We describe a spectral sequence for computing Leibniz cohomology for Lie algebras.

The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.

Dans cette note, nous montrons que la suite spectrale du coniveau associée à un spectre motivique sur un corps parfait coïncide avec sa suite spectrale d’hypercohomologie pour la t-structure homotopique.

On calcule dans cet article l’homologie stable des groupes orthogonaux et symplectiques sur un corps fini $k$ à coefficients tordus par un endofoncteur usuel $F$ des $k$-espaces vectoriels (puissance extérieure, symétrique, divisée...). Par homologie stable, on entend, pour tout entier naturel $i$, les colimites des espaces vectoriels $H_i(O_{n,n}\left(k\right);F\left(k^{2n}\right))$ et $H_i({\mathrm{Sp}}_{2n}\left(k\right);F\left(k^{2n}\right))$ — dans cette situation, la stabilisation (avec une borne explicite en fonction de $i$ et $F$) est un résultat classique de Charney. Tout d’abord, nous donnons un cadre...

We consider Taylor approximation for functors from the small category of finite pointed sets $\Gamma$ to modules and give an explicit description for the homology of the layers of the Taylor tower. These layers are shown to be fibrant objects in a suitable closed model category structure. Explicit calculations are presented in characteristic zero including an application to higher order Hochschild homology. A spectral sequence for the homology of the homotopy fibres of this approximation is provided.

We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra....

We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical $\mathrm{Ext}$-computations as well as new results. In particular, we get a cohomological version of the “fundamental theorems” from classical invariant theory for $G{L}_n$ for $n$ big enough (and we give a conjecture for smaller values of $n$). We also study the “twisting spectral sequence” $E^{s,t}(F,G,r)$ converging to the extension groups ${\mathrm{Ext}}_{{𝒫}_{𝕜}}^{*}(F^{\left(r\right)},G^{\left(r\right)})$ between the...

Let p be a prime, and let ℱ be the category of functors from the finite ${}_p$-vector spaces to all ${}_p$-vector spaces. The object Id of ℱ is the inclusion functor. Let F and G be two objects in ℱ. If F and G satisfy suitable conditions, the main result of this paper allows one to compute $Ext{*}_{ℱ}(Id,G\circ F)$ from the knowledge of $Ext{*}_{ℱ}(Id,F)$ and $Ext{*}_{ℱ}(Id,G)$.