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On établit une décomposition de l’homologie stable des groupes d’automorphismes des groupes libres à coefficients polynomiaux contravariants en termes d’homologie des foncteurs. Elle permet plusieurs calculs explicites, qui recoupent des résultats établis de manière indépendante par O. Randal-Williams et généralisent certains d’entre eux. Nos méthodes reposent sur l’examen d’extensions de Kan dérivées associées à plusieurs catégories de groupes libres, la généralisation d’un critère d’annulation homologique à coefficients polynomiaux dû à Scorichenko, le théorème de Galatius identifiant l’homologie stable des groupes d’automorphismes des groupes libres à celle des groupes symétriques, la machinerie des
-espaces et le scindement de Snaith.
Let R be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of R-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and décalage.
We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and recover known results when the twisting comes from finite order elements in discrete torsion.
The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme
there exists a motivic
-structure for the category
of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If
is of finite type over a field, then the heart of this
-structure (the category of mixed motivic sheaves over
) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over
in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic
-structure is transversal to the Chow weight structure for
(that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.
We prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. No direct combinatorial proof is known as yet. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.
Low dimensional algebraic K-groups of a commutative ring are described in terms of the homology of its elementary matrix group. This approach is prompted by recent successful computations of low-dimensional K-groups using group homology methods, and it builds on the identity K2(R)=H2(ER).
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