Let M be a closed orientable manifold of dimension dand ${𝒞}^{*}\left(M\right)$ be the usual cochain algebra on M with coefficients in a fieldk. The Hochschild cohomology of M, $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is a graded commutative and associative algebra. The augmentation map $\epsilon :{𝒞}^{*}\left(M\right)\to 𝑘$ induces a morphism of algebras $I:H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))\to H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$. In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$, which is in general quite small. The algebra $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is expected to be isomorphic...

This paper studies the Hochschild cohomology of finite-dimensional monomial algebras. If Λ = K/I with I an admissible monomial ideal, then we give sufficient conditions for the existence of an embedding of $K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$ into the Hochschild cohomology ring HH*(Λ). We also introduce stacked algebras, a new class of monomial algebras which includes Koszul and D-Koszul monomial algebras. If Λ is a stacked algebra, we prove that $HH*\left(\Lambda \right)/\cong K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$, where is the ideal in HH*(Λ) generated by the homogeneous nilpotent elements. In...

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 give a new proof of the main result of [1] which does not use the classification of the finite simple groups.

We describe the representation-infinite blocks B of the group algebras KG of finite groups G over algebraically closed fields K for which all simple modules are periodic with respect to the action of the syzygy operators. In particular, we prove that all such blocks B are periodic algebras of period 4. This confirms the periodicity conjecture for blocks of group algebras.

The aim of this paper is to study the pro-algebraic fundamental group of a compact Kähler manifold. Following work by Simpson, the structure of this group’s pro-reductive quotient is already well understood. We show that Hodge-theoretic methods can also be used to establish that the pro-unipotent radical is quadratically presented. This generalises both Deligne et al.’s result on the de Rham fundamental group, and Goldman and Millson’s result on deforming representations of Kähler groups, and can...

Let Γ be a finite-dimensional hereditary basic algebra. We consider the radical rad Γ as a Γ-bimodule. It is known that there exists a quasi-hereditary algebra 𝓐 such that the category of matrices over rad Γ is equivalent to the category of Δ-filtered 𝓐-modules ℱ(𝓐,Δ). In this note we determine the quasi-hereditary algebra 𝓐 and prove certain properties of its module category.