A multiplicative Tate spectral sequence for compact Lie group actions / Alice Hedenlund, John Rognes.

Format:

Book

Author/Creator:

Hedenlund, Alice, author.

Rognes, John, author.

Series:

Memoirs of the American Mathematical Society ; v. 294, no. 1468.

Memoirs of the American Mathematical Society, 0065-9266 ; v. 294, no. 1468

Language:

English

Subjects (All):

Algebraic topology.

Physical Description:

v, 134 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2024]

Summary:

Given a compact Lie group G and a commutative orthogonal ring spectrum R such that R[G]*=π*(R^G+) is finitely generated and projective over π*(R), we construct a multiplicative G-Tate spectral sequence for each R-module X in orthogonal G-spectra, with E2-page given by the Hopf algebra Tate cohomology of R[G]* with coefficients in π*(X). Under mild hypotheses, such as X being bounded below and the derived page RE∞ vanishing, this spectral sequence converges strongly to the homotopy π*(XtG) of the G-Tate construction XtG = [EG^F(EG+,X)]G.

Contents:

Chapter 1. Introduction ; Background and aim ; Main results ; Organisations ; Acknowledgements

Chapter 2. Tate Cohomology for Hopf Algebras ; 2.1. Modules over Hopf algebras ; 2.2. Chain complexes of T-modules ; 2.3. Tate complexes ; 2.4. Complete resolutions ; 2.5. Multiplicative structure of Tate cohomology ; 2.6. Computation

Chapter 3. Homotopy Groups of Orthogonal G-Spectra ; 3.1. Equivariant homotopy groups ; 3.2. A cocommutative Hopf algebra ; 3.3. A restriction homomorphism

Chapter 4. Sequences of Spectra and Spectral Sequences ; 4.1. Cartan-Eilenberg systems ; 4.2. Sequences ; 4.3. Filtrations ; 4.4. Pairings of sequences ; 4.5. Pairings of Cartan-Eilenberg systems, I ; 4.6. Pairings of Cartan-Eilenberg systems, II ; 4.7. The convolution product

Chapter 5. The G-Homotopy Fixed Point Spectral Sequence ; 5.1. The filtered G-space EG ; 5.2. G-homotopy fixed points ; 5.3. Algebraic description of the E1- and E2-pages ; 5.4. The odd spheres filtration

Chapter 6. The G-Tate Spectral Sequence ; 6.1. The filtered G-space EG ; 6.2. The G-Tate construction ; 6.3. The Hesselholt-Madsen filtration ; 6.4. Algebraic description of E1 and E2 ; 6.5. The Greenlees-May filtration ; Convergence ; Summary: The T-Tate spectral sequence

Bibliography

Index.

Notes:

"February 2024, volume 294, number 1468 (fifth of 5 numbers)."

Includes bibliographical references (pages 131-132) and index.

ISBN:

1470468786

9781470468781

OCLC:

1426995130