functors for G. These results generalize the classical and extremely useful universal coefficient and Künneth theorems of Rosenberg-Schochet, which are the case G=1.

Dmitry Kaledin
Hochschild-Witt complex

Abstract:
The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves WΩ*(X) on a smooth algebraic variety X over a finite field, computing the cristalline cohomology of X. I am going to present a non-commutative generalization of this: even for a non-commutative ring A, one can define a functorial "Hochschild-Witt complex" with homology WHH_*(A); if A is commutative, then WHH_i(A)=WΩⁱ(X), X = Spec A (this is analogous to the isomorphism HH_i(A)=Ωⁱ(X) discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

Nick Kuhn
A polynomial functor approach to some equivariant K-theory algebras.

Abstract:
Given a finite complex X, we consider the direct sum over r of the equivariant K-theory groups K_Σr(X^r). We discuss how this highly structured object, and some others, arise as Grothendieck groups of categories of polynomial functors.

Geoffrey Powell
On the connective K-theory of elementary abelian 2-groups

Abstract:
We explain how the usage of functorial techniques provides a new and conceptual approach to calculating the connective K-cohomology and homology of the classifying space BV of an elementary abelian 2-group, V. This provides new insight into the structure, for instance facilitating the analysis of the local cohomology spectral sequence.

Christine Vespa
Polynomial functors from groups to abelian groups

Abstract:
Although many areas of algebra study only additive functors between abelian categories, many functors are obviously non additive. For example, if we consider the category of abelian groups Ab, we can define a functor ⊗ⁿ : Ab → Ab which associates to an abelian group G the n-th tensor product ⊗ⁿ (G) = G ⊗ . . . ⊗ G. This functor is not additive but it is a polynomial functor of degree n. In general, combinatorial description of polynomial functors is a very intricate problem. In 2001 Baues, Dreckmann, Franjou and Pirashvili gave a description of polynomial functors from (finitely generated free) abelian groups to abelian groups in terms of non-linear Mackey functors. The aim of this talk is to present a generalization of this result for polynomial functors from (finitely generated free) groups to abelian groups (obtained in collaboration with Hartl and Pirashvili). This description is a consequence of the two following results. First we obtain a description of polynomial functors from P-algebras (for P a set-operad) to abelian groups. Secondly we obtain an isomorphism between polynomial functors on monoids and those on groups.