Coverings, the graded center and Hochschild cohomology

Suppose that R is a group graded K-algebra, where K is a commutative ring and R is graded by a group G. The G-grading of R leads to a G-grading of certain Ext-algebras of R. On the other hand, with the G-grading of R, one associates a ‘covering’ algebra S. This paper begins by studying the relationship between Ext-algebras of the covering S and the covering of the Ext-algebras of R. We investigate the fixed ring SG and obtain an explicit K-splitting of S as SGcircled plusI, for some K-submodule I of S. We also study the relationship between the graded centers of R and S. 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 G acts on the Hochschild cohomology ring of S, image, and that there are monomorphisms image, for n≥0. We provide explicit descriptions of these maps for n=0 and 1.