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.