acyclic. Let us show that this complex is exact inT⊗R(k+2) and choose a cycle x=P
jt0,j⊗Rt1,j⊗R· · · ⊗Rtk+1,j ∈T⊗R(k+2). Then X
j
t0,j⊗R1⊗Rt1,j⊗Rt2,j⊗R· · · ⊗Rtk+1,j
=X
j
t0,j⊗Rt1,j⊗R1⊗Rt2,j⊗R· · · ⊗Rtk+1,j
−X
j
t0,j⊗Rt1,j⊗Rt2,j⊗R1⊗R· · · ⊗Rtk+1,j
± · · · (−1)k+1X
j
t0,j⊗Rt1,j⊗Rt2,j⊗R· · · ⊗R1⊗Rtk+1,j Multiplying the first two factors we obtainx= (idT⊗R∂⊗RidT)(P
jt0,jt1,j⊗R
· · · ⊗Rtk+1,j) where∂:T⊗R(k−1) →T⊗R(k) is the boundary map in the cochain complex associated with C•(T /R). So the cycle x is a boundary which proves
the proposition. 2
Given a Hopf-Galois extensionR→T, the two mapsδ0, δ1:T →T⊗RT from the Amitsur complex correspond to the mapsβ and T⊗Rη from the equalizer diagram (5.1). So the cofixed points occur as the zeroth cohomology group of the Amitsur complex and when the extension is faithfully flat, this is the only non-vanishing Amitsur cohomology group by proposition 5.2. We now turn to our topological definitions.
5.2 Definitions
5.2.1 Hopf-Algebras in stable homotopy theory
Definition 5.4 (Bialgebras). Let A be an associative S-algebra. We define a bialgebra H under A to be anA-unca that is also a coalgebra, hence equipped with a unit η: A → H and an associative multiplication µ:H ∧AH → H, together with a counit:H→Aand a coproduct ∆ : H→H∧AH in the cat-egory ofS-algebras underAsatisfying the usual coassociativity and counitality assumptions.
ForA=S a bialgebra H with structure maps in the category of commutative S-algebras was called a Hopf algebra in [75]. In algebra, a Hopf algebra by definition is a bialgebra with antipode where an antipode is a mapλ:H →H such thatµ(H∧Aλ)∆ =η=µ(λ∧AH)∆. The antipode condition was dropped in [75] as this would restrict the number of examples. Although in the examples we will usually have maps that satisfy the antipode condition up to homotopy we cannot always strictify these maps to obtain an antipode in the category of S-algebras under A. So we do not include this condition in our point-set-level definition. We might call theA-bialgebras defined aboveA-Hopf algebras when we agree that in stable homotopy theory the existence of an antipode is usually
not included in the definition. We will do so and instead always emphasize when a Hopf algebra has a (homotopy) antipode. In this sense, for A = S the definition above is the definition of [75] with the difference, that we work with associative, not commutative S-algebras. Note that in general it is no longer automatic that H∧AH is an S-algebra. It is of course an S-algebra if A is commutative and H is an associative A-algebra or if the map A → H is central or more generally if H is of the form H=A∧He for some S-bialgebra H. In any of our examples we will be in one of these situations. Moreovere the multiplicative structure of the Hopf algebra is not needed for many of the statements we will make. We can often work with objects of the following type.
Definition 5.5 (Unital coalgebra). Let A be an associative S-algebra. We define aunital coalgebraH(orcoalgebra under A) to be anA-bimodule underA with a counit:H →Aand a coassociative, counital comultiplication ∆ : H→ H∧AH in the category of A-bimodules under A.
Here are some examples:
(a) First note that for any (pointed) space X the group ringS[X] :=S∧X+
has the structure of a unital coalgebra. This structure is induced by the inclusion of the basepoint ∗ → X, the diagonal X → X ×X and the projection X→ ∗. Moreover ifX is ann-fold loop space, S[X] inherits the En-operad action from X and for n≥ 1 we obtain a Hopf algebra in the sense above, see proposition 6.3.
(b) For a finite group G the functional dual DG+ := F(G+, S) ' Q
GS can be rigidified to a commutative Hopf algebra underS. Again smashing with any unca A provides a Hopf-algebra H 'F(G+, S)∧A under A which is equivalent to F(S[G], A) by the dualizability ofG.
(c) We can obtain new Hopf algebras from old ones: The smash product of Hopf algebras under some commutative S-algebra is again a Hopf algebra under the same commutativeS-algebra. And for any Hopf algebraHunder S also A∧H is a Hopf algebra under A for any S-algebra A.
5.2.2 Coaction of a Hopf-algebra and Hopf-Galois extensions Definition 5.6. (Coaction) Let A be an associative S-algebra, B an A-unca and H a Hopf-algebra under A. We say thatH coacts on B under A if there is a map
β:B →B∧AH
of S-algebras under A, again satisfying coassociativity and counitality. More generally, H can just be a coalgebra in the category of A-bimodules under A in which case β is just required to be a coassociative and counital map of A-bimodules under A.
We will now approach the definition of Hopf-Galois extension and see that this definition also makes sense in case a unital coalgebraH coacts on some uncaB
5.2. DEFINITIONS 65
underA. In this case, note that we can immediately define the morphism h as h:B∧AB −−−−→B∧Aβ B∧AB∧AH−−−−→µ∧AH B∧AH (5.2) which is a map of (B, A)-bimodules underA.
Formulating a cofixed point condition in analogy to the algebraic situation needs a little more work. First note that there is a cosimplicial object, theHopf-cobar complex CA•(H;B) which underA is defined by
CAq(H;B) :=B∧AH∧A· · · ∧AH
| {z }
q
with coface mapsδ0 :=β∧1q,δi:= 1i∧A∆∧A1q−ifor 0< i < qandδq:= 1q∧η.
The codegeneracies are given byσi := 1i∧∧1q−i. We denote the totalization of this cosimplicial object asCA(H;B) and have a coaugmentation
i:A→CA(H;B). (5.3)
We will require some cofibrancy assumptions that make the theory homotopy invariant. Having a map ofS-algebrasA→B and a unital coalgebraH coact-ing on B under A we will first require that A is cofibrant as an S-algebra.
Second we will require that smashing withB respectively H overA represents the derived smash product. If these two requirements are satisfied, we will say that our data satisfy good cofibrancy assumptions. Of course we have good cofibrancy assumptions ifB and H are cofibrant objects in the category of A-uncas by corollary 3.17 but we have to allow more flexibility here to include the coalgebra case. Moreover, as we do not have model structures on the category of comodules or coalgebras, at least we do not know that such structures exist, it might not be possible to keep the comodule and coalgebra structure when cofibrantly replacingB respectivelyH in the category of A-uncas. So even in the Hopf algebra situation we might not be able to force cofibrancy in this sense if it is not given anyway. It is therefore worth noting that by theorem 3.16 we have good cofibrancy assumptions whenever B and H are in the class FA of extended cellA-bimodules as introduced in definition 3.14.
Under good cofibrancy assumptions, C(H;B) is a good replacement for the homotopy cofixed points as we will now explain. An investigation of the derived functor of the cofixed point functor in general model categories based on cobar constructions and the investigation of Hopf-Galois extensions in this context has been announced by Kathryn Hess, see [44, 45]. We mentioned that in the algebraic context the cofixed points are exactly given by the equalizer of the unit and coaction that correspond to the mapsδ0, δ1:C0 → C1 in the Hopf-cobar complex. This equalizer is just Hom∆(∗, C•) where∗is the constant cosimplicial object with a point in each degree and it is clear that a homotopy invariant version should be Hom∆ evaluated at a cofibrant and fibrant replacement, and this is Hom∆(∆, C•f) = Tot(C•f). We will show below that in the situation
of the next definitionC• itself is good enough i.e. Tot(C•)'Tot(C•f) has the correct homotopy type. Here is the basic definition of this chapter:
Definition 5.7(Hopf-Galois extensions and coalgebra extensions of associative S-algebras). LetA be a cofibrant S-algebra andH a Hopf algebra that coacts onB overAas above. Assume that smashing withB respectivelyHrepresents the derived smash product over A, e.g. this is the case when B and H are cofibrant S-algebras under A. We call B an H-Hopf-Galois extension of A, if the maps h and i from equations (5.2) and (5.3) are both weak equivalences.
More generally, if H is just a coalgebra coacting onB under A such that the morphisms hand iare weak equivalences we callA→B acoalgebra extension.
There is a nice criterion in [75, prop. 12.1.8] for when a mapS→B of commu-tative S-algebras is a Hopf-Galois extension. For a map of S-algebrasA → B define the Amitsur complex to be the cosimplicial objectC•(B/A), analogously defined to the object C•(T /R) from section 5.1, see [75, 8.2.1]. Then define A∧B := TotC•(B/A) to be the completion of A along B formed in A-modules, [75, 8.2.1]. This is one example of a G-completion as defined in [17]. More pre-ciselyA∧B is the G-completion of Adefined as the totalization of a cosimplicial resolution of A in the G-model structure on the category of cosimplicial A-modules (MA)∆ for the class of injective models G ={G|Ga leftB-module}.
At least for commutative A this completionA∧B is equivalent to Bousfield’s B-nilpotent completion LbABA from [15] understood to be defined in the category of A-modules. This follows by comparison of [17] with [15, 5.6,5.8], a proof if A → B is a map of commutative S-algebras is given in [75, 8.2.3]. The proof applies equally well if B is an associativeS-algebra. IfA is just inAS we have to require that A → B is central in order that the nilpotent resolution given in [75, 8.2.2] is one of A-modules. We will need to compare these completions only in the case whereA is a commutative S-algebra.
Criterion 5.8. Let H coact onB over A as above with appropriate cofibrancy assumptions. Then B is an H-Hopf-Galois-extension of A if and only if
1. the canonical maph:B∧AB →B∧AH is a weak equivalence and 2. the canonical mapA→A∧Bis a weak equivalence, i.e. Ais complete along
B.
Proof of criterion 5.8: We have to show that Tot(C•(B, H)) defines the G-completion ofA for the class of injective modelsG as above. In the terminol-ogy of [17] we can prove the criterion by showing thatC•(B, H) defines a weak G-resolution of Aas we will now recall.
First look at the Amitsur complex C•(B/A) defined as Ck(B/A) := B∧Ak+1 with the obvious coface and codegeneracy maps, i.e. given by unit and multipli-cation maps as in the algebraic case. The Amitsur complex is exactly the triple resolution for the triple (B∧A−, η, ν). This is a weakG-resolution ofAby [17, 7.4] and by [17, 6.5] weak resolutions can be used to calculate completions.
Now let us show that also C•(B, H) defines a weak G-resolution of A. Note that there is a cosimplicial weak equivalence C•(B/A) → C•(B, H) defined