The equality of Hochschild cohomology group and module cohomology group for semigroup algebras

‎Let $S$ be a commutative inverse semigroup with idempotent set $E$‎. ‎In this paper‎, ‎we show that for every $n\in \mathbb{N}_0$‎, ‎$n$-th Hochschild cohomology group of semigroup algebra $\ell^1(S)$ with coefficients in $\ell^\infty(S)$ and its $n$-th $\ell^1(E)$-module cohomology group‎, ‎are equal‎. ‎Indeed‎, ‎we prove that‎ ‎\[ \HH^{n}(\ell^1(S),\ell^\infty(S))=\HH^{n}_{\ell^1(E)}(\ell^1(S),\ell^\infty(S)),\] for all $n\geq 0$‎.

G-Algebra Structure on the Higher Order Hochschild Cohomology HS2∗(A,A)

We present a deformation theory associated to the higher Hochschild cohomology [Formula: see text]. We also study a [Formula: see text]-algebra structure associated to this deformation theory.

Deformation and Hochschild cohomology of coisotropic algebras

Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.

On the geometry of lattices and finiteness of Picard groups

Let ( K , 𝒪 , k ) {(K,\mathcal{O},k)} be a p-modular system with k algebraically closed and 𝒪 {\mathcal{O}} unramified, and let Λ be an 𝒪 {\mathcal{O}} -order in a separable K-algebra. We call a Λ-lattice L rigid if Ext Λ 1 ⁡ ( L , L ) = 0 {{\operatorname{Ext}}^{1}_{\Lambda}(L,L)=0} , in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into “varieties of lattices”, we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular, the Picard groups of blocks of finite groups defined over 𝒪 {\mathcal{O}} are always finite.