Coverings, the graded center and Hochschild cohomology

Abstract

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 S G and obtain an explicit K -splitting of S as S G ⊕ I , 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 , HH ∗ ( S ) , and that there are monomorphisms ( HH n ( S ) ) G → HH n ( R ) , for n ≥ 0 . We provide explicit descriptions of these maps for n = 0 and 1.