April2nd,2020 TommyLundemo OntherelationshipbetweenlogarithmicTAQandlogarithmicTHH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

On the relationship between logarithmic TAQ and logarithmic THH

Tommy Lundemo

Radboud University Nijmegen

April 2nd, 2020

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

1 THHandTAQ

2 Formally étale maps

3 Logarithmic ring spectra

4 LogTHH and logTAQ

5 Formally log étale maps

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Let(M,,1) be a cocomplete symmetric monoidal category, and letP →M be a map of commutative monoids in M.

Definition

Thecyclic bar constructionB_P^cy(M)• is the following simplicial commutative monoid inM: theq-simplices are the(1+q)-fold coproductMP · · ·P M inCM_P/. Face maps

d_i(m₀, . . . ,m_q) =

((m0, . . . ,m_imi+1, . . . ,m_q), if 0≤i <q, (m_qm₀,· · ·,mq−1), ifi =q

and degeneraciess_j(m0, . . . ,m_q) = (m0, . . . ,m_j−1,1,m_j, . . . ,m_q).

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Let(M,,1) be a cocomplete symmetric monoidal category, and letP →M be a map of commutative monoids in M.

Definition

Thecyclic bar constructionB_P^cy(M)• is the following simplicial commutative monoid inM: theq-simplices are the(1+q)-fold coproductMP · · ·P M inCM_P/. Face maps

d_i(m₀, . . . ,m_q) =

((m0, . . . ,m_imi+1, . . . ,m_q), if 0≤i <q, (m_qm₀,· · ·,mq−1), ifi =q

and degeneraciess_j(m0, . . . ,m_q) = (m0, . . . ,m_j−1,1,m_j, . . . ,m_q).

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Let(M,,1) be a cocomplete symmetric monoidal category, and letP →M be a map of commutative monoids in M.

Definition

Thecyclic bar constructionB_P^cy(M)• is the following simplicial commutative monoid inM:

theq-simplices are the(1+q)-fold coproductMP · · ·P M inCM_P/. Face maps

d_i(m₀, . . . ,m_q) =

((m0, . . . ,m_imi+1, . . . ,m_q), if 0≤i <q, (m_qm₀,· · ·,mq−1), ifi =q

and degeneraciess_j(m0, . . . ,m_q) = (m0, . . . ,m_j−1,1,m_j, . . . ,m_q).

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Let(M,,1) be a cocomplete symmetric monoidal category, and letP →M be a map of commutative monoids in M.

Definition

Thecyclic bar constructionB_P^cy(M)• is the following simplicial commutative monoid inM: theq-simplices are the(1+q)-fold coproductMP · · ·P M inCM_P/.

Face maps

d_i(m₀, . . . ,m_q) =

((m0, . . . ,m_imi+1, . . . ,m_q), if 0≤i <q, (m_qm₀,· · ·,mq−1), ifi =q

and degeneraciess_j(m0, . . . ,m_q) = (m0, . . . ,m_j−1,1,m_j, . . . ,m_q).

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Let(M,,1) be a cocomplete symmetric monoidal category, and letP →M be a map of commutative monoids in M.

Definition

Thecyclic bar constructionB_P^cy(M)• is the following simplicial commutative monoid inM: theq-simplices are the(1+q)-fold coproductMP · · ·P M inCM_P/. Face maps

d_i(m₀, . . . ,m_q) =

((m0, . . . ,m_imi+1, . . . ,m_q), if 0≤i <q, (m_qm₀,· · ·,mq−1), ifi =q

and degeneraciess_j(m0, . . . ,m_q) = (m0, . . . ,m_j−1,1,m_j, . . . ,m_q).

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Let(M,,1) be a cocomplete symmetric monoidal category, and letP →M be a map of commutative monoids in M.

Definition

Thecyclic bar constructionB_P^cy(M)• is the following simplicial commutative monoid inM: theq-simplices are the(1+q)-fold coproductMP · · ·P M inCM_P/. Face maps

d_i(m₀, . . . ,m_q) =

((m0, . . . ,m_imi+1, . . . ,m_q), if 0≤i <q, (m_qm₀,· · ·,mq−1), ifi =q

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Lemma

B_P^cy(M)• ∼=P B₁^cy(P)•B₁^cy(M)• =P B^cy(P)•B^cy(M)•.

Definition

LetX• be a finite simplicial set.

Let X•⊗_PM be the simplicial commutative monoid [q]7→M^P|Xq|=MP· · ·PM, the |X_q|-fold coproduct. AssumeX• is pointed and let M →N→M be an object of CM_M//M. LetX•_M N=colim(M ←−N −→X•⊗_MN). Lemma

FixP →M. Then B_P^cy(M)•∼=S_•¹⊗_PM ∼=S_•¹_M(M P M).

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Lemma

B_P^cy(M)• ∼=P B₁^cy(P)•B₁^cy(M)• =P B^cy(P)•B^cy(M)•.

Definition

LetX• be a finite simplicial set.

Let X•⊗_PM be the simplicial commutative monoid [q]7→M^P|Xq|=MP· · ·PM, the |X_q|-fold coproduct.

AssumeX• is pointed and let M →N→M be an object of CM_M//M. LetX•_M N=colim(M ←−N−→X•⊗_MN).

Lemma

FixP →M. Then B_P^cy(M)•∼=S_•¹⊗_PM ∼=S_•¹_M(M P M).

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Lemma

B_P^cy(M)• ∼=P B₁^cy(P)•B₁^cy(M)• =P B^cy(P)•B^cy(M)•.

Definition

LetX• be a finite simplicial set.

Let X•⊗_PM be the simplicial commutative monoid [q]7→M^P|Xq|=MP· · ·PM, the |X_q|-fold coproduct.

AssumeX• is pointed and let M →N→M be an object of CM_M//M. LetX•_M N=colim(M ←−N−→X•⊗_MN).

Lemma

FixP →M. Then B_P^cy(M)•∼=S_•¹⊗_P M

∼=S_•¹_M(M P M).

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

The cyclic bar construction

Lemma

B_P^cy(M)• ∼=P B₁^cy(P)•B₁^cy(M)• =P B^cy(P)•B^cy(M)•.

Definition

LetX• be a finite simplicial set.

Let X•⊗_PM be the simplicial commutative monoid [q]7→M^P|Xq|=MP· · ·PM, the |X_q|-fold coproduct.

AssumeX• is pointed and let M →N→M be an object of CM_M//M. LetX•_M N=colim(M ←−N−→X•⊗_MN).

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological Hochschild homology

Definition

LetR→A be a map of commutative (symmetric) ring spectra. DefineTHH^R(A) =|B_R^cy(A)•|, where the cyclic bar construction is taken in(Sp^Σ,∧,S).

Corollary

THH^R(A)∼=R∧_THH(R)THH(A)∼=S¹⊗_RA∼=S¹_A(A∧_RA). The tensors now participate in simplicial model structures. In particular,S¹_A(A∧_R A) models the suspension ofA∧_R Ain CSp^Σ_A//A.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological Hochschild homology

Definition

LetR→A be a map of commutative (symmetric) ring spectra.

DefineTHH^R(A) =|B_R^cy(A)•|, where the cyclic bar construction is taken in(Sp^Σ,∧,S).

Corollary

THH^R(A)∼=R∧_THH(R)THH(A)∼=S¹⊗_RA∼=S¹_A(A∧_RA). The tensors now participate in simplicial model structures. In particular,S¹_A(A∧_R A) models the suspension ofA∧_R Ain CSp^Σ_A//A.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological Hochschild homology

Definition

LetR→A be a map of commutative (symmetric) ring spectra.

DefineTHH^R(A) =|B_R^cy(A)•|, where the cyclic bar construction is taken in(Sp^Σ,∧,S).

Corollary

THH^R(A)∼=R∧_THH(R)THH(A)∼=S¹⊗_RA∼=S¹_A(A∧_RA). The tensors now participate in simplicial model structures. In particular,S¹_A(A∧_R A) models the suspension ofA∧_R Ain CSp^Σ_A//A.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological Hochschild homology

Definition

LetR→A be a map of commutative (symmetric) ring spectra.

DefineTHH^R(A) =|B_R^cy(A)•|, where the cyclic bar construction is taken in(Sp^Σ,∧,S).

Corollary

THH^R(A)∼=R∧_THH(R)THH(A)∼=S¹⊗_RA

∼=S¹_A(A∧_RA). The tensors now participate in simplicial model structures. In particular,S¹_A(A∧_R A) models the suspension ofA∧_R Ain CSp^Σ_A//A.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological Hochschild homology

Definition

LetR→A be a map of commutative (symmetric) ring spectra.

DefineTHH^R(A) =|B_R^cy(A)•|, where the cyclic bar construction is taken in(Sp^Σ,∧,S).

Corollary

THH^R(A)∼=R∧_THH(R)THH(A)∼=S¹⊗_RA∼=S¹_A(A∧_RA).

The tensors now participate in simplicial model structures. In particular,S¹_A(A∧_R A) models the suspension ofA∧_R Ain CSp^Σ_A//A.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological Hochschild homology

Definition

LetR→A be a map of commutative (symmetric) ring spectra.

DefineTHH^R(A) =|B_R^cy(A)•|, where the cyclic bar construction is taken in(Sp^Σ,∧,S).

Corollary

THH^R(A)∼=R∧_THH(R)THH(A)∼=S¹⊗_RA∼=S¹_A(A∧_RA).

The tensors now participate in simplicial model structures.

In particular,S¹_A(A∧_R A) models the suspension ofA∧_R Ain CSp^Σ_A//A.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological Hochschild homology

Definition

LetR→A be a map of commutative (symmetric) ring spectra.

DefineTHH^R(A) =|B_R^cy(A)•|, where the cyclic bar construction is taken in(Sp^Σ,∧,S).

Corollary

THH^R(A)∼=R∧_THH(R)THH(A)∼=S¹⊗_RA∼=S¹_A(A∧_RA).

The tensors now participate in simplicial model structures. In particular,S¹_A(A∧_R A) models the suspension ofA∧_R Ain CSp^Σ_A//A.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological André–Quillen homology

LetR→A be a map of discrete commutative rings. There is an A-moduleΩ¹_A|R such that

Mod_A(Ω¹_A|R,M)∼=Der_R(A,M) =CRing_R//A(A,A⊕M).

Quillen: Ω¹_A|R ∼=IA(A⊗_RA)/(−)² is the abelianization of A. Theorem (Basterra–Mandell)

There are Quillen adjunctions

Mod_A ^QA Nuca_A CSp^Σ_A//A

IA

which become Quillen equivalences after stabilization.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological André–Quillen homology

LetR→A be a map of discrete commutative rings. There is an A-moduleΩ¹_A|R such that

Mod_A(Ω¹_A|R,M)∼=Der_R(A,M) =CRing_R//A(A,A⊕M).

Quillen: Ω¹_A|R ∼=IA(A⊗_RA)/(−)² is the abelianization of A.

Theorem (Basterra–Mandell) There are Quillen adjunctions

Mod_A ^QA Nuca_A CSp^Σ_A//A

IA

which become Quillen equivalences after stabilization.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological André–Quillen homology

LetR→A be a map of discrete commutative rings. There is an A-moduleΩ¹_A|R such that

Mod_A(Ω¹_A|R,M)∼=Der_R(A,M) =CRing_R//A(A,A⊕M).

Quillen: Ω¹_A|R ∼=IA(A⊗_RA)/(−)² is the abelianization of A.

Theorem (Basterra–Mandell) There are Quillen adjunctions

Mod_A ^QA Nuca_A CSp^Σ_A//A

which become Quillen equivalences after stabilization.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological André–Quillen homology

LetR→A be a map of discrete commutative rings. There is an A-moduleΩ¹_A|R such that

Mod_A(Ω¹_A|R,M)∼=Der_R(A,M) =CRing_R//A(A,A⊕M).

Quillen: Ω¹_A|R ∼=IA(A⊗_RA)/(−)² is the abelianization of A.

Theorem (Basterra–Mandell) There are Quillen adjunctions

Mod_A ^QA Nuca_A CSp^Σ_A//A

IA

which become Quillen equivalences after stabilization.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological André–Quillen homology

Definition

Thetopological André–Quillen homologyof Ais the A-module TAQ^R(A) =Q_A^LI_A^R(A∧_RA).

Proposition

There is a natural weak equivalence Map_Mod

A(TAQ^R(A),X)'Map_CSpΣ

R//A(A,A∨X), and the right-hand side is by definition the space of derivations Der_R(A,X).

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Topological André–Quillen homology

Definition

Thetopological André–Quillen homologyof Ais the A-module TAQ^R(A) =Q_A^LI_A^R(A∧_RA).

Proposition

There is a natural weak equivalence Map_Mod

A(TAQ^R(A),X)'Map_CSpΣ

R//A(A,A∨X),

and the right-hand side is by definition the space of derivations Der_R(A,X).

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Étale morphisms of ring spectra

A mapR →Aof commutative ring spectra isétale if

π0(R)→π0(A) is étale andπ0(A)⊗_π0(R)π∗(R)−→π∗(A) is an isomorphism.

Theorem (Lurie)

LetR be anE∞-ring. The functorπ0(−)induces an equivalence CAlg^ét_R/→CRing^ét_π

0(R)/ between the category of étaleR-algebras to the (nerve of the) category of étale algebras overπ₀(R).

Z[1/2]→Z[1/2,i] is étale,Z→Z[i] is not. S[1/2]→S[1/2,i]exists, S→S[i]does not.

This notion of étaleness is particularly well-behaved between maps ofconnective ring spectra.

The map KO→KUfails to be étale, despite enjoying many of the formal properties of étale maps.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Étale morphisms of ring spectra

A mapR →Aof commutative ring spectra isétale if

π0(R)→π0(A) is étale andπ0(A)⊗_π0(R)π∗(R)−→π∗(A) is an isomorphism.

Theorem (Lurie)

LetR be anE∞-ring. The functorπ0(−)induces an equivalence CAlg^ét_R/→CRing^ét_π

0(R)/ between the category of étaleR-algebras to the (nerve of the) category of étale algebras overπ₀(R).

Z[1/2]→Z[1/2,i] is étale,Z→Z[i] is not. S[1/2]→S[1/2,i]exists, S→S[i]does not.

This notion of étaleness is particularly well-behaved between maps ofconnective ring spectra.

The map KO→KUfails to be étale, despite enjoying many of the formal properties of étale maps.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Étale morphisms of ring spectra

A mapR →Aof commutative ring spectra isétale if

π0(R)→π0(A) is étale andπ0(A)⊗_π0(R)π∗(R)−→π∗(A) is an isomorphism.

Theorem (Lurie)

LetR be anE∞-ring. The functorπ0(−)induces an equivalence CAlg^ét_R/→CRing^ét_π

0(R)/ between the category of étaleR-algebras to the (nerve of the) category of étale algebras overπ₀(R).

Z[1/2]→Z[1/2,i] is étale,Z→Z[i] is not.

S[1/2]→S[1/2,i]exists, S→S[i]does not.

This notion of étaleness is particularly well-behaved between maps ofconnective ring spectra.

The map KO→KUfails to be étale, despite enjoying many of the formal properties of étale maps.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Étale morphisms of ring spectra

A mapR →Aof commutative ring spectra isétale if

π0(R)→π0(A) is étale andπ0(A)⊗_π0(R)π∗(R)−→π∗(A) is an isomorphism.

Theorem (Lurie)

LetR be anE∞-ring. The functorπ0(−)induces an equivalence CAlg^ét_R/→CRing^ét_π

0(R)/ between the category of étaleR-algebras to the (nerve of the) category of étale algebras overπ₀(R).

Z[1/2]→Z[1/2,i] is étale,Z→Z[i] is not.

S[1/2]→S[1/2,i]exists, S→S[i]does not.

This notion of étaleness is particularly well-behaved between maps ofconnective ring spectra.

The map KO→KUfails to be étale, despite enjoying many of the formal properties of étale maps.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Étale morphisms of ring spectra

A mapR →Aof commutative ring spectra isétale if

π0(R)→π0(A) is étale andπ0(A)⊗_π0(R)π∗(R)−→π∗(A) is an isomorphism.

Theorem (Lurie)

LetR be anE∞-ring. The functorπ0(−)induces an equivalence CAlg^ét_R/→CRing^ét_π

0(R)/ between the category of étaleR-algebras to the (nerve of the) category of étale algebras overπ₀(R).

Z[1/2]→Z[1/2,i] is étale,Z→Z[i] is not.

S[1/2]→S[1/2,i]exists, S→S[i]does not.

This notion of étaleness is particularly well-behaved between

The map KO→KUfails to be étale, despite enjoying many of the formal properties of étale maps.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Étale morphisms of ring spectra

A mapR →Aof commutative ring spectra isétale if

π0(R)→π0(A) is étale andπ0(A)⊗_π0(R)π∗(R)−→π∗(A) is an isomorphism.

Theorem (Lurie)

LetR be anE∞-ring. The functorπ0(−)induces an equivalence CAlg^ét_R/→CRing^ét_π

0(R)/ between the category of étaleR-algebras to the (nerve of the) category of étale algebras overπ₀(R).

Z[1/2]→Z[1/2,i] is étale,Z→Z[i] is not.

S[1/2]→S[1/2,i]exists, S→S[i]does not.

This notion of étaleness is particularly well-behaved between maps ofconnective ring spectra.

The map KO→KUfails to be étale, despite enjoying many of the formal properties of étale maps.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for HH

LetR→A be an étale morphism of discrete commutative rings.

Weibel-Geller show thatA⊗_RHH(R)−→^' HH(A)in this case.

They relate this to descent forHH along R→A.

From this it is formal to see thatA−→^' HH^R(A).

Sinceπ1HH^R(A)∼= Ω¹_A|R, this in turn implies that Ω¹_A|R ∼=0. If Ω¹_A|R ∼=0, thenR→A is étale as soon as it is flat and finitely presented.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for HH

LetR→A be an étale morphism of discrete commutative rings.

Weibel-Geller show thatA⊗_RHH(R)−→^' HH(A)in this case. They relate this to descent forHH along R→A.

From this it is formal to see thatA−→^' HH^R(A).

Sinceπ1HH^R(A)∼= Ω¹_A|R, this in turn implies that Ω¹_A|R ∼=0. If Ω¹_A|R ∼=0, thenR→A is étale as soon as it is flat and finitely presented.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for HH

LetR→A be an étale morphism of discrete commutative rings.

Weibel-Geller show thatA⊗_RHH(R)−→^' HH(A)in this case. They relate this to descent forHH along R→A.

From this it is formal to see thatA−→^' HH^R(A).

Sinceπ1HH^R(A)∼= Ω¹_A|R, this in turn implies that Ω¹_A|R ∼=0. If Ω¹_A|R ∼=0, thenR→A is étale as soon as it is flat and finitely presented.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for HH

LetR→A be an étale morphism of discrete commutative rings.

Weibel-Geller show thatA⊗_RHH(R)−→^' HH(A)in this case. They relate this to descent forHH along R→A.

From this it is formal to see thatA−→^' HH^R(A).

Sinceπ1HH^R(A)∼= Ω¹_A|R, this in turn implies that Ω¹_A|R ∼=0.

If Ω¹_A|R ∼=0, thenR→A is étale as soon as it is flat and finitely presented.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for HH

LetR→A be an étale morphism of discrete commutative rings.

Weibel-Geller show thatA⊗_RHH(R)−→^' HH(A)in this case. They relate this to descent forHH along R→A.

From this it is formal to see thatA−→^' HH^R(A).

Sinceπ1HH^R(A)∼= Ω¹_A|R, this in turn implies that Ω¹_A|R ∼=0.

If Ω¹_A|R ∼=0, thenR→A is étale as soon as it is flat and finitely presented.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

LetR−→^f A be a map of commutative symmetric ring spectra. We say thatf

is étaleifπ0(f)is étale and π0(A)⊗_π0(R)π∗(R)−→^∼= π∗(A);

satisfies étale descentifA∧_RTHH(R)−→^' THH(A); is formallyTHH-étale ifA−→^' THH^R(A);

is formallyTAQ-étale ifTAQ^R(A) is contractible. Conclusion of forthcoming discussion: downwards implications always hold, upwards under connectivity (and finiteness) hypotheses.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

LetR−→^f A be a map of commutative symmetric ring spectra. We say thatf

is étaleifπ0(f)is étale and π0(A)⊗_π0(R)π∗(R)−→^∼= π∗(A);

satisfies étale descentifA∧_RTHH(R)−→^' THH(A);

is formallyTHH-étale ifA−→^' THH^R(A);

is formallyTAQ-étale ifTAQ^R(A) is contractible. Conclusion of forthcoming discussion: downwards implications always hold, upwards under connectivity (and finiteness) hypotheses.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

LetR−→^f A be a map of commutative symmetric ring spectra. We say thatf

is étaleifπ0(f)is étale and π0(A)⊗_π0(R)π∗(R)−→^∼= π∗(A);

satisfies étale descentifA∧_RTHH(R)−→^' THH(A);

is formallyTHH-étale ifA−→^' THH^R(A);

is formallyTAQ-étale ifTAQ^R(A) is contractible. Conclusion of forthcoming discussion: downwards implications always hold, upwards under connectivity (and finiteness) hypotheses.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

LetR−→^f A be a map of commutative symmetric ring spectra. We say thatf

is étaleifπ0(f)is étale and π0(A)⊗_π0(R)π∗(R)−→^∼= π∗(A);

satisfies étale descentifA∧_RTHH(R)−→^' THH(A);

is formallyTHH-étale ifA−→^' THH^R(A);

is formallyTAQ-étale ifTAQ^R(A) is contractible.

Conclusion of forthcoming discussion: downwards implications always hold, upwards under connectivity (and finiteness) hypotheses.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

LetR−→^f A be a map of commutative symmetric ring spectra. We say thatf

is étaleifπ0(f)is étale and π0(A)⊗_π0(R)π∗(R)−→^∼= π∗(A);

satisfies étale descentifA∧_RTHH(R)−→^' THH(A);

is formallyTHH-étale ifA−→^' THH^R(A);

is formallyTAQ-étale ifTAQ^R(A) is contractible.

Conclusion of forthcoming discussion: downwards implications always hold, upwards under connectivity (and finiteness) hypotheses.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Theorem (Mathew)

IfR →Ais étale, then A∧_R THH(R)−→^' THH(A).

Proof sketch.

Lurie shows thatR →A étale implies that, for anyC, Map_CAlg(A,C) Hom_CRing(π0(A), π0(C)) Map_CAlg(R,C) Hom_CRing(π₀(R), π₀(C))

is cartesian. Mathew deducesA∧_R(S¹⊗R)−^'→S¹⊗A from this.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Theorem (Mathew)

IfR →Ais étale, then A∧_R THH(R)−→^' THH(A).

Proof sketch.

Lurie shows thatR→A étale implies that, for anyC, Map_CAlg(A,C) Hom_CRing(π0(A), π0(C)) Map_CAlg(R,C) Hom_CRing(π₀(R), π₀(C))

is cartesian. Mathew deducesA∧_R(S¹⊗R)−^'→S¹⊗A from this.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Still formal to see that A∧_RTHH(R)−^'→THH(A) implies A−→^' THH^R(A).

Proposition

IfA−→^' THH^R(A), thenTAQ^R(A) is contractible.

Proof.

The assumption is thatTHH^R(A)∼=S¹_A(A∧_R A)is weakly trivial inCSp^Σ_A//A. Hence{Sⁿ_A(A∧_RA)}= Σ^∞(A∧_RA) is stably trivial inSp(CSp^Σ_A//A). By Basterra-Mandell, this implies thatTAQ^R(A) is contractible.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Still formal to see that A∧_R THH(R)−^'→THH(A) implies A−→^' THH^R(A).

Proposition

IfA−→^' THH^R(A), thenTAQ^R(A) is contractible.

Proof.

The assumption is thatTHH^R(A)∼=S¹_A(A∧_R A)is weakly trivial inCSp^Σ_A//A. Hence{Sⁿ_A(A∧_RA)}= Σ^∞(A∧_RA) is stably trivial inSp(CSp^Σ_A//A). By Basterra-Mandell, this implies thatTAQ^R(A) is contractible.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Still formal to see that A∧_R THH(R)−^'→THH(A) implies A−→^' THH^R(A).

Proposition

IfA−→^' THH^R(A), thenTAQ^R(A) is contractible.

Proof.

The assumption is thatTHH^R(A)∼=S¹_A(A∧_R A)is weakly trivial inCSp^Σ_A//A. Hence{Sⁿ_A(A∧_RA)}= Σ^∞(A∧_RA) is stably trivial inSp(CSp^Σ_A//A). By Basterra-Mandell, this implies thatTAQ^R(A) is contractible.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Still formal to see that A∧_R THH(R)−^'→THH(A) implies A−→^' THH^R(A).

Proposition

IfA−→^' THH^R(A), thenTAQ^R(A) is contractible.

Proof.

The assumption is thatTHH^R(A)∼=S¹_A(A∧_R A)is weakly trivial inCSp^Σ_A//A.

Hence{Sⁿ_A(A∧_RA)}= Σ^∞(A∧_RA) is stably trivial inSp(CSp^Σ_A//A). By Basterra-Mandell, this implies thatTAQ^R(A) is contractible.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Still formal to see that A∧_R THH(R)−^'→THH(A) implies A−→^' THH^R(A).

Proposition

IfA−→^' THH^R(A), thenTAQ^R(A) is contractible.

Proof.

The assumption is thatTHH^R(A)∼=S¹_A(A∧_R A)is weakly trivial inCSp^Σ_A//A. Hence{Sⁿ_A(A∧_RA)}= Σ^∞(A∧_RA) is

Σ

By Basterra-Mandell, this implies thatTAQ^R(A) is contractible.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Still formal to see that A∧_R THH(R)−^'→THH(A) implies A−→^' THH^R(A).

Proposition

IfA−→^' THH^R(A), thenTAQ^R(A) is contractible.

Proof.

The assumption is thatTHH^R(A)∼=S¹_A(A∧_R A)is weakly trivial inCSp^Σ_A//A. Hence{Sⁿ_A(A∧_RA)}= Σ^∞(A∧_RA) is stably trivial inSp(CSp^Σ_A//A). By Basterra-Mandell, this implies thatTAQ^R(A) is contractible.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Example

ConsiderKO→KU. Then

THH(KO)'KO∨ΣKO_Q, THH(KU)'KU∨ΣKU_Q, and it is the case thatKU∧_KOTHH(KO)−^'→THH(KU).

Hence KU−^'→THH^KO(KU)andTAQ^KO(KU)' ∗

Similarly forL_p →KU_p.

In general,TAQ^R(A)' ∗ does not implyA−→^' THH^R(A), which in turn does not implyA∧_RTHH(R)−→^' THH(A).

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Example

ConsiderKO→KU. Then

THH(KO)'KO∨ΣKO_Q, THH(KU)'KU∨ΣKU_Q,

and it is the case thatKU∧_KOTHH(KO)−^'→THH(KU). Hence KU−^'→THH^KO(KU)andTAQ^KO(KU)' ∗

Similarly forL_p →KU_p.

In general,TAQ^R(A)' ∗ does not implyA−→^' THH^R(A), which in turn does not implyA∧_RTHH(R)−→^' THH(A).

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Example

ConsiderKO→KU. Then

THH(KO)'KO∨ΣKO_Q, THH(KU)'KU∨ΣKU_Q,

and it is the case thatKU∧_KOTHH(KO)−^'→THH(KU). Hence KU−^'→THH^KO(KU)andTAQ^KO(KU)' ∗

Similarly forL_p →KU_p.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Lurie shows thatLA|R =TAQ^R(A)' ∗implies étale for R andA connective andπ0(A) is finitely presented over π0(R).

Theorem (Mathew)

LetR→A is a map of connective commutative symmetric ring spectra withTAQ^R(A)' ∗. ThenA∧_R THH(R)−^'→THH(A).

Mathew shows that it is enough to show that

Map_CAlg(A,C) Hom_CRing(π0(A), π0(C)) Map_CAlg(R,C) Hom_CRing(π0(R), π0(C)) is homotopy cartesian. By the proof in the connective and étale case by Lurie (HA.7.5.1.15), our hypotheses imply this.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Lurie shows thatLA|R =TAQ^R(A)' ∗implies étale for R andA connective andπ0(A) is finitely presented over π0(R).

Theorem (Mathew)

LetR→A is a map of connective commutative symmetric ring spectra withTAQ^R(A)' ∗. ThenA∧_R THH(R)−^'→THH(A).

Mathew shows that it is enough to show that

Map_CAlg(A,C) Hom_CRing(π0(A), π0(C)) Map_CAlg(R,C) Hom_CRing(π0(R), π0(C)) is homotopy cartesian. By the proof in the connective and étale case by Lurie (HA.7.5.1.15), our hypotheses imply this.

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Lurie shows thatLA|R =TAQ^R(A)' ∗implies étale for R andA connective andπ0(A) is finitely presented over π0(R).

Theorem (Mathew)

LetR→A is a map of connective commutative symmetric ring spectra withTAQ^R(A)' ∗. ThenA∧_R THH(R)−^'→THH(A).

Mathew shows that it is enough to show that

Map_CAlg(A,C) Hom_CRing(π0(A), π0(C)) Map_CAlg(R,C) Hom_CRing(π₀(R), π₀(C)) is homotopy cartesian.

By the proof in the connective and étale case by Lurie (HA.7.5.1.15), our hypotheses imply this.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

Lurie shows thatLA|R =TAQ^R(A)' ∗implies étale for R andA connective andπ0(A) is finitely presented over π0(R).

Theorem (Mathew)

LetR→A is a map of connective commutative symmetric ring spectra withTAQ^R(A)' ∗. ThenA∧_R THH(R)−^'→THH(A).

Mathew shows that it is enough to show that

Map_CAlg(A,C) Hom_CRing(π0(A), π0(C)) Map_CAlg(R,C) Hom_CRing(π₀(R), π₀(C))

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

In conclusion: there are always downwards implications, while the converse statements hold ifR andAare connective:

étale descent: A∧_R THH(R)−^'→THH(A).

formally THH-étale: A−^'→THH^R(A).

formally TAQ-étale: TAQ^R(A)' ∗.

Non-examples include the connective covers of our examples: the maps

ko−→ku, `_p−→ku_p do not satisfy any of these properties.

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Formal étaleness properties for THH

In conclusion: there are always downwards implications, while the converse statements hold ifR andAare connective:

étale descent: A∧_R THH(R)−^'→THH(A).

formally THH-étale: A−^'→THH^R(A).

formally TAQ-étale: TAQ^R(A)' ∗.

Non-examples include the connective covers of our examples: the maps

ko−→ku, `_p−→ku_p

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Logarithmic rings

Idea from logarithmic geometry: rigidifying rings with the extra data of alogarithmic structureallows one to treat sometamely ramified extensions as if unramified.

Definition

Apre-log ring (A,M) consists of a commutative ring A; a commutative monoid M;

a map of commutative monoids α:M →(A,·). It is alog ring ifα⁻¹GL₁(A)−→^∼= GL₁(A).

Triviallog ring (A,GL₁(A)). (A,M) gives a localizationA[M⁻¹].

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Logarithmic rings

Idea from logarithmic geometry: rigidifying rings with the extra data of alogarithmic structureallows one to treat sometamely ramified extensions as if unramified.

Definition

Apre-log ring (A,M) consists of a commutative ring A; a commutative monoid M;

a map of commutative monoids α:M →(A,·). It is alog ring ifα⁻¹GL₁(A)−→^∼= GL₁(A).

Triviallog ring (A,GL₁(A)). (A,M) gives a localizationA[M⁻¹].

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Logarithmic rings

Idea from logarithmic geometry: rigidifying rings with the extra data of alogarithmic structureallows one to treat sometamely ramified extensions as if unramified.

Definition

Apre-log ring (A,M) consists of a commutative ring A;

a commutative monoidM;

a map of commutative monoids α:M →(A,·).

It is alog ring ifα⁻¹GL₁(A)−→^∼= GL₁(A).

Triviallog ring (A,GL₁(A)). (A,M) gives a localizationA[M⁻¹].

Tommy Lundemo Log TAQ and log THH

THHandTAQ Formally étale maps Logarithmic ring spectra LogTHHand logTAQ Formally log étale maps

Logarithmic rings

Idea from logarithmic geometry: rigidifying rings with the extra data of alogarithmic structureallows one to treat sometamely ramified extensions as if unramified.

Definition

Apre-log ring (A,M) consists of a commutative ring A;

a commutative monoidM;

a map of commutative monoids α:M →(A,·).

It is alog ring ifα⁻¹GL₁(A)−→^∼= GL₁(A).

Triviallog ring (A,GL₁(A)). (A,M) gives a localizationA[M⁻¹].