The Tate construction for finite groups

We start with some brief recollections on norm maps in ∞-categorical situations. For this, we follow closely [71,§6.1.6]. The following classes of∞-categories will be relevant

to us.

Definition I.1.1. LetC be an∞-category.

(i) The∞-categoryCispointed if it admits an object which is both initial and final;

such objects are called zero objects.

(ii) The ∞-category C is preadditive if it is pointed, finite products and finite co-products exist, and for any two objectsX, Y∈C, the map

id_X 0 0 id_Y

:XtY −!X×Y

is an equivalence. Here, 0∈Hom_C(X, Y) denotes the compositionX!0!Y for any zero object 0∈C.

If C is preadditive, we write X⊕Y for XtY'X×Y. If C is preadditive, then π0Hom_C(X, Y) acquires the structure of a commutative monoid for allX, Y∈C.

(iii) The∞-categoryCisadditive if it is preadditive andπ0Hom_C(X, Y) is a group for allX, Y∈C.

(iv) The∞-category C isstable if it is additive, all finite limits and colimits exist, and the loop functor Ω:C!C,X7!0×X0, is an equivalence.

We refer to [71, Chapter 1] for an extensive discussion of stable∞-categories. These notions are also discussed in [37]. Note that in [71, §6.1.6] Lurie uses the term semiad-ditive in place of preadsemiad-ditive.

Definition I.1.2. LetG be a group, and C an∞-category. A G-equivariant object in C is a functorBG!C, where BGis a fixed classifying space forG. The∞-category C^BGofG-equivariant objects inC is the functor∞-category Fun(BG,C).

Remark I.1.3. We are tempted to write C^G in place of C^BG, which is closer to standard usage by algebraists. However, this leads to conflicts with the notationC^X for a general Kan complexXthat we will use momentarily. This conflict is related to the fact that algebraists writeH^∗(G, M) for group cohomology of G acting on a G-module M, where topologists would rather writeH^∗(BG, M).

Remark I.1.4. Applying this definition in the case of the ∞-category of spectra C=Sp, one gets a notion of G-equivariant spectrum. This notion is different from the notions usually considered inequivariant stable homotopy theory,(⁷)and we discuss their relation in§II.2below. To avoid possible confusion, we will refer toG-equivariant objects in Sp asspectra with G-action instead ofG-equivariant spectra.

(⁷) It is even more naive than what is usually called naive.

Definition I.1.5. LetGbe a group, andC an∞-category.

(i) Assume thatCadmits all colimits indexed byBG. Thehomotopy orbits functor is given by

−hG:C^BG−!C,

(F:BG!C)7−!colim_BGF.

(ii) Assume that C admits all limits indexed by BG. The homotopy fixed points functor is given by

−^hG:C^BG−!C, (F:BG!C)7−!lim_BGF.

Remark I.1.6. Note that, in the setting of DefinitionI.1.5, it might be tempting to drop the word “homotopy”, i.e. to refer to these objects as “orbits” and “fixed points”

and denote them by−G and−^G, since in an∞-categorical setting this is the only thing that makes sense. However, later in the paper we will need some elements of equivariant homotopy theory, so that there is another spectrum referred to as “fixed points”. In order to avoid confusion we use the prefix “homotopy”.

Now, assume thatGis finite, and thatCis a sufficiently nice∞-category, in a sense that will be made precise as we go along. We construct a norm map

NmG:XhG−!X^hG

as a natural transformation of functors−hG!−^hG:C^BG!C. The construction will be carried out in several steps.

For any Kan complex X, let C^X=Fun(X,C) be the functor ∞-category. For any mapf:X!Y of Kan complexes, there is a pullback functor

f^∗:C^Y = Fun(Y,C)−!Fun(X,C) =C^X.

We denote the left (resp. right) adjoint off^∗ by f_! (resp. f_∗), if it exists. In light of [69, Proposition 4.3.3.7], we will also refer tof_! (resp. f_∗) as the left (resp. right) Kan extension alongf:X!Y.

As an example, note that, iff:BG!∗is the projection to a point, then the resulting functorsf_!, f_∗:C^BG!C are given by−_hG and−^hG, respectively.

We will often use the following construction, where we make the implicit assumption that all functors are defined, i.e. thatChas sufficiently many (co)limits.

Construction I.1.7. Letf:X!Y be a map of Kan complexes, andδ:X!X×YX be the diagonal. Assume that there is a natural transformation

Nmδ:δ!−!δ_∗ of functorsC^X!C^X×YX, and that Nmδ is an equivalence.

Letp0, p1:X×YX!X denote the projections onto the first and second factor. We get a natural transformation

p^∗₀−!δ∗δ^∗p^∗₀'δ∗ Nm⁻¹_δ

' δ!'δ!δ^∗p^∗₁−!p^∗₁, and, by adjunction, a map id_CX!p0∗p^∗₁. Now, consider the diagram

X×_YX ^p0 //

p₁

X

f X ^f //Y.

By [71, Lemma 6.1.6.3] (cf. [71, Definition 4.7.4.13] for the definition of right adjointable diagrams), the natural transformationf^∗f∗!p0∗p^∗₁ is an equivalence. We get a natural transformation id_CX!f^∗f∗ of functorsC^X!C^X, which is adjoint to a natural transfor-mation Nmf:f!!f∗ of functorsC^X!C.

Recall that a mapf:X!Y of Kan complexes is (−1)-truncated if all fibers off are either empty or contractible.(⁸) Equivalently, δ:X!X×YX is an equivalence. In this case, Nmδ exists tautologically.

Lemma I.1.8. ([71, Proposition 6.1.6.7]) If C is a pointed ∞-category, then the functors f! and f_∗ exist for all (−1)-truncated maps f:X!Y, and Nmf:f!!f_∗ is an equivalence.

Thus, ifC is pointed, we can now play the game for 0-truncated mapsf:X!Y, as thenδ:X!X×YX is (−1)-truncated, and so Nmδ exists and is an equivalence. We say that a 0-truncated mapf:X!Y has finite fibers if all fibers off are equivalent to finite sets.

Lemma I.1.9. ([71, Proposition 6.1.6.12]) If C is a preadditive ∞-category, then the functors f! and f_∗ exist for all 0-truncated maps f:X!Y with finite fibers, and Nmf:f!!f_∗ is an equivalence.

(⁸) Here and in the following, all fibers of maps of Kan complexes are understood to be homotopy fibers.

Therefore, if C is preadditive, we can go one step further, and pass to 1-truncated mapsf:X!Y. We say that a 1-truncated mapf:X!Y is arelative finite groupoidif all fibers off have finitely many connected components, and each connected component is the classifying space of a finite group. Note that, iff:X!Y is a relative finite groupoid, thenδ:X!X×YX is a 0-truncated map with finite fibers.

Definition I.1.10. Let C be a preadditive ∞-category which admits limits and co-limits indexed by classifying spaces of finite groups. Let f:X!Y be a map of Kan complexes which is a relative finite groupoid. The norm map

Nmf:f!−!f_∗

is the natural transformation of functorsC^X!C^Y given by ConstructionI.1.7.

Example I.1.11. LetCbe a preadditive∞-category which admits limits and colimits indexed byBGfor some finite groupG. Applying the previous definition in the special case of the projectionf:BG!∗, we get a natural transformation

NmG:XhG!X^hG of functorsC^BG!C.

Example I.1.12. LetCbe a preadditive∞-category which admits limits and colimits indexed by BGfor some finite group G. Assume thatG is a normal subgroup of some (topological) group H. In this case, for any H-equivariant object X∈C^BH, the G-homotopy orbits andG-homotopy fixed points acquire a remainingH/G-action. More precisely, consider the projectionf:BH!B(H/G). Then, by [71, Proposition 6.1.6.3], the functors

f!, f_∗:C^BH−!C^B(H/G) sit in commutative diagrams

C^{BH f!} //

C^B(H/G)

C^{BG −hG} //C

and

C^{BH f∗} //

C^B(H/G)

C^{BG −}

hG //C,

where the vertical functors are the forgetful functors. By abuse of notation, we sometimes denote these functors simply by

−hG,−^hG:C^BH−!C^B(H/G).

We claim that, in this situation, the natural transformation Nm_G:−_hG!−^hGof functors C^BG!C refines to a natural transformation Nm_f:f_!!f_∗ of functors C^BH!C^B(H/G), i.e. Nm_G isH/G-equivariant. Indeed, Nm_f is a special case of DefinitionI.1.10.

Definition I.1.13. LetCbe a stable∞-category which admits all limits and colimits indexed byBGfor some finite groupG. The Tate construction is the functor

−^tG:C^BG−!C,

X7−!cofib(NmG:XhG!X^hG).

IfGis a normal subgroup of a (topological) groupH, we also write−^tGfor the functor

−^tG:C^BH−!C^B(H/G),

X7−!cofib(Nmf:f!X!f_∗X), wheref:BH!B(H/G) denotes the projection.

We will use this definition in particular in the case whereC=Sp is the∞-category of spectra. More generally, one can apply it to the∞-categoryR-Mod of module spectra over any associative ring spectrumR∈Alg(Sp); this includes the case of the ∞-derived category of S-modules for some usual associative ring S by taking R=HS. However, limits and colimits inR-Mod are compatible with the forgetful functorR-Mod!Sp, as are the norm maps, so that the resulting Tate constructions are also compatible.

If one takes the Tate spectrum of an Eilenberg–MacLane spectrum HM for a G-module M, then one gets back classical Tate cohomology, more precisely π_i(HM^tG)∼= Hb⁻ⁱ(G, M). As an example, we have Hb^∗(C_p,Z)∼=Fp[t^±1], withtof degree 2. We see in this example that there is a canonical ring structure on Tate (co)homology. This is in fact a general phenomenon, which is encoded in the statement that the functor

−^tG: Sp^BG−!Sp

admits a canonical lax symmetric monoidal structure. We will discuss this further below in§I.3. For now, we will ignore all multiplicative structure.