The Twisted Partial Group Algebra and (Co)homology of Partial Crossed Products

Given a group G and a partial factor set \(\sigma \) of G, we introduce the twisted partial group algebra \({\kappa }_{\textrm{par}}^\sigma G,\) which governs the partial projective \(\sigma \)-representations of G into algebras over a field \(\kappa .\) Using the relation between partial projective representations and twisted partial actions we endow \({\kappa }_{\textrm{par}}^\sigma G\) with the structure of a crossed product by a twisted partial action of G on a commutative subalgebra of \({\kappa }_{\textrm{par}}^\sigma G.\) Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product \(A*_{\Theta } G,\) involving the Hochschild homology of A and the partial homology of G, where \({\Theta }\) is a unital twisted partial action of G on a \(\kappa \)-algebra A with a \(\kappa \)-based twist. An analogous third quadrant cohomological spectral sequence is also obtained.