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A Strong Künneth Theorem for Periodic TopologicalCyclic Homology
Michael A. Mandell
Indiana University
Shanks Workshop on Homotopy Theory
Vanderbilt University
March 25, 2017
M.A.Mandell (IU) Periodic Topological Cyclic Homology Mar 2017 1 / 16
A Strong Künneth Theorem for Topological PeriodicCyclic Homology
Michael A. Mandell
Indiana University
Shanks Workshop on Homotopy Theory
Vanderbilt University
March 25, 2017
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 1 / 16
Overview
Topological periodic cyclic homology (TP) is the analogue of periodiccyclic homology (HP) using THH in place of HH. If k is a finite field,then smooth and proper d.g. categories over k satisfy a strongKünneth theorem:
TP(X ) ∧LTP(k) TP(Y )→ TP(X ⊗k Y )
is an isomorphism in the derived category of TP(k)-modules.
Joint work with Andrew BlumbergPreprint Soon
Outline1 Introduction to TP2 Structure and properties of TP3 The Künneth theorem
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 2 / 16
Overview
Topological periodic cyclic homology (TP) is the analogue of periodiccyclic homology (HP) using THH in place of HH. If k is a finite field,then smooth and proper varieties over k satisfy a strongKünneth theorem:
TP(X ) ∧LTP(k) TP(Y )→ TP(X ⊗k Y )
is an isomorphism in the derived category of TP(k)-modules.
Joint work with Andrew BlumbergPreprint Soon
Outline1 Introduction to TP2 Structure and properties of TP3 The Künneth theorem
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 2 / 16
Overview
Topological periodic cyclic homology (TP) is the analogue of periodiccyclic homology (HP) using THH in place of HH. If k is a finite field,then smooth and proper d.g. algebras over k satisfy a strongKünneth theorem:
TP(X ) ∧LTP(k) TP(Y )→ TP(X ⊗k Y )
is an isomorphism in the derived category of TP(k)-modules.
Joint work with Andrew BlumbergPreprint Soon
Outline1 Introduction to TP2 Structure and properties of TP3 The Künneth theorem
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 2 / 16
Overview
Topological periodic cyclic homology (TP) is the analogue of periodiccyclic homology (HP) using THH in place of HH. If k is a finite field,then smooth and proper d.g. algebras over k satisfy a strongKünneth theorem:
TP(X ) ∧LTP(k) TP(Y )→ TP(X ⊗k Y )
is an isomorphism in the derived category of TP(k)-modules.
Joint work with Andrew BlumbergPreprint Soon
Outline
1 Introduction to TP2 Structure and properties of TP3 The Künneth theorem
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 2 / 16
Overview
Topological periodic cyclic homology (TP) is the analogue of periodiccyclic homology (HP) using THH in place of HH. If k is a finite field,then smooth and proper d.g. algebras over k satisfy a strongKünneth theorem:
TP(X ) ∧LTP(k) TP(Y )→ TP(X ⊗k Y )
is an isomorphism in the derived category of TP(k)-modules.
Joint work with Andrew BlumbergPreprint Soon
Outline1 Introduction to TP
2 Structure and properties of TP3 The Künneth theorem
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 2 / 16
Overview
Topological periodic cyclic homology (TP) is the analogue of periodiccyclic homology (HP) using THH in place of HH. If k is a finite field,then smooth and proper d.g. algebras over k satisfy a strongKünneth theorem:
TP(X ) ∧LTP(k) TP(Y )→ TP(X ⊗k Y )
is an isomorphism in the derived category of TP(k)-modules.
Joint work with Andrew BlumbergPreprint Soon
Outline1 Introduction to TP2 Structure and properties of TP
3 The Künneth theorem
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 2 / 16
Overview
Topological periodic cyclic homology (TP) is the analogue of periodiccyclic homology (HP) using THH in place of HH. If k is a finite field,then smooth and proper d.g. algebras over k satisfy a strongKünneth theorem:
TP(X ) ∧LTP(k) TP(Y )→ TP(X ⊗k Y )
is an isomorphism in the derived category of TP(k)-modules.
Joint work with Andrew BlumbergPreprint Soon
Outline1 Introduction to TP2 Structure and properties of TP3 The Künneth theorem
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 2 / 16
Introduction
Hochschild Homology
and Cyclic Homology
Cyclic bar construction
Ncyq R = R ⊗ · · · ⊗ R︸ ︷︷ ︸
q factors
⊗R
R ⊗ · · · ⊗ R
⊗ ⊗
R
Chain complex
Construct Double Complex:
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HNHPHHHC
Cyclic structure =⇒ Connes’ B operator B : NcyR → NcyR[−1]
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 3 / 16
Introduction
Hochschild Homology
and Cyclic Homology
Cyclic bar construction
Ncyq R = R ⊗ · · · ⊗ R︸ ︷︷ ︸
q factors
⊗R
R ⊗ · · · ⊗ R
⊗ ⊗
R
Chain complex
Construct Double Complex:
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HNHPHHHC
Cyclic structure =⇒ Connes’ B operator B : NcyR → NcyR[−1]
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 3 / 16
Introduction
Hochschild Homology
and Cyclic Homology
Cyclic bar construction
Ncyq R = R ⊗ · · · ⊗ R︸ ︷︷ ︸
q factors
⊗R
R ⊗ · · · ⊗ R
⊗ ⊗
R
Chain complex
Construct Double Complex:
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HNHPHHHC
Cyclic structure =⇒ Connes’ B operator B : NcyR → NcyR[−1]
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 3 / 16
Introduction
Hochschild Homology and Cyclic Homology
Cyclic bar construction
Ncyq R = R ⊗ · · · ⊗ R︸ ︷︷ ︸
q factors
⊗R
R ⊗ · · · ⊗ R
⊗ ⊗
R
Chain complex
Construct Double Complex:
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HNHP
HH
HC
Cyclic structure =⇒ Connes’ B operator B : NcyR → NcyR[−1]
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 3 / 16
Introduction
Hochschild Homology and Cyclic Homology
Cyclic bar construction
Ncyq R = R ⊗ · · · ⊗ R︸ ︷︷ ︸
q factors
⊗R
R ⊗ · · · ⊗ R
⊗ ⊗
R
Chain complex
Construct Double Complex:
···
······
······
↓
↓ ↓ ↓ ↓ ···
· · · ← • ←
• ← • ← • ← •
↓
↓ ↓ ↓
· · · ← • ←
• ← • ← •
↓
↓ ↓
· · · ← • ←
• ← •
↓
↓
· · · ← • ←
•
↓· · · ← •
···
HNHPHH
HC
Cyclic structure =⇒ Connes’ B operator B : NcyR → NcyR[−1]
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 3 / 16
Introduction
Hochschild Homology and Cyclic Homology
Cyclic bar construction
Ncyq R = R ⊗ · · · ⊗ R︸ ︷︷ ︸
q factors
⊗R
R ⊗ · · · ⊗ R
⊗ ⊗
R
Chain complex
Construct Double Complex:···
···
······
···
↓ ↓
↓ ↓ ↓ ···
· · · ← • ← •
← • ← • ← •
↓ ↓
↓ ↓
· · · ← • ← •
← • ← •
↓ ↓
↓
· · · ← • ← •
← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
HN
HPHHHC
Cyclic structure =⇒ Connes’ B operator B : NcyR → NcyR[−1]
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 3 / 16
Introduction
Hochschild Homology and Cyclic Homology
Cyclic bar construction
Ncyq R = R ⊗ · · · ⊗ R︸ ︷︷ ︸
q factors
⊗R
R ⊗ · · · ⊗ R
⊗ ⊗
R
Chain complex
Construct Double Complex:···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
HN
HP
HHHC
Cyclic structure =⇒ Connes’ B operator B : NcyR → NcyR[−1]
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 3 / 16
Introduction
Topological Hochschild Homology
Cyclic bar construction (Bökstedt)
Ncyq R = R ∧ · · · ∧ R︸ ︷︷ ︸
q factors
∧R
R ∧ · · · ∧ R
∧ ∧R
Spectrum
Construction
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HH corresponds to THH
Cyclic structure =⇒ circle group action
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 4 / 16
Introduction
Topological Hochschild Homology
Cyclic bar construction (Bökstedt)
Ncyq R = R ∧ · · · ∧ R︸ ︷︷ ︸
q factors
∧R
R ∧ · · · ∧ R
∧ ∧R
Spectrum
Construction
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HH corresponds to THH
Cyclic structure =⇒ circle group action
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 4 / 16
Introduction
Topological Hochschild Homology
Cyclic bar construction (Bökstedt)
Ncyq R = R ∧ · · · ∧ R︸ ︷︷ ︸
q factors
∧R
R ∧ · · · ∧ R
∧ ∧R
Spectrum
Construction
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HH corresponds to THH
Cyclic structure =⇒ circle group action
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 4 / 16
Introduction
Topological Hochschild Homology
Cyclic bar construction (Bökstedt)
Ncyq R = R ∧ · · · ∧ R︸ ︷︷ ︸
q factors
∧R
R ∧ · · · ∧ R
∧ ∧R
Spectrum
Construction
···
···
······
···↓
↓
↓ ↓ ↓ ···· · · ← • ←
•
← • ← • ← •↓
↓
↓ ↓· · · ← • ←
•
← • ← •↓
↓
↓· · · ← • ←
•
← •↓
↓
· · · ← • ←
•
↓· · · ← •
···
HH corresponds to THH
Cyclic structure =⇒ circle group action
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 4 / 16
Introduction
Topological Hochschild Homology
Cyclic bar construction (Bökstedt)
Ncyq R = R ∧ · · · ∧ R︸ ︷︷ ︸
q factors
∧R
R ∧ · · · ∧ R
∧ ∧R
Spectrum
Construction
···
······
······
↓
↓ ↓ ↓ ↓ ···
· · · ← • ←
• ← • ← • ← •
↓
↓ ↓ ↓
· · · ← • ←
• ← • ← •
↓
↓ ↓
· · · ← • ←
• ← •
↓
↓
· · · ← • ←
•
↓· · · ← •
···
HH corresponds to THHHC corresponds to THHhT
Cyclic structure =⇒ circle group action
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 4 / 16
Introduction
Topological Hochschild Homology
Cyclic bar construction (Bökstedt)
Ncyq R = R ∧ · · · ∧ R︸ ︷︷ ︸
q factors
∧R
R ∧ · · · ∧ R
∧ ∧R
Spectrum
Construction···
···
······
···
↓ ↓
↓ ↓ ↓ ···
· · · ← • ← •
← • ← • ← •
↓ ↓
↓ ↓
· · · ← • ← •
← • ← •
↓ ↓
↓
· · · ← • ← •
← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
HH corresponds to THHHN corresponds to THHhT
Cyclic structure =⇒ circle group action
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 4 / 16
Introduction
Topological Hochschild Homology
Cyclic bar construction (Bökstedt)
Ncyq R = R ∧ · · · ∧ R︸ ︷︷ ︸
q factors
∧R
R ∧ · · · ∧ R
∧ ∧R
Spectrum
Construction···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
HH corresponds to THHHP corresponds to THH tT
Cyclic structure =⇒ circle group action
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 4 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)
(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)
(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)
(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)
(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Topological Periodic Cyclic Homology
DefinitionFor a ring spectrum R, define the Topological Periodic CyclicHomology of R by TP(R) = THH(R)tT.
HighlightsMajor player in trace method K -theory calculations
Characteristic p replacement for HP (?)(2014–) Hasse-Weil zeta function: Connes-Consani Hesselholt(2011–) Non-commutative motives: Kontsevich, Marcolli-Tabuadanon-commutative homological motives ????
Realization functor / Weil cohomology theory
HP∗(X )⊗k [t ,t−1] HP∗(Y )→ HP∗(X ⊗k Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 5 / 16
Introduction
Künneth Theorem
TheoremLax symmetric monoidal functor
TP(X ) ∧LTP(R) TP(Y )→ TP(X ⊗L
R Y )
is an isomorphism when X and Y are smooth and proper over k.
DefinitionA k -algebra X is smooth when it is compact as an X ⊗k X op-module,i.e., when R HomX⊗k X op
(X ,−) commutes with direct sums.
DefinitionA k -algebra X is proper when it is compact as a k -module.
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 6 / 16
Introduction
Künneth Theorem
TheoremLet k be finite field. The lax symmetric monoidal functor
TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗
L
k Y )
is an isomorphism when X and Y are smooth and proper over k.
DefinitionA k -algebra X is smooth when it is compact as an X ⊗k X op-module,i.e., when R HomX⊗k X op
(X ,−) commutes with direct sums.
DefinitionA k -algebra X is proper when it is compact as a k -module.
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 6 / 16
Introduction
Künneth Theorem
TheoremLet k be finite field. The lax symmetric monoidal functor
TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗
L
k Y )
is an isomorphism when X and Y are smooth and proper over k.
DefinitionA k -algebra X is smooth when it is compact as an X ⊗k X op-module,i.e., when R HomX⊗k X op
(X ,−) commutes with direct sums.
DefinitionA k -algebra X is proper when it is compact as a k -module.
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 6 / 16
Introduction
Künneth Theorem
TheoremLet k be finite field. The lax symmetric monoidal functor
TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗
L
k Y )
is an isomorphism when X and Y are smooth and proper over k.
DefinitionA k -algebra X is smooth when it is compact as an X ⊗k X op-module,i.e., when R HomX⊗k X op
(X ,−) commutes with direct sums.
DefinitionA k -algebra X is proper when it is compact as a k -module.
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 6 / 16
Introduction
Künneth Theorem
TheoremLet k be finite field. The lax symmetric monoidal functor
TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗
L
k Y )
is an isomorphism when X and Y are smooth and proper over k.
DefinitionA k -algebra X is smooth when it is compact as an X ⊗k X op-module,i.e., when R HomX⊗k X op
(X ,−) commutes with direct sums.
DefinitionA k -algebra X is proper when it is compact as a k -module.
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 6 / 16
Introduction
Künneth Theorem
TheoremLet k be finite field. The lax symmetric monoidal functor
TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗
L
k Y )
is an isomorphism when X and Y are smooth and proper over k.
DefinitionA k -algebra X is smooth when it is compact as an X ⊗k X op-module,i.e., when R HomX⊗k X op
(X ,−) commutes with direct sums.
DefinitionA k -algebra X is proper when it is compact as a k -module.
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 6 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
Review of Tate Construction
ET ET+ → S0 → ET
Smash with Z ET and take fixed points
(Z ET ∧ ET+)T → (Z ET)T → (Z ET ∧ ET)T
(X ET ∧ ET+)T ' Σ(X ET)hT ' ΣXhT (Adams Isomorphism)
Definition
For Z a T-equivariant spectrum Z tT = (Z ET ∧ ET)T.(Composite of derived functors.)
ΣZhT → Z hT → Z tT → Σ2ZhT
TP(X ) = THH(X )tT
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 7 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET
←− This is coherent
THH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET
←− This can be made coherent!
Use diagonal map ET→ ET× ET ←− This is coherentTHH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Multiplication
TP(X ) ∧ TP(Y )→ TP(X ∧ Y )
TP(X ) = (THH(X )ET ∧ ET)T
ET ∧ ET ' ET ←− This can be made coherent!Use diagonal map ET→ ET× ET ←− This is coherentTHH(X ) ∧ THH(Y ) ∼= THH(X ∧ Y )
TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
TP(X ) ∧ TP(R) ∧ TP(Y )→ TP(X ∧ R ∧ Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 8 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET
/ on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . .
/ S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET / on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . . / S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET / on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . . / S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET / on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . . / S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Filtration···
······
······
↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •
↓ ↓ ↓ ↓· · · ← • ← • ← • ← •
↓ ↓ ↓· · · ← • ← • ← •
↓ ↓· · · ← • ← •
↓· · · ← •
···
···�� �� �� ��
T× T× T�� �� ��
OOOOOO
T× T�� ��
OOOO
T
OO
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
TP(X ) = (THH(X )ET ∧ ET)T
Simplicial filtration on ET / on ET
T+ ∧ (T× T/(T ∨ T)) ∧∆[2]/∂∆[2]
T+ ∧ (T/{1}) ∧∆[1]/∂∆[1]
T+
T+, Σ2T+, Σ4T+,. . . / S0, ΣT+, Σ3T+,. . .
Filtration on TP(X ) : F iTP(X ) =
{(THH(X )(ET,ET−i−1) ∧ S0)T i ≤ 0(THH(X )ET ∧ ETi)
T i > 0
F i/F i−1 =
{(THH(X )(Σ2iT+))T i ≤ 0(THH(X )ET ∧ Σ2i−1T+)T i > 0
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 9 / 16
Structure and Properties
The Spectral Sequence···
······
······ ···
· · · ← • ← • ← • ← • ← •
· · · ← • ← • ← • ← •
· · · ← • ← • ← •
· · · ← • ← •
· · · ← •···
(1,1) periodic on E1
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
Spectral sequence
E1i,j = πi+jΣ
2iTHH(X ) = THHj−i(X )
Renumber: Double filtration degree
E2r2i,j = (E r
i,i+j)old, d2r = (dr )old
Spectral SequenceConditionally convergent spectral sequence
E22i,j = THHj(X ) =⇒ TP2i+j(X ). (E r
2i+1,j = 0)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 10 / 16
Structure and Properties
The Spectral Sequence···
······
· · · • • • · · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
(2,0) periodic on E2
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
Spectral sequence
E1i,j = πi+jΣ
2iTHH(X ) = THHj−i(X )
Renumber: Double filtration degree
E2r2i,j = (E r
i,i+j)old, d2r = (dr )old
Spectral SequenceConditionally convergent spectral sequence
E22i,j = THHj(X ) =⇒ TP2i+j(X ). (E r
2i+1,j = 0)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 10 / 16
Structure and Properties
The Spectral Sequence···
······
· · · • • • · · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
(2,0) periodic on E2
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
Spectral sequence
E1i,j = πi+jΣ
2iTHH(X ) = THHj−i(X )
Renumber: Double filtration degree
E2r2i,j = (E r
i,i+j)old, d2r = (dr )old
Spectral SequenceConditionally convergent spectral sequence
E22i,j = THHj(X ) =⇒ TP2i+j(X ). (E r
2i+1,j = 0)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 10 / 16
Structure and Properties
The Spectral Sequence···
······
· · · • • • · · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
· · · • •
hh
•
hh
· · ·
(2,0) periodic on E2
Filtration on TP(X ) with associated graded
F i/F i−1 ' Σ2iTHH(X )
Spectral sequence
E1i,j = πi+jΣ
2iTHH(X ) = THHj−i(X )
Renumber: Double filtration degree
E2r2i,j = (E r
i,i+j)old, d2r = (dr )old
Spectral SequenceConditionally convergent spectral sequence
E22i,j = THHj(X ) =⇒ TP2i+j(X ). (E r
2i+1,j = 0)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 10 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map
? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map
? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map
? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map
? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry.
What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal?
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Structure and Properties
Combining the Multiplication and Filtration
Multiplicative spectral sequence:
TP(X ) ∧ TP(Y )→ TP(X ∧ Y ) a filtered map? Coherent model?
In homotopy category, easy obstruction theory cellular approximationto diagonal & multiplication.
What about TP(X ) ∧TP(R) TP(X )→ TP(X ∧R Y )?=⇒ map of spectral sequences
Definitely not with symmetry. What about just associativity?
Coherently homotopy associative cellular approximation to diagonal!
Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 11 / 16
Digression
Approximation of the diagonal for simplicial spaces
Let X• be a simplicial space, |X•| its geometric realization. |X n• | vs |X•|n
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Find an A∞ operad A and a map of operadsA(n)→ Filt(|X•|, |X•|n) ⊂ T (|X•|, |X•|n).
Barycentric Coordinates and Milnor Coordinates on ∆[m]
Barycentric t0, . . . , tm,∑
ti = 1↔ Milnor 0 ≤ u0 ≤ u1 ≤ · · · ≤ um−1 ≤ 1
An element in |X•| is specified by (x ∈ Xm,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 12 / 16
Digression
Approximation of the diagonal for simplicial spaces
Let X• be a simplicial space, |X•| its geometric realization. |X n• | vs |X•|n
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Find an A∞ operad A and a map of operadsA(n)→ Filt(|X•|, |X•|n) ⊂ T (|X•|, |X•|n).
Barycentric Coordinates and Milnor Coordinates on ∆[m]
Barycentric t0, . . . , tm,∑
ti = 1↔ Milnor 0 ≤ u0 ≤ u1 ≤ · · · ≤ um−1 ≤ 1
An element in |X•| is specified by (x ∈ Xm,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 12 / 16
Digression
Approximation of the diagonal for simplicial spaces
Let X• be a simplicial space, |X•| its geometric realization. |X n• | vs |X•|n
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Find an A∞ operad A and a map of operadsA(n)→ Filt(|X•|, |X•|n) ⊂ T (|X•|, |X•|n).
Barycentric Coordinates and Milnor Coordinates on ∆[m]
Barycentric t0, . . . , tm,∑
ti = 1↔ Milnor 0 ≤ u0 ≤ u1 ≤ · · · ≤ um−1 ≤ 1
An element in |X•| is specified by (x ∈ Xm,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 12 / 16
Digression
Approximation of the diagonal for simplicial spaces
Let X• be a simplicial space, |X•| its geometric realization. |X n• | vs |X•|n
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Find an A∞ operad A and a map of operadsA(n)→ Filt(|X•|, |X•|n) ⊂ T (|X•|, |X•|n).
Barycentric Coordinates and Milnor Coordinates on ∆[m]
Barycentric t0, . . . , tm,∑
ti = 1↔ Milnor 0 ≤ u0 ≤ u1 ≤ · · · ≤ um−1 ≤ 1
An element in |X•| is specified by (x ∈ Xm,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 12 / 16
Digression
Approximation of the diagonal for simplicial spaces
Let X• be a simplicial space, |X•| its geometric realization. |X n• | vs |X•|n
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Find an A∞ operad A and a map of operadsA(n)→ Filt(|X•|, |X•|n) ⊂ T (|X•|, |X•|n).
Barycentric Coordinates and Milnor Coordinates on ∆[m]
Barycentric t0, . . . , tm,∑
ti = 1↔ Milnor 0 ≤ u0 ≤ u1 ≤ · · · ≤ um−1 ≤ 1
An element in |X•| is specified by (x ∈ Xm,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 12 / 16
Digression
Approximation of the diagonal for simplicial spaces (ii)
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Solution
1 The overlapping little 1-cubes operad CΞ1
2 The map CΞ1 → T (|X•|, |X•|n).
3 little 1-cubes operad C1 ⊂ CΞ1 .
An element c ∈ CΞ(n) specifies n monotonic PL maps gi : I → I
(x ,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1) 7→((x ,0 ≤ g1(u0) ≤ · · · ≤ g1(um−1) ≤ 1), . . . ,
(x ,0 ≤ gn(u0) ≤ · · · ≤ gn(um−1) ≤ 1))
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 13 / 16
Digression
Approximation of the diagonal for simplicial spaces (ii)
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Solution1 The overlapping little 1-cubes operad CΞ
1
2 The map CΞ1 → T (|X•|, |X•|n).
3 little 1-cubes operad C1 ⊂ CΞ1 .
An element c ∈ CΞ(n) specifies n monotonic PL maps gi : I → I
(x ,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1) 7→((x ,0 ≤ g1(u0) ≤ · · · ≤ g1(um−1) ≤ 1), . . . ,
(x ,0 ≤ gn(u0) ≤ · · · ≤ gn(um−1) ≤ 1))
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 13 / 16
Digression
Approximation of the diagonal for simplicial spaces (ii)
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Solution1 The overlapping little 1-cubes operad CΞ
12 The map CΞ
1 → T (|X•|, |X•|n).
3 little 1-cubes operad C1 ⊂ CΞ1 .
An element c ∈ CΞ(n) specifies n monotonic PL maps gi : I → I
(x ,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1) 7→((x ,0 ≤ g1(u0) ≤ · · · ≤ g1(um−1) ≤ 1), . . . ,
(x ,0 ≤ gn(u0) ≤ · · · ≤ gn(um−1) ≤ 1))
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 13 / 16
Digression
Approximation of the diagonal for simplicial spaces (ii)
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Solution1 The overlapping little 1-cubes operad CΞ
12 The map CΞ
1 → T (|X•|, |X•|n).
3 little 1-cubes operad C1 ⊂ CΞ1 .
An element c ∈ CΞ(n) specifies n monotonic PL maps gi : I → I
(x ,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1) 7→((x ,0 ≤ g1(u0) ≤ · · · ≤ g1(um−1) ≤ 1), . . . ,
(x ,0 ≤ gn(u0) ≤ · · · ≤ gn(um−1) ≤ 1))
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 13 / 16
Digression
Approximation of the diagonal for simplicial spaces (ii)
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Solution1 The overlapping little 1-cubes operad CΞ
12 The map CΞ
1 → T (|X•|, |X•|n).
3 little 1-cubes operad C1 ⊂ CΞ1 .
An element c ∈ CΞ(n) specifies n monotonic PL maps gi : I → I
(x ,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1) 7→((x ,0 ≤ g1(u0) ≤ · · · ≤ g1(um−1) ≤ 1), . . . ,
(x ,0 ≤ gn(u0) ≤ · · · ≤ gn(um−1) ≤ 1))
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 13 / 16
Digression
Approximation of the diagonal for simplicial spaces (ii)
ProblemParametrize a contractible spaces of filtered approximations of thediagonal maps |X•| → |X•|n for all n that compose appropriately.
Solution1 The overlapping little 1-cubes operad CΞ
12 The map CΞ
1 → T (|X•|, |X•|n).3 little 1-cubes operad C1 ⊂ CΞ
1 .
An element c ∈ CΞ(n) specifies n monotonic PL maps gi : I → I
(x ,0 ≤ u0 ≤ · · · ≤ um−1 ≤ 1) 7→((x ,0 ≤ g1(u0) ≤ · · · ≤ g1(um−1) ≤ 1), . . . ,
(x ,0 ≤ gn(u0) ≤ · · · ≤ gn(um−1) ≤ 1))
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 13 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Applied to TP
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Applied to TP
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Applied to TP
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Applied to TP
F−iTP(X ) ∧ F−jTP(X )→ F−i−jTP(X ∧ Y )
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
Little 1-cubes: Moore construction (Moore loop space)
Use length parameter to make fully associative
F−iTP(X ) ∧ R>0+ ∧ F−jTP(Y ) ∧ R>0
+ → F−i−jTP(X ∧ Y ) ∧ R>0+
Filtered monoidal TPM(X ) ∧ TPM(Y )→ TPM(X ∧ Y )
TPM(X ) ∧TPM (R) TPM(Y )→ TPM(X ∧R Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
Little 1-cubes: Moore construction (Moore loop space)
Use length parameter to make fully associative
F−iTP(X ) ∧ R>0+ ∧ F−jTP(Y ) ∧ R>0
+ → F−i−jTP(X ∧ Y ) ∧ R>0+
Filtered monoidal TPM(X ) ∧ TPM(Y )→ TPM(X ∧ Y )
TPM(X ) ∧TPM (R) TPM(Y )→ TPM(X ∧R Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
Little 1-cubes: Moore construction (Moore loop space)
Use length parameter to make fully associative
F−iTP(X ) ∧ R>0+ ∧ F−jTP(Y ) ∧ R>0
+ → F−i−jTP(X ∧ Y ) ∧ R>0+
Filtered monoidal TPM(X ) ∧ TPM(Y )→ TPM(X ∧ Y )
TPM(X ) ∧TPM (R) TPM(Y )→ TPM(X ∧R Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
Little 1-cubes: Moore construction (Moore loop space)
Use length parameter to make fully associative
F−iTP(X ) ∧ R>0+ ∧ F−jTP(Y ) ∧ R>0
+ → F−i−jTP(X ∧ Y ) ∧ R>0+
Filtered monoidal TPM(X ) ∧ TPM(Y )→ TPM(X ∧ Y )
TPM(X ) ∧TPM (R) TPM(Y )→ TPM(X ∧R Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
Little 1-cubes: Moore construction (Moore loop space)
Use length parameter to make fully associative
F−iTP(X ) ∧ R>0+ ∧ F−jTP(Y ) ∧ R>0
+ → F−i−jTP(X ∧ Y ) ∧ R>0+
Filtered monoidal TPM(X ) ∧ TPM(Y )→ TPM(X ∧ Y )
TPM(X ) ∧TPM (R) TPM(Y )→ TPM(X ∧R Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
Little 1-cubes: Moore construction (Moore loop space)
Use length parameter to make fully associative
F−iTP(X ) ∧ R>0+ ∧ F−jTP(Y ) ∧ R>0
+ → F−i−jTP(X ∧ Y ) ∧ R>0+
Filtered monoidal TPM(X ) ∧ TPM(Y )→ TPM(X ∧ Y )
TPM(X ) ∧TPM (R) TPM(Y )→ TPM(X ∧R Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Digression
Filtered Monoidal Structure
A filtered approximation of the diagonal gives a mapETi+j−1 → (ETi−1 × ET) ∪ (ET× ETj−1) ⊂ ET× ET
Hence a map (ET,ETi+j−1)→ (ET,ETi−1)× (ET,ETj−1)
Parametrized
C1(n)+ ∧ F−i1TP(X1) ∧ · · · ∧ F−in TP(Xn)→ F−i1−···−in TP(X1 ∧ · · · ∧ Xn)
Little 1-cubes: Moore construction (Moore loop space)
Use length parameter to make fully associative
F−iTP(X ) ∧ R>0+ ∧ F−jTP(Y ) ∧ R>0
+ → F−i−jTP(X ∧ Y ) ∧ R>0+
Filtered monoidal TPM(X ) ∧ TPM(Y )→ TPM(X ∧ Y )
TPM(X ) ∧TPM (R) TPM(Y )→ TPM(X ∧R Y )
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 14 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences
preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences
preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences
preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences
preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences
preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences
preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
Also map of filtered TP(R)-modules
=⇒ map of spectral sequences preserving periodicity op. on E2
Righthand spectral sequence is Tate spectral sequence for
THH(X ∧R Y ) ∼= THH(X ) ∧THH(R) THH(Y )
E2 periodic with π∗(THH(X ) ∧THH(R) THH(Y )) in each even column
Lefthand spectral sequence has (renumbered) E2-term
π∗Gr(TP(X ) ∧TP(R) TP(Y )) ∼= π∗(Gr TP(X ) ∧Gr TP(R) Gr TP(Y ))
E2-term is π∗Gr TP(R)-module =⇒ (2,0)-periodic
Proposition
Map of spectral sequences is an isomorphism on E2
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 15 / 16
Künneth Theorem
Outline of Proof of Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
induces isomorphism of E2-terms of spectral sequences
RHSS: Tate spectral sequence =⇒ conditionally convergent.
TheoremIf R = Hk and X and Y are smooth and proper, then the LHSS isconditionally convergent.
In fact, strongly convergent.
TheoremIf R is an E∞ ring spectrum and A is a smooth and proper R-algebra,then THH(A) is a compact THH(R)-module.
TP∗(k) = Wk [v , v−1] is periodicπ∗(F 0TP(k)) = Wk [v ,pv−1] has finite global dimension
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 16 / 16
Künneth Theorem
Outline of Proof of Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
induces isomorphism of E2-terms of spectral sequences
RHSS: Tate spectral sequence =⇒ conditionally convergent.
TheoremIf R = Hk and X and Y are smooth and proper, then the LHSS isconditionally convergent.
In fact, strongly convergent.
TheoremIf R is an E∞ ring spectrum and A is a smooth and proper R-algebra,then THH(A) is a compact THH(R)-module.
TP∗(k) = Wk [v , v−1] is periodicπ∗(F 0TP(k)) = Wk [v ,pv−1] has finite global dimension
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 16 / 16
Künneth Theorem
Outline of Proof of Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
induces isomorphism of E2-terms of spectral sequences
RHSS: Tate spectral sequence =⇒ conditionally convergent.
TheoremIf R = Hk and X and Y are smooth and proper, then the LHSS isconditionally convergent.
In fact, strongly convergent.
TheoremIf R is an E∞ ring spectrum and A is a smooth and proper R-algebra,then THH(A) is a compact THH(R)-module.
TP∗(k) = Wk [v , v−1] is periodicπ∗(F 0TP(k)) = Wk [v ,pv−1] has finite global dimension
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 16 / 16
Künneth Theorem
Outline of Proof of Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
induces isomorphism of E2-terms of spectral sequences
RHSS: Tate spectral sequence =⇒ conditionally convergent.
TheoremIf R = Hk and X and Y are smooth and proper, then the LHSS isconditionally convergent.
In fact, strongly convergent.
TheoremIf R is an E∞ ring spectrum and A is a smooth and proper R-algebra,then THH(A) is a compact THH(R)-module.
TP∗(k) = Wk [v , v−1] is periodicπ∗(F 0TP(k)) = Wk [v ,pv−1] has finite global dimension
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 16 / 16
Künneth Theorem
Outline of Proof of Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
induces isomorphism of E2-terms of spectral sequences
RHSS: Tate spectral sequence =⇒ conditionally convergent.
TheoremIf R = Hk and X and Y are smooth and proper, then the LHSS isconditionally convergent. In fact, strongly convergent.
TheoremIf R is an E∞ ring spectrum and A is a smooth and proper R-algebra,then THH(A) is a compact THH(R)-module.
TP∗(k) = Wk [v , v−1] is periodicπ∗(F 0TP(k)) = Wk [v ,pv−1] has finite global dimension
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 16 / 16
Künneth Theorem
Outline of Proof of Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
induces isomorphism of E2-terms of spectral sequences
RHSS: Tate spectral sequence =⇒ conditionally convergent.
TheoremIf R = Hk and X and Y are smooth and proper, then the LHSS isconditionally convergent. In fact, strongly convergent.
TheoremIf R is an E∞ ring spectrum and A is a smooth and proper R-algebra,then THH(A) is a compact THH(R)-module.
TP∗(k) = Wk [v , v−1] is periodic
π∗(F 0TP(k)) = Wk [v ,pv−1] has finite global dimension
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 16 / 16
Künneth Theorem
Outline of Proof of Künneth Theorem
Filtered map TP(X ) ∧TP(R) TP(Y )→ TP(X ∧R Y )
induces isomorphism of E2-terms of spectral sequences
RHSS: Tate spectral sequence =⇒ conditionally convergent.
TheoremIf R = Hk and X and Y are smooth and proper, then the LHSS isconditionally convergent. In fact, strongly convergent.
TheoremIf R is an E∞ ring spectrum and A is a smooth and proper R-algebra,then THH(A) is a compact THH(R)-module.
TP∗(k) = Wk [v , v−1] is periodicπ∗(F 0TP(k)) = Wk [v ,pv−1] has finite global dimension
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 16 / 16
Questions and Answers
M.A.Mandell (IU) Topological Periodic Cyclic Homology Mar 2017 17 / 17
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![THE HOMOLOGY OF CYCLIC PRODUCTS · 2018. 11. 16. · the homology modulo p of p-fold cyclic products was determined by Richard-son and Smith [18]. More recently, Nakaoka [16] determined](https://img.pdfslide.net/doc/110x75/6015673b32b6a72436789439/the-homology-of-cyclic-2018-11-16-the-homology-modulo-p-of-p-fold-cyclic-products.jpg)
![arXiv:1707.01799v2 [math.AT] 7 Sep 2018 · Abstract. Topological cyclic homology is a refinementof Connes-Tsygan’s cyclic homology which was introduced by B¨okstedt–Hsiang–Madsen](https://img.pdfslide.net/doc/110x75/6015682f85fbeb6c746f0c8d/arxiv170701799v2-mathat-7-sep-2018-abstract-topological-cyclic-homology-is.jpg)
