Reflexive and dihedral (co)homology of a pre‐additive category

Abstract

The group dihedral homology of an algebra over a field with characteristic zero was introduced by Tsygan (1983). The dihedral homology and cohomology of an algebra with involution over commutative ring with identity, associated with the small category, were studied by Krasauskas et al. (1988), Loday (1987), and Lodder (1993). The aim of this work is concerned with dihedral and reflexive (co)homology of small pre-additive category. We also define the free product of involutive algebras associated with this category and study its dihedral homology group. Finally, following Perelygin (1990), we show that a small pre-additive category is Morita equivalence

On the Cohomology of Relative Babach Algebras

“…In this part, we defined the Banach relative dihedral cohomology and we study some of its properties and its relations with other cohomologies with it`s proves. We use [ (Alaa Hassan Noreldeen (2014), (2015))], [Gouda,Y. Gh., Alaa, H. N. & M. Saad,(2017)], and in this part.…”