Coverings, the graded center and Hochschild cohomology

Suppose that RR is a group graded KK-algebra, where KK is a commutative ring and RR is graded by a group GG. The GG-grading of RR leads to a GG-grading of certain Ext-algebras of RR. On the other hand, with the GG-grading of RR, one associates a ‘covering’ algebra SS. This paper begins by studying the relationship between Ext-algebras of the covering SS and the covering of the Ext-algebras of RR.We investigate the fixed ring SGSG and obtain an explicit KK-splitting of SS as SG⊕ISG⊕I, for some KK-submodule II of SS. We also study the relationship between the graded centers of RR and SS.Finally, it has been noted by a number of authors [C. Cibils, M.J. Redondo, Cartan–Leray spectral sequence for Galois coverings of linear categories, J. Algebra 284 (2005) 310–325; E.N. Marcos, R. Martínez-Villa, Ma.I.R. Martins, Hochschild cohomology of skew group rings and invariants, Cent. Eur. J. Math. 2 (2) (2004) 177–190 (electronic)], that GG acts on the Hochschild cohomology ring of SS, HH∗(S), and that there are monomorphisms (HHn(S))G→HHn(R), for n≥0n≥0. We provide explicit descriptions of these maps for n=0n=0 and 1.