  We prove a general form of the statement that the cohomology of a quotient
stack can be computed by the Borel construction. It also applies to the lisse
extensions of generalized cohomology theories like motivic cohomology and
algebraic cobordism. We use this to prove a (higher) equivariant
Grothendieck-Riemann-Roch theorem, comparing Borel-equivariant G-theory and
equivariant Chow groups. We also give a Bernstein-Lunts-type gluing description
of the infinity-category of equivariant sheaves on a scheme X, in terms of
nonequivariant sheaves on X and sheaves on its Borel construction.
