We now review the construction of the map ν from above in order to make visible that it extends to bimodules over associative algebras.
We start with the evaluation map
M∧R0 FR(M, P)−→P (4.4)
which is adjoint to the identity onFR(M, P). It is a map of leftR-modules and ifP is an (R, K)-bimodule, the map is one of (R, K)-bimodules as can be seen by the adjunction
MR∧Kop(M∧R0FR(M, P), P)∼=MR0∧Kop(FR(M, P), FR(M, P)).
We assume thatP is an (R, K)-bimodule and smash (4.4) with a leftK-module Z and obtain a map
M∧R0FR(M, P)∧KZ −→P∧KZ. (4.5) The map ν is defined as the adjoint to (4.5) where R=R0 =K =A. Dualiz-ability is a condition in case R=R0=K=P =A and Z =M:
Definition 4.14 (Dualizable bimodules). Let A be an associative S-algebra.
We say that an A-bimodule M is dualizable over A if the canonical map ν:FA(M, A)∧AM −→FA(M, M) is a weak equivalence.
The following proposition says that duality theory makes sense in any closed monoidal category. The monoidal structure needs not to be symmetric as as-sumed in [33] and [64]. IfAis an associativeS-algebra the statements from e.g.
[64, III.1.3.] have the following generalizations.
Proposition 4.15. Let A be an associative S-algebra and M anA-bimodule.
1. IfM is dualizable overA the canonical map of A-bimodules ρ:M →DAopDAM
is a weak equivalence.
2. IfM is dualizable overA then the canonical map
ν:FA(M, Y)∧AZ −→FA(M, Y ∧AZ) (4.6) is a weak equivalence for all A-bimodules Y and leftA-modules Z.
3. IfM is dualizable overAthenDAM :=FA(M, A)is dualizable overAop. 4. IfM, Y andZ areA-bimodules then (4.6) is also a weak equivalence ifZ
is dualizable over A.
Proof: The canonical mapρis defined to be the adjoint to the mapDAM∧Aop M ∼=M∧ADAM →Awhere the last map is the evaluation, i.e. adjoint to the identity on DAM. To prove the first statement, we give an inverse of ρ in the
4.5. DUALIZABILITY OVER ASSOCIATIVE ALGEBRAS 43
stable homotopy category. Note that asM is assumed to be dualizable there is a weak equivalence FA(M, M)←−ν DAM∧AM. This gives a map
η:A→FA(M, M)→DAM∧AM (4.7) in the homotopy category which can be choosen to be A-bilinear.
An inverse ofρ is then given as
DAopDAM ∼=A∧AopDAopDAM −−−−−→η∧Aopid M∧AopDAM∧AopDAopDAM
M∧Aopeval
−−−−−−−→M.
The proof that this gives an inverse of ρ is left to the reader. One can also consult [33].
Now let us show that the map ν from the second part of the statement is a weak equivalence if M is dualizable. Again we construct an inverse in the stable homotopy category. This inverse is given as
FA(M, Y ∧AZ)−−−−→η∧Aid DAM∧AM∧AFA(M, Y ∧AZ)−−→eval
DAM∧AY ∧AZ −ν−−−∧A→Z FA(M, Y)∧AZ. (4.8) In order to prove the third part of the statement it suffices to prove the analogue of the second statement for theAop-bimodule DAM. Note that in the proof of the second part, we used the fact that the mapν in the composition (4.8) was a weak equivalence only for the construction of the mapη:A→ DAM∧AM. Hence for the proof of the third statement it suffices to find an analogue of this map, i.e. a map A → DAopDAM ∧Aop DAM so that the proof of the second statement can be copied. Using (4.7) from above which exists asM is supposed to be dualizable overA this analogue is given as
A η //DAM ∧AM ∼=M∧AopDAM ρ //DAopDAM∧AopDAM also using the mapρ from part 1.
We come to the last part of the statement. The parts of the proposition we have already proved, provide the following chain of equivalences:
FA(M, Y)∧AZ ' FA(M, Y)∧ADAopDAZ ' DAopDAZ∧AopFA(M, Y) ' FAop(DAZ, FA(M, Y)) ' FA(M, FAop(DAZ, Y)) ' FA(M, DAopDAZ∧AopY) ' FA(M, Y ∧AZ)
This proves the last point of the proposition. 2
Following [76] we call a topological group (stably) dualizable when it is dual-izable as anS-module. A first hint that dualizability might play a role in the topological theory of Galois extensions comes from the following observations:
First recall that for every algebraic Galois-extension R → T the extension T is a finitely generated R-module, see lemma 1.3. This corresponds to the no-tion of dualizability defined above. Now, topologically we have the following characterization from [36, III.7.9].
Lemma 4.16. Let A be a commutative S-algebra. Then M is a dualizable A-module if and only if it is weakly equivalent to a retract of a finite cellA-module, i.e. if and only if it is semi-finite.
If A is connective this is the case if and only if M is a retract of a finite CW A-spectrum. In a local category MS,E of E-local S-modules where S 6= E however, there may be dualizable spectra that are not weakly equivalent to a retract of a finite cell module. The K(1)-local Galois extension LK(1)S → J U2∧ into the complex image-of-J spectrum J U2∧ is an example for this. The spectrum J U2∧ is a dualizable but not a semi-finite LK(1)S-module, see [75, 6.2.2]. Galois extensions of commutative S-algebras are always dualizable by [75, 6.2.1.] and we will show in section 4.5.2 that this holds for Galois extensions of associativeS-algebras as well.
4.5.1 The norm map
For a spectrumXwith leftG-action recall the definition of the homotopy orbit spectrum
XhG:=EG+∧GX
where EG = B(∗, G, G) is the standard contractible space with free right G-action given by a bar construction. Also recall the dualizing spectrum
SadG:=S[G]hG=F(EG+, S[G])G
formed with respect to the rightG–actions [76, 2.5.]. For any stably dualizable group and anyS-moduleX with left G–action, there is a map
N: (X∧SadG)hG−→XhG
called thenorm map ([75, 3.6], [76, ch.5]). We refer the reader to these sources for details. We just recall for later use that the norm map is a weak equivalece whenever X is of the form W ∧G+, see [76, 5.2.5].
4.5.2 Associative Galois extensions and dualizability
Proposition 4.17. Let A be an S-algebra and A → B a Galois extension of S-algebras with Galois group G. Then B is dualizable over A and over Aop.
4.5. DUALIZABILITY OVER ASSOCIATIVE ALGEBRAS 45
Proof: We prove that B is dualizable over Aop with the following diagram whereM is aB-bimodule.
FAop(B, A)∧AopM ν //
i∗
FAop(B, A∧AopM)
i∗
FAop(B, BhG)∧Aop M ν //
FAop(B, BhG∧Aop M)
FAop(B, BhG)∧Aop M
FAop(B,(B∧Aop M)hG)
FAop(B, B)hG∧Aop M //FAop(B, B∧Aop M)hG
(B∧G+)hG∧Aop M //
jhG
OO
(B∧G+∧AopM)hG
(jM∧AB)hG
OO
M∧A(B∧G+)hG //
∼=
OO
(M∧AB∧G+)hG
∼=
OO
M∧A(B∧G+∧SadG)hG ∼= //
M∧AN
OO
(M∧AB∧G+∧SadG)hG
N
OO
It is clear that the diagram commutes and that the lower horizontal map is an isomorphism. We will prove that all the vertical maps are weak equivalences.
The statement then follows from the caseM =B. The top vertical maps are weak equivalences asi:A→BhG is an equivalence by assumption. The second top vertical map on the right is an equivalence by lemma 4.13. Then, one level further below, we have weak equivalences by a general property for homotopy fixed points that can be checked easily. The maps labeledjhG and (jM∧AB)hG are weak equivalences by lemma 4.12. The isomorphisms are clear and the norm mapsN are weak equivalences as the spectra in sight are of the formX∧G+. The dualizability of B overA can be proved analogously using the mapsej. 2
Lemma 4.18. Let B be an A-algebra with an action of a topological group G underA. Let M be dualizable over Aop. Then the natural map
M ∧ABhK →(M∧AB)hK
is a weak equivalence for any subgroupK of G. Analogously, ifN is dualizable over A then the canonical maps
BhK ∧AN →(B∧AN)hK are weak equivalences.
Proof: AsM is supposed to be dualizable overAopthere is a weak equivalence M →DADAopM by proposition 4.15. We then get a commutative diagram
M∧ABhK ρ∧id //
DADAopM∧ABhK
ν //FA(DAopM, BhK)
'
(M∧AB)hK(ρ∧id)
hK//(DADAopM∧AB)hK νhK //FA(DAopM, B)hK
in which all the horizontal and hence also the vertical maps are weak equiva-lences. Again the statements involvingN are proved analogously. 2 Corollary 4.19. For any Galois extension A → B of associative S-algebras any subgroup K of the Galois group, the canonical maps
B∧ABhK →(B∧AB)hK BhK∧AB→(B∧AB)hK
are weak equivalences. Here the actions on B∧AB are on the second factor in the first line and on the first factor in the second line.
Proof: This follows from lemma 4.18 asB is dualizable over A and overAop
by proposition 4.17. 2