Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 5, Number 2 (2005), 653-690.
Differentials in the homological homotopy fixed point spectral sequence
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.
Algebr. Geom. Topol., Volume 5, Number 2 (2005), 653-690.
Received: 2 June 2004
Revised: 3 June 2005
Accepted: 21 June 2005
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55S12: Dyer-Lashof operations 55T05: General
Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55P91: Equivariant homotopy theory [See also 19L47]
homotopy fixed points Tate spectrum homotopy orbits commutative $S$–algebra Dyer–Lashof operations differentials topological Hochschild homology topological cyclic homology algebraic $K$–theory
Bruner, Robert R; Rognes, John. Differentials in the homological homotopy fixed point spectral sequence. Algebr. Geom. Topol. 5 (2005), no. 2, 653--690. doi:10.2140/agt.2005.5.653. https://projecteuclid.org/euclid.agt/1513796425