We analyze in homological terms the homotopy fixed point spectrum of a –equivariant commutative –algebra . There is a homological homotopy fixed point spectral sequence with , converging conditionally to the continuous homology of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class in the –term of the spectral sequence there are other classes in the –term (obtained mostly by Dyer–Lashof operations on ) that are infinite cycles, ie survive to the –term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra of many –algebras, including , , , and . Similar results apply for all finite subgroups , and for the Tate and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic –theory of commutative –algebras.
"Differentials in the homological homotopy fixed point spectral sequence." Algebr. Geom. Topol. 5 (2) 653 - 690, 2005. https://doi.org/10.2140/agt.2005.5.653