Andrew J. Blumberg, Columbia University, New York, New York.Michael A. Mandell, Indiana University, Bloomington, Indiana.

• 1. Introduction2. Orthogonal $G$-spectra and the Tate fixed points3. A lax Kunneth theorem for Tate fixed points4. The Tate spectral sequences5. Topological periodic cyclic homology6. A filtration argument (Proof of Theorem A)7. Filtered modules over filtered ring orthogonal spectra8. Comparison of the lefthand and righthand spectral sequences (Proof of Theorem 6.3)9. Conditional convergence of the lefthand spectral sequence (Proof of Lemma 6.5)10. Constructing the filtered model: The positive filtration11. Constructing the filtered model: The negative filtration12. Constructing the filtered model and verifying the hypotheses of Chapter 613. The $E^1$-term of the Hesselholt-Madsen $\mathbb {T}$-Tate spectral sequence14. Comparison of the Hesselholt-Madsen and Greenlees $\mathbb {T}$-Tate Spectral Sequences15. Coherence of the equivalences $E\mathbb {T}/E\mathbb {T}_{2n-1}\simeq E\mathbb {T}_{+}\wedge S^{\mathbb {C}(1)^{n}}$ (Proof of Lemma 13.3)16. The strong Kunneth theorem for $THH$17. $THH$ of smooth and proper algebras (Proof of Theorem C)18. The finiteness theorem for $TP$ (Proof of Theorem B)19. Comparing monoidal models20. Identification of the enveloping algebras and $\operatorname {Bal}$21. A topologically enriched lax symmetric monoidal fibrant replacement functor for equivariant orthogonal spectra (Proof of Lemma 3.1)

Andrew J. Blumberg, Columbia University, New York, New York.Michael A. Mandell, Indiana University, Bloomington, Indiana.

• 1. Introduction2. Orthogonal $G$-spectra and the Tate fixed points3. A lax Kunneth theorem for Tate fixed points4. The Tate spectral sequences5. Topological periodic cyclic homology6. A filtration argument (Proof of Theorem A)7. Filtered modules over filtered ring orthogonal spectra8. Comparison of the lefthand and righthand spectral sequences (Proof of Theorem 6.3)9. Conditional convergence of the lefthand spectral sequence (Proof of Lemma 6.5)10. Constructing the filtered model: The positive filtration11. Constructing the filtered model: The negative filtration12. Constructing the filtered model and verifying the hypotheses of Chapter 613. The $E^1$-term of the Hesselholt-Madsen $\mathbb {T}$-Tate spectral sequence14. Comparison of the Hesselholt-Madsen and Greenlees $\mathbb {T}$-Tate Spectral Sequences15. Coherence of the equivalences $E\mathbb {T}/E\mathbb {T}_{2n-1}\simeq E\mathbb {T}_{+}\wedge S^{\mathbb {C}(1)^{n}}$ (Proof of Lemma 13.3)16. The strong Kunneth theorem for $THH$17. $THH$ of smooth and proper algebras (Proof of Theorem C)18. The finiteness theorem for $TP$ (Proof of Theorem B)19. Comparing monoidal models20. Identification of the enveloping algebras and $\operatorname {Bal}$21. A topologically enriched lax symmetric monoidal fibrant replacement functor for equivariant orthogonal spectra (Proof of Lemma 3.1)

Andrew J. Blumberg, Columbia University, New York, New York.Michael A. Mandell, Indiana University, Bloomington, Indiana.

• 1. Introduction2. Orthogonal $G$-spectra and the Tate fixed points3. A lax Kunneth theorem for Tate fixed points4. The Tate spectral sequences5. Topological periodic cyclic homology6. A filtration argument (Proof of Theorem A)7. Filtered modules over filtered ring orthogonal spectra8. Comparison of the lefthand and righthand spectral sequences (Proof of Theorem 6.3)9. Conditional convergence of the lefthand spectral sequence (Proof of Lemma 6.5)10. Constructing the filtered model: The positive filtration11. Constructing the filtered model: The negative filtration12. Constructing the filtered model and verifying the hypotheses of Chapter 613. The $E^1$-term of the Hesselholt-Madsen $\mathbb {T}$-Tate spectral sequence14. Comparison of the Hesselholt-Madsen and Greenlees $\mathbb {T}$-Tate Spectral Sequences15. Coherence of the equivalences $E\mathbb {T}/E\mathbb {T}_{2n-1}\simeq E\mathbb {T}_{+}\wedge S^{\mathbb {C}(1)^{n}}$ (Proof of Lemma 13.3)16. The strong Kunneth theorem for $THH$17. $THH$ of smooth and proper algebras (Proof of Theorem C)18. The finiteness theorem for $TP$ (Proof of Theorem B)19. Comparing monoidal models20. Identification of the enveloping algebras and $\operatorname {Bal}$21. A topologically enriched lax symmetric monoidal fibrant replacement functor for equivariant orthogonal spectra (Proof of Lemma 3.1)