show that this difference does not matter on the level of THH. Note that the mapM∧AB →M∧B∧ABB equalizes the two mapsµ∧AB andM∧Aµ. There is hence the following commutative diagram where the vertical compositions are coequalizers.
M∧A(H∧AH)∧AH
Φ(M∧AB∧AB)◦Φ(M∧
AB)◦ΦM
//
M∧Aα∧A3
//M ∧AB∧AB∧AB
M∧AH
//M ∧AB∧AB
µ∧AB //
M∧Aµ //M∧AB
M ∧H∧AHH (ΦM)∗=α∗ //M∧B∧ABB
The upper horizontal maps in the diagram induce a map (ΦM)∗on coequalizers and the lower horizontal maps in the diagram induce a map α∗. As the lower right vertical map equalizesµ∧AB and M∧Aµ, the maps Φ∗ andα∗ coincide.
We obtain the following commutative diagram.
thhA(H;M) Φ'∗ //
'
thhA(B;M)
' THHA(H;M) THHA(B;M)
M∧LH∧
AH H α∗ //M∧LB∧
ABB
Using thatM is a cellB-module andB is an extended cellA-module it follows that the vertical maps in the diagram are weak equivalences, the crucial prop-erty being thatM∧AXrepresents the derived smash product for every extended cell A-module X. It follows from the last diagram that α∗: THHA(H;M) → THHA(B;M) is a weak equivalence as well. 2
10.2 Hochschild homology of Hopf algebras with
10.2. THH OF HOPF ALGEBRAS WITH HOMOTOPY ANTIPODE 121
M regarded as an H-bimodule with this structure. Also note that there is a homotopy commutative diagram
α:Hη∧AH//H∧AH Φ //H∧AH µ //H
H η∧AH//H∧AH Ψ //H∧AH µ //H
H //A η //H
(10.5)
The square in the middle of the upper part of the diagram commutes up to homotopy as both Φ and Ψ are homotopy inverses to the canonical maph. The lower part of the diagram commutes up to homotopy due to the antipode prop-erty. Note that the bottom mapη◦is also a map of commutative A-algebras and so induces yet anotherH-bimodule structure onM. We will writeMforM with theH-bimodule structure induced byη◦. As the diagram above commutes up to homotopy there is a weak equivalence THHA(H;Mα) ' THHA(H;M) as follows by a K¨unneth spectral sequence argument.
In order to proceed, we have to recall the bar construction, see [36, IV.7.]. Let A be a commutative S-algebra, B an associative A-algebra, M a right andN a leftB-module. We define a simplicial S-moduleBA∗(M, B, N) by
BAp(M, B, N) :=M∧AB∧Ap∧AN, with face and degeneracy operators
di =
µ∧Aidp−1∧AidN ifi= 0 idM∧Aidi−1∧Aµ∧Aidp−i−1∧AidN if 1≤i < p
idM∧Aidp−1◦µ ifi=p
andsi = idM∧Aidi∧Aη∧Aidp−i∧AidN.
Theorem 10.5. LetA be a commutativeS-algebra and H a commutative Hopf algebra under A with homotopy antipode. Assume that A is cofibrant as a commutative S-algebra and H is cofibrant as a commutative A-algebra. Let M = M be an extended cell A-module which we see as a central H-bimodule via the map :H→A. Then
thhA(H, M)'M∧ABA(A, H, A). (10.6) Proof: Comparing the definitions of the simplicial spectra thh∗ and B∗ we see that thhA∗(H;M) ∼= BA∗(M, H, A) where A is seen as a left H-module via : H → A. Here we are using that also the H-module structures on M are defined via the map . Analogously it follows that BA∗(M, H, A) ∼= M∧ABA∗(A, H, A). Passing to realizations then gives the stated result. 2
Proposition 10.6. Let A be a commutative S-algebra and H a commutative Hopf algebra under Awith homotopy antipode. Assume thatA is cofibrant as a commutative S-algebra and H is cofibrant as a commutative A-algebra. Let M be an H-bimodule relative A which is central and an extended cell A-module.
Then
THHA(H;M)'M∧ABA(A, H, A).
In particular there is an equivalence
THHA(H)'H∧ABA(A, H, A).
Proof: We can see A→H as a Hopf-Galois extension and have the following chain of weak equivalences.
THHA(H, M) ' THHA(H;Mα) lemma 10.4 ' THHA(H;M) diagram (10.5)
' thhA(H;M) M =M ext. cellA-module ' M∧ABA(A, H, A) theorem 10.5
This proves the first equivalence. The second equivalence is just the case M =
H. 2
Theorem 10.7. Let A → B be a Hopf-Galois extension of commutative S-algebras with Hopf algebra H which has a homotopy antipode. Assume that A is a cofibrant commutative S-algebra and B and H are cofibrant commutative A-algebras. Let M be a central B-bimodule relative Awhich is an extended cell A-module. Then
THHA(B;M)'M∧ABA(A, H, A).
If H is a of the form H=A∧K then
THHA(B;M)'M∧ BS(S, K, S).
Proof: There is the following chain of weak equivalences.
THHA(B;M) ' THHA(H;Mα) lemma 10.4
' M∧ABA(A, H, A) as in the proof above This proves the first equivalence and the second in case H =A∧K is a direct
consequence. 2
In the following corollary we implicitely assume that the usual cofibrancy as-sumptions are satisfied.
Corollary 10.8. Let M f be a Thom spectrum associated with an infinite loop map f:X→BGL1(A). Then
THHA(M f)'M f∧BX+ (10.7)
10.3. A SPECTRAL SEQUENCE CONVERGING TO THE HOCHSCHILD
HOMOLOGY OF A HOPF-GALOIS EXTENSION 123
Proof: Recall from chapter 6 that the group ringH :=A[X] is a Hopf algebra with homotopy antipode coacting on M f such that the canonical map h is a weak equivalence of commutativeA-algebras. It follows as in the last theorem that there is a weak equivalence THHA(M f) 'M ∧ BS(S, S[X], S). Then as the space level and the spectrum level bar constructions commute, e.g. by [36, X.1.3.], we see thatBS(S,Σ∞X, S)∼= Σ∞B(∗, X,∗) =S[BX]. 2 In case A = S the statement was formulated and proven in [13, 12] where [13] also includes Thom spectra arising from at least 3-fold loop maps X → BGL1(S). We can not reproduce the result for finite loop maps by our approach since we used that the canonical morphism h is a morphism of commutative S-algebras which is only the case when starting with an infinite loop map.
However, none of the references deals withA different fromS.
10.3 A spectral sequence converging to the Hoch-schild homology of a Hopf-Galois extension
So far we have only dealt with topological Hochschild homology relative to the base spectrum A of a Hopf-Galois extension A → B. We also want to work relative S and express THHS(B;M) in terms of the topological Hochschild homology ofAand the Hopf algebraH. More generally, letR→Abe a map of commutativeS-algebras mapping a commutativeS-algebra Rto a Hopf-Galois extension A → B. The next statement describes the topological Hochschild homology ofB relative R in a different way. We assume that R is a cofibrant commutativeS-algebra,A is a cofibrant commutativeR-algebra andB andH are cofibrant commutativeA-algebras.
Theorem 10.9.LetR→Abe a map of commutativeS-algebras and letA→B be a Hopf-Galois extension with respect to a Hopf algebra H. Assume that the above cofibrancy hypotheses hold. Let M be a B-bimodule relative R and let Mf→M be a cell approximation in the category of B∧RB-modules. Then
THHR(B;M)'THHA(H;Mf∧A∧RAA). (10.8) If moreoverH is of the form H =A∧RK for a Hopf algebraK underR then this equivalence takes the form
THHR(B;M)'THHR(K;Mf∧A∧RAA). (10.9) Proof: There is the following chain of weak equivalences.
THHR(B;M)'Mf∧B∧RBB MfcellB∧RB-module
∼=Mf∧B∧RB(B∧AB)∧B∧ABB
'THHA(B;Mf∧B∧RB(B∧AB)) N :=Mf∧B∧RB(B∧AB) 'thhA(B;Mf∧B∧RB(B∧AB)) cellB∧AB-module 'thhA(H;Mf∧B∧RB(B∧AB)) proposition 10.3
'THHA(H;Mf∧B∧RB(B∧AB)) N extended cell A-module 'THHA(H;Mf∧A∧RAA)
The last equivalence is due to isomorphisms
Mf∧B∧RBB∧AB ∼= (B∧BMf)∧B∧RB(B∧AB)
∼= (B∧BB)∧B∧RA(Mf∧BB)
∼= (B∧AA)∧B∧RA(Mf∧AA)
∼= (B∧BMf)∧A∧RA(A∧AA)
∼= Mf∧A∧RAA
where the second and fourth isomorphism is due to a comparison of coequalizer diagrams, compare with [36, III.3.10.]. This proves (10.8). Equation (10.9) is a
direct consequence. 2
The last theorem allows the construction of spectral sequences.
Theorem 10.10. Let R, A, B, K, M andMf be as in theorem 10.9. There is a spectral sequence
Torπ∗∗(K∧RK)(π∗THHR(A;M), K∗) =⇒π∗THHR(B;M).
Proof: This is the usual spectral sequence converging toπ∗THHR(B;M) from [36, IX.1.6.] and nothing but a K¨unneth spectral sequence. Here we are using theorem 10.9 which says that π∗T HHR(K;Mf∧A∧RAA) ∼= π∗THHR(B;M).
We are also using thatMf∧A∧RAA'THHR(A;M). To prove this equivalence, we have to show thatMf∧A∧RAA represents the derived smash product. This follows from our cofibrancy assumptions as the following chain of equivalences shows. We write Ae for a cell approximation of A in the category of A∧R A-modules.
Mf∧A∧RAA ∼= Mf∧B∧RB(B∧RB)∧A∧RAA
∼= Mf∧B∧RB(B∧AA∧AB) ' Mf∧B∧RB(B∧AAe∧AB)
∼= Mf∧B∧RB(B∧RB)∧A∧RAAe
∼= Mf∧A∧RAAe
This completes the proof. 2
Under flatness assumptions the spectral sequence simplifies, e.g. to a spectral sequence
HH∗(K∗;π∗THHR(A;M)) =⇒π∗THHR(B;M).
Moreover,π∗THHR(A;M) may be given by ordinary Hochschild homology. For instance, assume that R,A and K are Eilenberg-McLane spectra andA∗ and K∗ are flat asR∗-modules. Then the spectral sequences specializes to a spectral sequence of the form
HH∗R∗(K∗;HH∗R∗(A∗;M∗)) =⇒π∗THHR(B;M)
and generalizes the spectral sequence for Hopf-Galois extensions of ordinary rings from [80].
Appendix
125
Appendix A
A Galois correspondence for associative ring extensions
The classical Galois correspondence for field extensions has a generalization to commutative ring theory. In this more general situation, the subgroups of the Galois groupG correspond to the intermediate rings that have the additional properties of being separable andG-strong [24], [42], see definitions 1.2 and A.4.
In non-commutative ring theory it is not clear how to characterize those inter-mediate rings that correspond to subgroups of the Galois group in general and Galois correspondences have only been obtained under additional assumptions.
Ferrero obtains a Galois correspondence in [37]. He supposes some so-called property (H) which in the commutative case is fulfilled if and only if there are no nontrivial idempotents, see definition A.2.
The aim of this appendix is to set up a Galois correspondence for Galois exten-sions R =SG → S of associative rings under the assumptions that the trace tr :S → R is surjective and that all idempotents of S lie in the center of S.
The intermediate rings corresponding to subgroups ofGthen are exactly those which are G-strong and fulfill a certain separability condition, see definition A.7. This condition equals the usual condition of separability ifR and S are commutative or under the assumptions of the main theorem of [37]. Our main theorem is hence a generalization of those of [24],[42] and [37].
A.1 A short review
The easy part of a Galois correspondence for ring extensions reads as follows, see e.g. [37] for a reference including the associative case.
Proposition A.1([37] prop. 3.1 p.83).
Let R → S be G-Galois. Then for any subgroup H < G the inclusion T :=
SH →S is an H-Galois extension andH is the subset ofGleavingT pointwise fixed. If tr(S) =R, then T is separable and if H is normal in G, then R→T is aG/H-Galois extension.
To obtain the converse implication of a main-theorem, Ferrero supposes some property called (H) in [37].
127
Definition A.2 (Property (H)). Let µ:S⊗ZSop → S be the multiplication.
We say that S verifies (H) if:
z∈S⊗ZSop, µ(z) =µ(z2) =⇒ µ(z) = 0 or µ(z) = 1.
Obviously, if S verifies (H), it has no idempotents other than 0 or 1. Also for the converse statement of a Galois theorem, recall the following definitions:
Definition A.3 (strongly distinct morphisms). Let f, g : A → B be two ho-momorphisms into a ring B. Then f and g are called strongly distinct, if for every non-zero idempotente∈B there is ana∈A such thatf(a)e6=g(a)e.
IfB only has the trivial idempotents 0 and 1, then obviouslyf andg as above are strongly distinct if and only if they are distinct.
Definition A.4 (G-strong intermediate rings). LetS be a G-Galois extension of R and T an intermediate ring. Then T is called G-strong if the restrictions toT of any two elements ofGare either equal or strongly distinct as maps from T toS.
The parts from the main theorems from [24],[42] and [37] which are interesting for our purposes can be stated as follows:
Theorem A.5 ([24] theorem 2.3).
Let R→S be aG-Galois extension of commutativerings. Then there is a one to one correspondence between subgroups of G and R-separable intermediate rings that are G-strong.
In the non-commutative case of [37] it takes the following form:
Theorem A.6 ([37] theorem 3.3).
Let S be a G-Galois extension ofR. IfS verifies (H) and tr(S) =R, there is a one to one correspondence between subgroups ofGandR-separable intermediate rings.
Note that tr(S) =Ris always satisfied ifRandSare commutative ([24, Lemma 1.6]) and that (H) implies that all intermediate rings areG-strong.