Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[P]\in H^3(X,\mathbb{Z})$, the twisted K-theory $K_P(X)$ can be abutted by a spectral sequence with $E^{pq}_2=H^p(X;K^q(*))$, but the coefficient is twisted by some $\xi_P\in H^1(X,\mathbb{Z}/2)$.
Since twisted cohomology theory are not generalized cohomologies, there should be some additional machineries on why it would work. The references in Atiyah and Segal's paper were too vague for explaning this issue. I think I would like to have a more solid reference or explanation.
Thanks.
