Group cohomology based on partial representations - Usuga, Emmanuel Jerez (2020)
We consider the partial group cohomology $H_^n(G,M)$ of a group $G$ with values in $\\K_G$-module $M$, which is defined as the right derived functor of the functor of partial invariants. Showing that the functor of partial invariants is representable, we relate the partial group cohomology with the space of partial derivations and the partial augmentation ideal; next, we construct a projective resolution of the algebra $B$ as a $\\K_G$-module, where $B$ is a commutative subalgebra of $\\K_G$. This allows us to give another characterization of the partial group cohomology in terms of classes of functions that satisfy a certain identity of $n$-cocycles. We show the existence of a Grothendieck spectral sequence that relates the cohomology of the partial smash product with the partial group cohomology and the algebra cohomology. Given a unital partial action $\\alpha$ of $G$ on a algebra $\\mathcal$ we consider the $\\K_G$-module structure of $\\mathcal$ induced by $\\alpha$ and study the globalization problem for the partial cohomology with values in $\\mathcal$. The problem is reduced to an extendibility property of cocycles. Moreover, if $\\mathcal$ is a product of indecomposable blocks, we show that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous, whence we have that $H_^n(G,M)$ is isomorphic to the usual cohomology group $H^n(G, \\mathcal(\\mathcal))$, where $\\mathcal$ is the algebra under the enveloping action of $\\alpha$ and $\\mathcal(\\mathcal)$ is the multiplier algebra of $\\mathcal$.