A Strong Künneth Theorem for Topological PeriodicCyclic Homology

Michael A. Mandell

Indiana University

Workshop on K -Theory and Related Fields

Hausdorff Research Institute for Mathematics

June 27, 2017

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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 arXiv:1706.06846

Outline1 Non-commutative derived algebraic geometry2 Introduction to TP3 The Künneth theorem

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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. 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 arXiv:1706.06846

Outline1 Non-commutative derived algebraic geometry2 Introduction to TP3 The Künneth theorem

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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. 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 arXiv:1706.06846

Outline1 Non-commutative derived algebraic geometry2 Introduction to TP3 The Künneth theorem

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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. 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 arXiv:1706.06846

Outline1 Non-commutative derived algebraic geometry2 Introduction to TP3 The Künneth theorem

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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. 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 arXiv:1706.06846

Outline1 Non-commutative derived algebraic geometry2 Introduction to TP3 The Künneth theorem

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 2 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

Non-commutative derived algebraic geometry

Basic objects: [small] differential graded (or spectral) categoriesEquivalences: Morita equivalences

Example: A ∼−→ModA∼−→ModModA

Example: Tilting AMB, BNA with M ⊗LN N ' A, N ⊗L

B M ' B

A ∼−→ModA∼−→ModB

∼←− B

Goal: Construct/study invariants

Example: K theory of subcategory of compact objects

Example: algebraic variety X ↔ d.g. cat of perfect complexes Ddgperf(X )

K (X ) = K (Ddgperf(X ))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 3 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

How is this algebraic geometry?

Use D = Ddgperf(X ) for X

For reasonable X , [some] properties of X equivalent to properties of D

Example: X is proper over spec k if and only if D(a,b) is a compactd.g. k -module for all a,b ∈ D . (

∑dim Hn(D(a,b)) <∞)

Example: X is smooth over spec k if and only if D is a compactDop ⊗k D-module. (RHomDop⊗k D(D ,−) commutes with

⊕)

DefinitionLet A be a d.g. (or spectral) R-algebra. Then A is:

proper if it is compact as an R-modulesmooth if it is compact as an Aop ⊗L

R A-module(or Aop ∧L

R A-module)

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 4 / 16

Non-commutative derived algebraic geometry

Take away

For theorems in non-commutative derived algebraic geometry:Statements are in terms of d.g. (or spectral) categoriesResults are about algebraic varieties (and generalizations)Proofs often just need the case of d.g. algebras (or ring spectra)

Example:

X = Pm

End

(m⊕

r=0

O(−r)

)∼−→ Ddg

perf(Pn)

H−n End() is a matrix of Extn groups, Extn(O(−j),O(−i))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 5 / 16

Non-commutative derived algebraic geometry

Take away

For theorems in non-commutative derived algebraic geometry:Statements are in terms of d.g. (or spectral) categoriesResults are about algebraic varieties (and generalizations)Proofs often just need the case of d.g. algebras (or ring spectra)

Example:

X = Pm

End

(m⊕

r=0

O(−r)

)∼−→ Ddg

perf(Pn)

H−n End() is a matrix of Extn groups, Extn(O(−j),O(−i))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 5 / 16

Non-commutative derived algebraic geometry

Take away

For theorems in non-commutative derived algebraic geometry:Statements are in terms of d.g. (or spectral) categoriesResults are about algebraic varieties (and generalizations)Proofs often just need the case of d.g. algebras (or ring spectra)

Example:

X = Pm

End

(m⊕

r=0

O(−r)

)∼−→ Ddg

perf(Pn)

H−n End() is a matrix of Extn groups, Extn(O(−j),O(−i))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 5 / 16

Non-commutative derived algebraic geometry

Take away

For theorems in non-commutative derived algebraic geometry:Statements are in terms of d.g. (or spectral) categoriesResults are about algebraic varieties (and generalizations)Proofs often just need the case of d.g. algebras (or ring spectra)

Example:

X = Pm

End

(m⊕

r=0

O(−r)

)∼−→ Ddg

perf(Pn)

H−n End() is a matrix of Extn groups, Extn(O(−j),O(−i))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 5 / 16

Non-commutative derived algebraic geometry

Take away

For theorems in non-commutative derived algebraic geometry:Statements are in terms of d.g. (or spectral) categoriesResults are about algebraic varieties (and generalizations)Proofs often just need the case of d.g. algebras (or ring spectra)

Example:

X = Pm End

(m⊕

r=0

O(−r)

)∼−→ Ddg

perf(Pn)

H−n End() is a matrix of Extn groups, Extn(O(−j),O(−i))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 5 / 16

Non-commutative derived algebraic geometry

Take away

For theorems in non-commutative derived algebraic geometry:Statements are in terms of d.g. (or spectral) categoriesResults are about algebraic varieties (and generalizations)Proofs often just need the case of d.g. algebras (or ring spectra)

Example:

X = Pm End

(m⊕

r=0

O(−r)

)∼−→ Ddg

perf(Pn)

H−n End() is a matrix of Extn groups, Extn(O(−j),O(−i))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 5 / 16

Non-commutative derived algebraic geometry

Take away

For theorems in non-commutative derived algebraic geometry:Statements are in terms of d.g. (or spectral) categoriesResults are about algebraic varieties (and generalizations)Proofs often just need the case of d.g. algebras (or ring spectra)

Example:

X = Pm End

(m⊕

r=0

O(−r)

)∼−→ Ddg

perf(Pn)

H−n End() is a matrix of Extn groups, Extn(O(−j),O(−i))

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 5 / 16

Introduction to TP

Hochschild Homology

and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology

and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology

and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Morita Invariance

Tilting situation AMB, BNA

Dennis-Waldhausen Argument

A⊗ · · · ⊗ A⊗ ⊗

N⊗

M⊗

B ⊗ · · · ⊗ B

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology

and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Morita Invariance

Tilting situation AMB, BNA

Dennis-Waldhausen Argument

A⊗ · · · ⊗ A⊗ ⊗

N⊗

M⊗

B ⊗ · · · ⊗ B

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology

and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Morita Invariance

Tilting situation AMB, BNA

Dennis-Waldhausen Argument

A⊗ · · · ⊗ A⊗ ⊗

N⊗

M⊗

B ⊗ · · · ⊗ B

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology

and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Morita Invariance

Tilting situation AMB, BNA

Dennis-Waldhausen Argument

A⊗ · · · ⊗ A⊗ ⊗

N⊗

M⊗

B ⊗ · · · ⊗ B

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology

and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Construct Double Complex:

···

···

······

···↓

↓

↓ ↓ ↓ ···· · · ← • ←

•

← • ← • ← •↓

↓

↓ ↓· · · ← • ←

•

← • ← •↓

↓

↓· · · ← • ←

•

← •↓

↓

· · · ← • ←

•

↓· · · ← •

···

HNHP

HH

HC

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Construct Double Complex:

···

······

······

↓

↓ ↓ ↓ ↓ ···

· · · ← • ←

• ← • ← • ← •

↓

↓ ↓ ↓

· · · ← • ←

• ← • ← •

↓

↓ ↓

· · · ← • ←

• ← •

↓

↓

· · · ← • ←

•

↓· · · ← •

···

HNHPHH

HC

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Construct Double Complex:···

···

······

···

↓ ↓

↓ ↓ ↓ ···

· · · ← • ← •

← • ← • ← •

↓ ↓

↓ ↓

· · · ← • ← •

← • ← •

↓ ↓

↓

· · · ← • ← •

← •

↓ ↓· · · ← • ← •

↓· · · ← •

···

HN

HPHHHC

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Hochschild Homology and Cyclic Homology

Cyclic bar construction

Ncyq A = A⊗ · · · ⊗ A︸ ︷︷ ︸

q factors

⊗A

A⊗ · · · ⊗ A

⊗ ⊗

A

Chain complex

Construct Double Complex:···

······

······

↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •

↓ ↓ ↓ ↓· · · ← • ← • ← • ← •

↓ ↓ ↓· · · ← • ← • ← •

↓ ↓· · · ← • ← •

↓· · · ← •

···

HN

HP

HHHC

Cyclic structure =⇒ Connes’ B operator B : NcyA→ NcyA[−1]

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 6 / 16

Introduction to TP

Topological Hochschild Homology

Cyclic bar construction (Bökstedt)

Ncyq A = A ∧ · · · ∧ A︸ ︷︷ ︸

q factors

∧A

A ∧ · · · ∧ A

∧ ∧A

Spectrum

Construction

···

···

······

···↓

↓

↓ ↓ ↓ ···· · · ← • ←

•

← • ← • ← •↓

↓

↓ ↓· · · ← • ←

•

← • ← •↓

↓

↓· · · ← • ←

•

← •↓

↓

· · · ← • ←

•

↓· · · ← •

···

HH corresponds to THH

Cyclic structure =⇒ circle group action

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 7 / 16

Introduction to TP

Topological Hochschild Homology

Cyclic bar construction (Bökstedt)

Ncyq A = A ∧ · · · ∧ A︸ ︷︷ ︸

q factors

∧A

A ∧ · · · ∧ A

∧ ∧A

Spectrum

Construction

···

···

······

···↓

↓

↓ ↓ ↓ ···· · · ← • ←

•

← • ← • ← •↓

↓

↓ ↓· · · ← • ←

•

← • ← •↓

↓

↓· · · ← • ←

•

← •↓

↓

· · · ← • ←

•

↓· · · ← •

···

HH corresponds to THH

Cyclic structure =⇒ circle group action

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 7 / 16

Introduction to TP

Topological Hochschild Homology

Cyclic bar construction (Bökstedt)

Ncyq A = A ∧ · · · ∧ A︸ ︷︷ ︸

q factors

∧A

A ∧ · · · ∧ A

∧ ∧A

Spectrum

Construction

···

···

······

···↓

↓

↓ ↓ ↓ ···· · · ← • ←

•

← • ← • ← •↓

↓

↓ ↓· · · ← • ←

•

← • ← •↓

↓

↓· · · ← • ←

•

← •↓

↓

· · · ← • ←

•

↓· · · ← •

···

HH corresponds to THH

Cyclic structure =⇒ circle group action

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 7 / 16

Introduction to TP

Topological Hochschild Homology

Cyclic bar construction (Bökstedt)

Ncyq A = A ∧ · · · ∧ A︸ ︷︷ ︸

q factors

∧A

A ∧ · · · ∧ A

∧ ∧A

Spectrum

Construction

···

···

······

···↓

↓

↓ ↓ ↓ ···· · · ← • ←

•

← • ← • ← •↓

↓

↓ ↓· · · ← • ←

•

← • ← •↓

↓

↓· · · ← • ←

•

← •↓

↓

· · · ← • ←

•

↓· · · ← •

···

HH corresponds to THH

Cyclic structure =⇒ circle group action

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 7 / 16

Introduction to TP

Topological Hochschild Homology

Cyclic bar construction (Bökstedt)

Ncyq A = A ∧ · · · ∧ A︸ ︷︷ ︸

q factors

∧A

A ∧ · · · ∧ A

∧ ∧A

Spectrum

Construction

···

······

······

↓

↓ ↓ ↓ ↓ ···

· · · ← • ←

• ← • ← • ← •

↓

↓ ↓ ↓

· · · ← • ←

• ← • ← •

↓

↓ ↓

· · · ← • ←

• ← •

↓

↓

· · · ← • ←

•

↓· · · ← •

···

HH corresponds to THHHC corresponds to THHhT

Cyclic structure =⇒ circle group action

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 7 / 16

Introduction to TP

Topological Hochschild Homology

Cyclic bar construction (Bökstedt)

Ncyq A = A ∧ · · · ∧ A︸ ︷︷ ︸

q factors

∧A

A ∧ · · · ∧ A

∧ ∧A

Spectrum

Construction···

···

······

···

↓ ↓

↓ ↓ ↓ ···

· · · ← • ← •

← • ← • ← •

↓ ↓

↓ ↓

· · · ← • ← •

← • ← •

↓ ↓

↓

· · · ← • ← •

← •

↓ ↓· · · ← • ← •

↓· · · ← •

···

HH corresponds to THHHN corresponds to THHhT

Cyclic structure =⇒ circle group action

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 7 / 16

Introduction to TP

Topological Hochschild Homology

Cyclic bar construction (Bökstedt)

Ncyq A = A ∧ · · · ∧ A︸ ︷︷ ︸

q factors

∧A

A ∧ · · · ∧ A

∧ ∧A

Spectrum

Construction···

······

······

↓ ↓ ↓ ↓ ↓ ···· · · ← • ← • ← • ← • ← •

↓ ↓ ↓ ↓· · · ← • ← • ← • ← •

↓ ↓ ↓· · · ← • ← • ← •

↓ ↓· · · ← • ← •

↓· · · ← •

···

HH corresponds to THHHP corresponds to THH tT

Cyclic structure =⇒ circle group action

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 7 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Introduction to TP

Topological Periodic Cyclic Homology

DefinitionFor a ring spectrum A, define the Topological Periodic CyclicHomology of A by TP(A) = THH(A)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 Jun 2017 8 / 16

Künneth Theorem

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.

Corollary

There are short exact sequences of graded Wk-modules

0→ (TP∗(X )⊗TP∗(k)TP∗(Y ))n → TPn(X⊗k Y )→ TorTP∗(k)1,n−1 (TP∗(X ),TP∗(Y ))→ 0

for all n, which split but not naturally.

Corollary

TP∗(X )[1/p]⊗TP∗(k)[1/p] TP∗(Y )[1/p]→ TP∗(X ⊗k Y )[1/p] is an isomorphism.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 9 / 16

Künneth Theorem

Künneth Theorem

TheoremLet k be finite field. The lax symmetric monoidal functor

TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗k Y)

is an isomorphism when X and Y are smooth and proper over k.

Corollary

There are short exact sequences of graded Wk-modules

0→ (TP∗(X )⊗TP∗(k)TP∗(Y ))n → TPn(X⊗k Y )→ TorTP∗(k)1,n−1 (TP∗(X ),TP∗(Y ))→ 0

for all n, which split but not naturally.

Corollary

TP∗(X )[1/p]⊗TP∗(k)[1/p] TP∗(Y )[1/p]→ TP∗(X ⊗k Y )[1/p] is an isomorphism.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 9 / 16

Künneth Theorem

Künneth Theorem

TheoremLet k be finite field. The lax symmetric monoidal functor

TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗k Y)

is an isomorphism when X and Y are smooth and proper over k.

Corollary

There are short exact sequences of graded Wk-modules

0→ (TP∗(X )⊗TP∗(k)TP∗(Y ))n → TPn(X⊗k Y )→ TorTP∗(k)1,n−1 (TP∗(X ),TP∗(Y ))→ 0

for all n, which split but not naturally.

Corollary

TP∗(X )[1/p]⊗TP∗(k)[1/p] TP∗(Y )[1/p]→ TP∗(X ⊗k Y )[1/p] is an isomorphism.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 9 / 16

Künneth Theorem

Künneth Theorem

TheoremLet k be finite field. The lax symmetric monoidal functor

TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗k Y)

is an isomorphism when X and Y are smooth and proper over k.

Corollary

There are short exact sequences of graded Wk-modules

0→ (TP∗(X )⊗TP∗(k)TP∗(Y ))n → TPn(X⊗k Y )→ TorTP∗(k)1,n−1 (TP∗(X ),TP∗(Y ))→ 0

for all n, which split but not naturally.

Corollary

TP∗(X )[1/p]⊗TP∗(k)[1/p] TP∗(Y )[1/p]→ TP∗(X ⊗k Y )[1/p] is an isomorphism.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 9 / 16

Künneth Theorem

Künneth Theorem

TheoremLet k be finite field. The lax symmetric monoidal functor

TP(X ) ∧LTP(k ) TP(Y )→ TP(X ⊗k Y)

is an isomorphism when X and Y are smooth and proper over k.

Corollary

There are short exact sequences of graded Wk-modules

0→ (TP∗(X )⊗TP∗(k)TP∗(Y ))n → TPn(X⊗k Y )→ TorTP∗(k)1,n−1 (TP∗(X ),TP∗(Y ))→ 0

for all n, which split but not naturally.

Corollary

TP∗(X )[1/p]⊗TP∗(k)[1/p] TP∗(Y )[1/p]→ TP∗(X ⊗k Y )[1/p] is an isomorphism.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 9 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 10 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 11 / 16

Künneth Theorem

The Multiplication

TP(X ) ∧ TP(Y )→ TP(X ∧ Y )

TP(X ) = (THH(X )ET ∧ ET)T

ET ∧ ET ' ET ←− This can be made coherentUse 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 Jun 2017 11 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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 Jun 2017 12 / 16

Künneth Theorem

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

Greenlees Tate 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 Jun 2017 13 / 16

Künneth Theorem

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

Greenlees Tate 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 Jun 2017 13 / 16

Künneth Theorem

The Spectral Sequence···

······

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(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

Greenlees Tate 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 Jun 2017 13 / 16

Künneth Theorem

The Spectral Sequence···

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(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

Greenlees Tate 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 Jun 2017 13 / 16

Künneth Theorem

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

Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 14 / 16

Künneth Theorem

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

Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 14 / 16

Künneth Theorem

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

Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 14 / 16

Künneth Theorem

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

Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 14 / 16

Künneth Theorem

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

Multiplication FiltrationDiagonal ET→ ET× ET Simplicial/cellular filt. on ETMult. ET ∧ ET ' ET Filtration on ET

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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 Jun 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 Jun 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 Jun 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 Jun 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 Jun 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 Jun 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 Jun 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 Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 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, k a perfect field of characteristic p > 0, and X and Y aresmooth and proper over k, then the LHSS is conditionally convergent.

Where do we use hypotheses?X smooth and proper =⇒ THH(X ) compact THH(R)-module.TP∗(k) = Wk [v , v−1] finite global dimension.THH(X ) compact =⇒ TP(X ) compact.Equivariantly, Hk is a compact THH(Hk)-module.

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 16 / 16

Questions and Answers

M.A.Mandell (IU) Topological Periodic Cyclic Homology Jun 2017 17 / 17