Equivariant cohomology and sheaves

RO(G)-graded equivariant cohomology was originally defined by using equivariant spectra. In this paper we construct a sheaf version of RO(G)-graded equivariant cohomology theory. By introducing a complex of presheaves we obtain an isomorphism theorem of two different cohomology theories. We further apply the results to the study of real algebraic varieties. (C) 2014 Elsevier Inc. All rights reserved.