The spectrum of derived Mackey functors
TTHE SPECTRUM OF DERIVED MACKEY FUNCTORS
IRAKLI PATCHKORIA, BEREN SANDERS, AND CHRISTIAN WIMMER
Abstract.
We compute the spectrum of the category of derived Mackey func-tors (in the sense of Kaledin) for all finite groups. We find that this space cap-tures precisely the top and bottom layers (i.e. the height infinity and heightzero parts) of the spectrum of the equivariant stable homotopy category. Dueto this truncation of the chromatic information, we are able to obtain a com-plete description of the spectrum for all finite groups, despite our incompleteknowledge of the topology of the spectrum of the equivariant stable homotopycategory. From a different point of view, we show that the spectrum of derivedMackey functors can be understood as the space obtained from the spectrumof the Burnside ring by “ungluing” closed points. In order to compute thespectrum, we provide a new description of Kaledin’s category, as the derivedcategory of an equivariant ring spectrum, which may be of independent inter-est. In fact, we clarify the relationship between several different categories,establishing symmetric monoidal equivalences and comparisons between theconstructions of Kaledin, the spectral Mackey functors of Barwick, the ordi-nary derived category of Mackey functors, and categories of modules over cer-tain equivariant ring spectra. We also illustrate an interesting feature of theordinary derived category of Mackey functors that distinguishes it from otherequivariant categories relating to the behavior of its geometric fixed points.
Contents
1. Introduction 12. Computation of the spectrum 43. Construction of D(H Z G ) and equivariant spectra 194. Equivalence with the categories of Kaledin and Barwick 255. Modules over the Burnside ring Mackey functor 42References 461. Introduction
A compelling yet not completely understood phenomenon in hypertopical algebrais the impression that some stable homotopy theories appear (at least intuitively)to be “linearizations” of other stable homotopy theories. For example, the derivedcategory of the integers D( Z ) can intuitively be regarded as a kind of “linearization”of the stable homotopy category of spectra SH. Although it is not our present goal Date : August 7, 2020.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . A T ] A ug IRAKLI PATCHKORIA, BEREN SANDERS, AND CHRISTIAN WIMMER to make this notion of “linearization” precise, one can readily find additional exam-ples; consider, for example, the relationship between the derived category of mo-tives DMot( k ) over a field and the corresponding motivic stable homotopy categorySH( k ). In this paper we are interested in the “linearization” of the G -equivariantstable homotopy category SH( G ) for G a finite group.As the homotopy groups of a G -spectrum X ∈ SH( G ) naturally form a graded G -Mackey functor, it seems plausible that the linearization of SH( G ) would besome kind of derived category of Mackey functors. The category of G -Mackeyfunctors M ack ( G ) forms an abelian category, so we can certainly consider its de-rived category D( M ack ( G )), but Kaledin [Kal11] argues that D( M ack ( G )) is notthe “correct” definition of the derived category of Mackey functors. He introduces anew triangulated category of “derived Mackey functors” DMack( G ), which contains M ack ( G ) as a subcategory, but with better behavior than D( M ack ( G )). Part ofKaledin’s argument against D( M ack ( G )) is that it does not behave in a way anal-ogous to the equivariant stable homotopy category SH( G ). From our point of view,it is Kaledin’s category DMack( G ) that is the correct “linearization” of SH( G ),rather than the ordinary derived category D( M ack ( G )).In this paper we will compute the tensor triangular spectrum of the compactobjects in Kaledin’s category of derived G -Mackey functors and explain its close re-lationship with the spectrum of the stable homotopy category of compact G -spectra(of which we have a fairly good understanding due to [BS17, BHN + G ) c captures precisely the top and bottom chro-matic layers of the spectrum of SH( G ) c . For example, the following diagram depictsthe relationship between the two spaces for G = C p the cyclic group of order p : Spec(DMack( C p ) c ) Spec(SH( C p ) c ) (cid:31) (cid:127) (cid:47) (cid:47) (cid:31) (cid:127) (cid:47) (cid:47) •◦•◦•◦•◦ ... •◦ •◦•◦•◦•◦ ... •◦ ( q (cid:54) = p,n ≥ •◦•◦•◦•◦ ... •◦ •◦•◦•◦•◦ ... •◦ •◦•◦•◦•◦ ... •◦ •◦•◦•◦•◦ ... •◦ . . .. . .. . .. . .. . . ( q (cid:54) = p,n ≥ •◦•◦•◦•◦ ... •◦ •◦•◦•◦•◦ ... •◦ •◦•◦•◦•◦ ... •◦ •◦•◦•◦•◦ ... •◦ . . .. . .. . .. . .. . . •◦ •◦•◦•◦ •◦ •◦ •◦ •◦ . . . •◦ •◦ •◦ •◦ . . . •◦ •◦ Although there remain unresolved questions about the topology of Spec(SH( G ) c )for nonabelian groups G , we are able to obtain a complete description of the spaceSpec(DMack( G ) c ) for all finite groups because the unresolved chromatic interac-tions in the topology of Spec(SH( G ) c ) get truncated away at the top and bottomchromatic layers.Our description of Spec(DMack( G ) c ) is achieved in Theorem 2.22, Theorem 2.36and Proposition 2.38. The corresponding classification of thick tensor-ideals isincluded as Theorem 2.47. Moreover, the precise relationship with Spec(SH( G ) c )is formulated in Corollary 2.40.There is also a very satisfying relationship with the spectrum of the Burnsidering. Recall that Spec( A ( G )) consists of a number of copies of Spec( Z ), one for eachconjugacy class of subgroups, but with certain closed points glued together. Wewill see that the spectrum of the category of derived Mackey functors is preciselythe space obtained from the spectrum of the Burnside ring by ungluing these closed HE SPECTRUM OF DERIVED MACKEY FUNCTORS 3 points. More precisely, Spec(DMack( G ) c ) consists of a number of disjoint copiesof Spec( Z ), one for each conjugacy class of subgroups, with topological interactionbetween the closed points describing the gluing that occurs in Spec( A ( G )). Thefollowing picture illustrates this for G = C p : ρ (cid:15) (cid:15) Spec(DMack( C p ) c ) =Spec( A ( C p )) = . . . . . . •◦ •◦ •◦ •◦ •◦ •◦ •◦ •◦ •◦ •◦•◦ •◦ (cid:124)(cid:123)(cid:122)(cid:125) (cid:95) (cid:15) (cid:15) (cid:71)(cid:72)(cid:35) •◦ •◦ •◦ •◦ . . . •◦ •◦ •◦ •◦ . . . •◦ •◦ Here the two closed points for the prime number p (a green one for the trivialsubgroup and a red one for the whole group G ) are glued together in the spectrumof the Burnside ring but remain distinct in the spectrum of the category of derivedMackey functors. A precise statement of the relationship between the two spacesis provided by Corollary 2.42 and additional examples are illustrated in 2.43–2.45.The category of derived Mackey functors thus lies cleanly between the equivariantstable homotopy category and the Burnside ring. It is a chromatic truncation of theformer and an equivariant refinement of the latter. This clarifies the two distinctfeatures noticed in [BS17] that distinguish Spec(SH( G ) c ) from Spec( A ( G )): theappearance of the chromatic filtration and the ungluing of the closed points. ∗ ∗ ∗ One feature common to the examples of linearization mentioned aboveSH (cid:32) D( Z ) and SH( k ) (cid:32) DMot( k )is that the linearized category can be interpreted as the derived category of modulesover a suitable Eilenberg–MacLane spectrum. Indeed,SH = D( S ) → D(H Z ) ∼ = D( Z ) and SH( k ) → D(H Z mot ) ∼ = DMot( k )where H Z is the ordinary Eilenberg–MacLane spectrum of the integers and H Z mot is the motivic ring spectrum representing motivic cohomology.We will apply a similar perspective to the linearization of SH( G ) and give an al-ternative description of Kaledin’s category of derived Mackey functors as the derivedcategory D(H Z G ) of a commutative equivariant ring spectrum H Z G := triv G (H Z )(see Definition 3.9). In fact, we will establish symmetric monoidal equivalencesbetween three categories: the derived category of H Z G -modules, the category ofH Z -valued spectral G -Mackey functors in the sense of Barwick [Bar17], and thecategory of derived G -Mackey functors in the sense of Kaledin [Kal11]. Theseequivalences will all arise from equivalences of the underlying symmetric monoidal ∞ -categories (see Corollary 4.11, Proposition 4.41 and Theorem 4.50). The com-parison with Kaledin’s constructions requires some technical care but the mainkey is Theorem 4.19 and Corollary 4.32 which establish that Barwick’s effective IRAKLI PATCHKORIA, BEREN SANDERS, AND CHRISTIAN WIMMER
Burnside ∞ -category is an ∞ -categorical localization of Kaledin’s Waldhausen typeconstruction on the category of finite G -sets.A common theme throughout the second half of the paper is the power of ∞ -categorical monadicity theorems (e.g. the Barr–Beck–Lurie Theorem) and thesymmetric monoidal universal characterization of the ∞ -category of G -spectra es-tablished by Gepner–Meier [GM20] and Robalo [Rob15]. As part of the analy-sis, we will obtain a symmetric monoidal equivalence between the ∞ -category of G -spectra and the ∞ -category of spectral G -Mackey functors (see Proposition 4.1and Remark 4.4) which may be of independent interest. ∗ ∗ ∗ The details of the construction of D(H Z G ) will be given in Section 3 and theequivalence D(H Z G ) ∼ = DMack( G ) will be established in Section 4. The actual com-putation of Spec(D(H Z G ) c ) appears in Section 2. Finally, we discuss in Section 5what goes wrong if one attempts to apply our method to compute the spectrum ofthe ordinary derived category of the abelian category of G -Mackey functors. As ob-served by Greenlees and Shipley [GS14], this amounts to studying modules over theEilenberg–MacLane G -spectrum associated to the Burnside ring Mackey functor:D( M ack ( G )) ∼ = D(H A G ). Ultimately things break down because the Eilenberg–MacLane spectra H A G do not behave well with respect to geometric fixed points.In fact, our explicit computations in Section 5 illustrate (and give a different per-spective on) Kaledin’s comments in [Kal11] about the pathological behavior of theordinary derived category of Mackey functors. More precisely, we show that the tar-get category of the geometric fixed point functor Φ H for D( M ack ( G )) depends onthe subgroup H ≤ G . This is quite different than what happens for equivariant cat-egories like SH( G ) and DMack( G ) where the geometric fixed point functors Φ H allland in the same category, namely the category associated to the trivial group. Thisenables us, in those examples, to pull back information from the well-understoodnonequivariant world. Acknowledgements :
We thank Rune Haugseng, Denis Nardin and Thomas Niko-laus for helpful conversations. We also thank EPFL and the University of Bonn fortheir hospitality and for providing the venues where this project first got off theground. 2.
Computation of the spectrum
We will assume familiarity with the description of SH( G ) and the computationof its spectrum from [BS17]. Following the approach in that work, we will begin bylisting the essential features of the category of derived Mackey functors which areneeded for the computation of its spectrum. The main point is that there is a well-behaved notion of geometric fixed point functor which aligns with the correspondingnotion for the equivariant stable homotopy category. The crucial feature whichleads to such well-behaved geometric fixed point functors is presented in (F) below.This feature of Kaledin’s category of derived Mackey functors is not shared by theordinary derived category of the abelian category of Mackey functors (as will bediscussed in Section 5).(A) For each finite group G , we have a tensor triangulated category D(H Z G )and an adjunction F G : SH( G ) (cid:29) D(H Z G ) : U G where the left adjoint F G is HE SPECTRUM OF DERIVED MACKEY FUNCTORS 5 a tensor triangulated functor. The tensor triangulated category D(H Z G ) isrigidly compactly generated by (cid:8) F G ( G/H + ) (cid:12)(cid:12) H ≤ G (cid:9) . Consequently, U G is conservative (=reflects isomorphisms) and F G preserves compact (=rigid)objects. Since F G preserves compact objects, it induces a tensor triangu-lated functor SH( G ) c → D(H Z G ) c and hence a continuous mapSpec(D(H Z G ) c ) Spec( F G ) −−−−−−→ Spec(SH( G ) c ) . (B) For G = { } the trivial group, D(H Z G ) ∼ = D(H Z ) and the adjunction in (A)is the usual adjunction F : SH (cid:29) D(H Z ) : U induced by the unit S → H Z of the Eilenberg–MacLane spectrum H Z .(C) For any homomorphism α : G → G (cid:48) of finite groups, there is an associ-ated tensor triangulated functor α ∗ : D(H Z G (cid:48) ) → D(H Z G ) which preservescoproducts, such that both squares inSH( G (cid:48) ) SH( G )D(H Z G (cid:48) ) D(H Z G ) F G (cid:48) α ∗ F G U G (cid:48) α ∗ U G commute up to natural isomorphism. As α ∗ : D(H Z G (cid:48) ) → D(H Z G ) isa tensor triangulated functor between rigidly compactly generated tensortriangulated categories, it preserves compact (=rigid) objects and henceinduces a continuous mapSpec(D(H Z G ) c ) Spec( α ∗ ) −−−−−−→ Spec(D(H Z G (cid:48) ) c ) . For a quotient α : G → G/N , we call infl
GG/N := α ∗ the inflation functorand for an inclusion α : H (cid:44) → G , we call res GH := α ∗ the restriction functor.Moreover, we set triv G := infl GG/G , regarded as a functor D(H Z ) → D(H Z G ).(D) For a composition G α −→ G (cid:48) β −→ G (cid:48)(cid:48) of group homomorphisms, we have anatural isomorphism ( β ◦ α ) ∗ ∼ = α ∗ ◦ β ∗ . For example, res GK ∼ = res HK ◦ res GH for K ≤ H ≤ G .(E) For any H ≤ G , the restriction functor res GH : D(H Z G ) → D(H Z H ) has aleft adjoint ind GH : D(H Z H ) → D(H Z G ).(F) For any normal subgroup N (cid:2) G , the compositeD(H Z G/N ) infl GG/N −−−−−→
D(H Z G ) → D(H Z G ) / Loc ⊗ (cid:104) F G ( G/H + ) | H (cid:54)⊇ N (cid:105) is an equivalence. In other words, D(H Z G/N ) is a particular finite localiza-tion of D(H Z G ), obtained by killing the generators of D(H Z G ) associatedto those subgroups which do not contain N . Define the “geometric fixedpoint” functor (cid:101) Φ N,G : D(H Z G ) → D(H Z G/N ) to be the compositeD(H Z G ) → D(H Z G ) / Loc ⊗ (cid:104) F G ( G/H + ) | H (cid:54)⊇ N (cid:105) ∼ = D(H Z G/N ) . By construction, it is split by inflation: (cid:101) Φ N,G ◦ infl GG/N ∼ = Id G/N . In partic-ular, taking N = G , we have the geometric fixed point functorΦ G : D(H Z G ) (cid:101) Φ G,G −−−→
D(H Z G/G ) ∼ = D(H Z )which is (up to equivalence) the localization obtained by killing all genera-tors except F G ( G/G + ) = D(H Z G ) . IRAKLI PATCHKORIA, BEREN SANDERS, AND CHRISTIAN WIMMER (G) For H ≤ G , we define Φ H,G : D(H Z G ) → D(H Z ) as the compositeD(H Z G ) res GH −−−→ D(H Z H ) Φ H −−→ D(H Z ) . These are tensor triangulated functors (preserving compact objects). Inparticular, Φ
H,G induces a continuous mapSpec(D(H Z ) c ) Spec(Φ
H,G ) −−−−−−−→ Spec(D(H Z G ) c ) . For each prime ideal p ∈ Spec Z ∼ = Spec(D(H Z ) c ) and H ≤ G , define P G ( H, p ) ∈ Spec(D(H Z G ) c ) by P G ( H, p ) := Spec(Φ H,G )( p ) = (Φ H,G ) − ( p ) . (H) For an inner automorphism c g := ( − ) g : G ∼ −→ G , the induced functor c ∗ g : D(H Z G ) ∼ −→ D(H Z G ) is naturally isomorphic to the identity functor. Itthen follows from (C) that for any subgroup H ≤ G , the left-hand trianglein D(H Z H g )D(H Z G ) D(H Z )D(H Z H ) c ∗ g ∼ Φ Hg res GHg res GH Φ H commutes up to natural isomorphism. The right-hand triangle also com-mutes up to natural isomorphism since c ∗ g ( F H g ( H g /K g )) ∼ = F H ( H/K ) forany K ≤ H ≤ G (again by (C)). Thus the functor Φ H,G : D(H Z G ) → D(H Z )only depends, up to natural isomorphism, on the G -conjugacy class ofthe subgroup H . That is, Φ H g ,G ∼ = Φ H,G for any g ∈ G . As natu-rally isomorphic functors induces the same map on spectra, it follows that P G ( H, p ) = P G ( K, p ) if H ∼ G K .(I) Finally, hinting at the reasons behind our choice of notation, U G ( ) ∼ =triv G (H Z ) as commutative monoids in SH( G ), where the right-hand side isthe Eilenberg–MacLane spectrum of the integers regarded as a G -spectrumwith trivial action. This is the most “explicit” fact about our categoriesD(H Z G ) that we will need.2.1. Remark.
The details of the construction of D(H Z G ) and verification of theabove facts (A) through (I) will be give in Section 3. For the rest of the presentsection, we will use the above properties as a black-box in order to compute thespectrum of D(H Z G ) c and describe its relationship with the spectrum of SH( G ) c .2.2. Remark.
The “geometric fixed point” functor (cid:101) Φ N,G : D(H Z G ) → D(H Z G/N )in (F) is nothing but the finite localization associated to the thick tensor idealof compact objects generated by the F G ( G/H + ) for H (cid:54)⊇ N . As such it has anassociated idempotent triangle e F [ (cid:54)⊇ N ] → → f F [ (cid:54)⊇ N ] → Σ e F [ (cid:54)⊇ N ] in D(H Z G ) andcan be conveniently understood simply as tensoring with the right idempotent: f F [ (cid:54)⊇ N ] ⊗ − : D(H Z G ) → f F [ (cid:54)⊇ N ] ⊗ D(H Z G ) ∼ = D(H Z G/N ) . Moreover, the latter equivalence is explicitly given by f F [ (cid:54)⊇ N ] ⊗ D(H Z G ) (cid:44) → D(H Z G ) ( − ) N −−−→ D(H Z G/N ) HE SPECTRUM OF DERIVED MACKEY FUNCTORS 7 where ( − ) N denotes the right adjoint of inflation (which exists since inflation pre-serves coproducts by assumption). In other words, it follows formally from thedefinition (F) that the geometric fixed point functor is given as(2.3) (cid:101) Φ N,G ( X ) ∼ = ( f F [ (cid:54)⊇ N ] ⊗ X ) N . For further discussion of these tensor idempotents, see [BS17, Section 5] and [BF11].2.4.
Lemma.
For any N (cid:2) G , the diagram (2.5) SH( G ) SH( G/N )D(H Z G ) D(H Z G/N ) (cid:101) Φ N,G F G F G/N (cid:101) Φ N,G commutes up to isomorphism.Proof.
Let f ∈ SH( G ) denote the right idempotent for the finite localization (cid:101) Φ N,G :SH( G ) → SH(
G/N ). Applying [BS17, Proposition 5.11] to the functor F G :SH( G ) → D(H Z G ) and recalling the definitions in (F) and [BS17, (H)], we seethat F G ( f ) ∈ D(H Z G ) is the right idempotent for the finite localization (cid:101) Φ N,G :D(H Z G ) → D(H Z G/N ). Moreover, the middle square ofSH(
G/N ) SH( G ) f ⊗ SH( G ) SH( G/N )D(H Z G/N ) D(H Z G ) F G ( f ) ⊗ D(H Z G ) D(H Z G/N ) infl GG/N F G/N (cid:101) Φ N,G F G F G ∼ = F G/N infl
GG/N (cid:101) Φ N,G ∼ = evidently commutes. The left-hand square commutes by (C) and the horizontalcomposites are the identity. It then follows formally that the right-hand squarealso commutes. (cid:3) Remark.
It follows from Lemma 2.4, (B), (C) and the definitions in (G) that(2.7) SH( G ) SHD(H Z G ) D(H Z ) Φ H F G F Φ H commutes up to isomorphism for any H ≤ G .2.8. Lemma.
For any N (cid:2) G , we have Φ G ∼ = Φ G/N ◦ (cid:101) Φ N,G .Proof.
Consider the two localizing ⊗ -ideals L ⊂ L of D(H Z G ) given by L := Loc ⊗ (cid:104) F G ( G/H + ) | H (cid:54)⊇ N (cid:105) and L := Loc ⊗ (cid:104) F G ( G/H + ) | H (cid:12) G (cid:105) . As Verdier quotients can be “nested”, the localization(2.9) D(H Z G ) → D(H Z G ) / L coincides with the composite(2.10) D(H Z G ) q −→ D(H Z G ) / L → (D(H Z G ) / L ) /q ( L ) . IRAKLI PATCHKORIA, BEREN SANDERS, AND CHRISTIAN WIMMER
Note that D(H Z G ) → D(H Z G ) / L ∼ = D(H Z ) is Φ G while D(H Z G ) → D(H Z G ) / L ∼ =D(H Z G/N ) is (cid:101) Φ N,G . We just need to show that the quotientD(H Z G/N ) → D(H Z G/N ) / (cid:101) Φ N,G ( L )is nothing but the finite localization defining Φ G/N : D(H Z G/N ) → D(H Z ). Now (cid:101) Φ N,G ( L ) is the localizing tensor-ideal generated by (cid:8) (cid:101) Φ N,G ( F G ( G/H + )) (cid:12)(cid:12) H (cid:12) G (cid:9) (see [Ver96, Prop. 2.31] and [Nee01, Cor. 3.2.11]). This coincides with the localizingtensor-ideal generated by (cid:8) F G/N (( G/N ) / ( H/N ) + ) (cid:12)(cid:12) N ≤ H (cid:12) G (cid:9) since (cid:101) Φ N,G ( F G ( G/H + )) ∼ = F G/N ( (cid:101) Φ N,G ( G/H + )) ∼ = (cid:40) H (cid:54)⊇ NF G/N ( G/H + ) if H ⊇ N by Lemma 2.4 and [LMS86, Cor. II.9.9]. (cid:3) Lemma.
For N ≤ K ≤ G with N (cid:2) G , we have (cid:101) Φ N,K ◦ res GK ∼ = res G/NK/N ◦ (cid:101) Φ N,G .Proof.
Let f F [ (cid:54)⊇ N ] ,G denote the right idempotent in D(H Z G ) for (cid:101) Φ N,G as in Re-mark 2.2. By [BS17, Prop. 5.11], its restriction res GK ( f F [ (cid:54)⊇ N ] ,G ) is the right idem-potent for a finite localization of D(H Z K ), namely the localization associated tothe compact thick tensor ideal generated by (cid:8) res GK ( F G ( G/H + )) (cid:12)(cid:12) H ≤ G, H (cid:54)⊇ N (cid:9) .Using (C) and the Mackey formula, this coincides with the thick tensor idealgenerated by (cid:8) F K ( K/L + ) (cid:12)(cid:12) L ≤ K, L (cid:54)⊇ N (cid:9) . In other words, res GK ( f F [ (cid:54)⊇ N ] ,G ) ∼ = f F [ (cid:54)⊇ N ] ,K is the idempotent in D(H Z K ) for (cid:101) Φ N,K . Now, by (D), infl
KK/N ◦ res G/NK/N ∼ =res GK ◦ infl GG/N . Applying this equation to (cid:101) Φ N,G ( X ) and post-composing by (cid:101) Φ N,K we obtainres
G/NK/N ( (cid:101) Φ N,G ( X )) ∼ = (cid:101) Φ N,K (res GK (infl GG/N ( (cid:101) Φ N,G ( X )))) ∼ = λ N,K ( f F [ (cid:54)⊇ N ] ,K ⊗ res GK (infl GG/N ( (cid:101) Φ N,G ( X )))) ∼ = λ N,K (res GK ( f F [ (cid:54)⊇ N ] ,G ) ⊗ res GK (infl GG/N ( (cid:101) Φ N,G ( X )))) ∼ = λ N,K (res GK ( f F [ (cid:54)⊇ N ] ,G ⊗ infl GG/N ( (cid:101) Φ N,G ( X )))) ∼ = λ N,K (res GK ( f F [ (cid:54)⊇ N ] ,G ⊗ X )) ∼ = λ N,K ( f F [ (cid:54)⊇ N ] ,K ⊗ res GK ( X )) ∼ = (cid:101) Φ N,K (res GK ( X ))where λ N,K denotes the right adjoint of infl
KK/N . Here we have used that f F [ (cid:54)⊇ N ] ⊗ infl GG/N ( (cid:101) Φ N,G ( X )) ∼ = f F [ (cid:54)⊇ N ] ⊗ X for any X in D(H Z G ). This follows from the fact that f F [ (cid:54)⊇ N ] ⊗ D(H Z G ) (cid:44) → D(H Z G ) ( − ) N −−−→ D(H Z G/N )is an equivalence with quasi-inverseD(H Z G/N ) infl GG/N −−−−−→
D(H Z G ) → f F [ (cid:54)⊇ N ] ⊗ D(H Z G )as explained in Remark 2.2. (cid:3) HE SPECTRUM OF DERIVED MACKEY FUNCTORS 9
Lemma.
Let K ≤ H ≤ G . The map Spec(res GH ) : Spec(D(H Z H ) c ) → Spec(D(H Z G ) c ) sends P H ( K, p ) to P G ( K, p ) .Proof. This is immediate from the definition of Φ
K,H and Φ
K,G (see (G)) and therelation res GK ∼ = res HK ◦ res GH (see (D)). (cid:3) Proposition.
For any H ≤ G , the image of the map Spec(res GH ) : Spec(D(H Z H ) c ) → Spec(D(H Z G ) c ) coincides with supp( F G ( G/H + )) .Proof. Since the restriction functor res GH preserves coproducts (C), its left adjointind GH (E) necessarily preserves compact objects (see [Nee96, Thm. 5.1] for instance).Hence the adjunction ind GH (cid:97) res GH restricts to an adjunctionind GH : D(H Z H ) c (cid:29) D(H Z G ) c : res GH on the subcategories of compact objects. Moreover, as the category D(H Z G ) isrigidly compactly generated (A), its subcategory of compact objects D(H Z G ) c is a rigid category (i.e. all objects are dualizable) so the duality D provides an equiva-lence between D(H Z G ) c and its opposite category. It follows that D ind GH D is right adjoint to res GH on the categories of compact objects. We can then invoke [Bal18,Thm. 1.7] to conclude that the image of Spec(res GH ) equals supp( D ind GH D ). Thiscoincides with supp(ind GH ( )) since D = and supp( DX ) = supp( X ) (by [Bal07,Prop. 2.7] for instance). Finally, by (C) we have res GH ◦ U G ∼ = U H ◦ res GH . Takingleft adjoints, ind GH ◦ F H ∼ = F G ◦ ind GH so ind GH ( ) ∼ = ind GH ( F H ( )) ∼ = F G (ind GH ( )) ∼ = F G ( G/H + ). (cid:3) Lemma.
Let N ≤ K ≤ G with N (cid:2) G . The map Spec(infl
GG/N ) : Spec(D(H Z G ) c ) → Spec(D(H Z G/N ) c ) sends P G ( K, p ) to P G/N ( K/N, p ) .Proof. Unravelling the definitions and factoring the composite K → G → G/N as K → K/N → G/N , property (D) reduces our claim to the assertion that Φ G ◦ infl GG/N ∼ = Φ G/N . This follows from Lemma 2.8 since infl
GG/N splits (cid:101) Φ N,G . (cid:3) Lemma.
Let N ≤ K ≤ G with N (cid:2) G . The map Spec( (cid:101) Φ N,G ) : Spec(D(H Z G/N ) c ) → Spec(D(H Z G ) c ) sends P G/N ( K/N, p ) to P G ( K, p ) .Proof. This follows from Lemma 2.8 and Lemma 2.11 and the definitions. (cid:3)
Corollary.
Let N ≤ K ≤ G with N (cid:2) G . Then P G ( K, p ) ⊆ P G ( G, q ) if andonly if P G/N ( K/N, p ) ⊆ P G/N ( G/N, q ) .Proof. The induced maps on spectra preserve inclusions. Thus ( ⇒ ) follows fromLemma 2.14 and ( ⇐ ) follows from Lemma 2.15. (cid:3) Remark.
Recall the prime ideals of the nonequivariant stable homotopy cat-egory of finite spectra SH c . They are of the form C p,n where p is a prime numberand 1 ≤ n ≤ ∞ is a “chromatic” number. Recall that C p, = SH c, tor =: C , is thesubcategory of finite torsion spectra, independently of p . It is the unique genericpoint of Spec(SH c ), while the points C p, ∞ are the closed points. Similarly recallthat the prime ideals of SH( G ) c are of the form P ( H, p, n ) for H ≤ G , p a primenumber and 1 ≤ n ≤ ∞ . Again, P ( H, p,
1) is independent of p and sometimeswritten P ( H,
1) or P ( H, , Proposition.
The map
Spec(D(H Z ) c ) Spec( F ) −−−−−→ Spec(SH c ) sends a prime ideal p ∈ Spec Z ∼ = Spec(D(H Z ) c ) to C p, ∞ if p = ( p ) and to C , if p = (0) .Proof. It follows from the Hurewicz theorem that the functor H Z ∧ − : SH → SH isconservative on compact objects. That is, if X ∈ SH c then H Z ∧ X = 0 if and only if X = 0. As a corollary, the functor F : SH c → D(H Z ) c is conservative: if F ( X ) = 0then H Z ∧ X ∼ = U F ( X ) = 0, hence X = 0. Thus by [Bal18, Thm. 1.2], the inducedmap on spectra ϕ := Spec( F ) : Spec(D(H Z ) c ) → Spec(SH c ) hits all the closedpoints C p, ∞ of Spec(SH c ). Now, the unit map S → H Z induces an isomorphism ofrings π ( S ) → π (H Z ) which, under the usual identifications of both sides with thering of integers, is just the identity. This is precisely the map on endomorphismrings End SH ( ) → End
D(H Z ) ( ) induced by the functor F : SH c → D(H Z ) c . Sincethe comparison map ρ : Spec( K ) → Spec(End K ( )) of [Bal10a, Section 5] is natural,we have a commutative diagramSpec(D(H Z ) c ) Spec(SH c )Spec( π (H Z )) Spec( π ( S ))Spec( Z ) Spec( Z ) ρ ∼ = ϕ ρ ∼ = ∼ = ∼ = and the left-hand comparison map is just the usual identification of the spectrumof D(H Z ) c ∼ = D( Z ) c with the spectrum of the integers. Thus the top map ϕ sendsthe prime (0) in Spec(D(H Z ) c ) to a point in the fiber (with respect to ρ ) of (0) inSpec(SH c ). There is only one such point in the fiber, namely C , = SH c, tor . Onthe other hand, the prime ( p ) in Spec(D(H Z ) c ) maps to a point in the fiber of ( p )in Spec(SH c ). Since all the closed points of Spec(SH c ) are hit, the closed point C p, ∞ in the fiber over ( p ) must be hit. Since ( p ) in Spec(D(H Z ) c ) is the only pointmapping to the fiber over ( p ), the only possibility is that it maps to the closed point C p, ∞ . (cid:3) Corollary.
For any H ≤ G , the map Spec(D(H Z G ) c ) Spec( F G ) −−−−−−→ Spec(SH( G ) c ) sends P G ( H, p ) to P G ( H, p, ∞ ) and P G ( H, to P G ( H, , .Proof. This follows immediately from (2.7) and Proposition 2.18. (cid:3)
Remark.
By formal nonsense, the adjunction in (A) provides an isomorphismEnd
D(H Z G ) ( ) (cid:39) π ( U G ( )) HE SPECTRUM OF DERIVED MACKEY FUNCTORS 11 of commutative rings, where U G ( ) is regarded as a commutative monoid in SH( G )via the induced lax monoidal structure on the functor U G . Moreover, the map onendomorphism rings π ( ) = End SH( G ) ( ) → End
D(H Z G ) ( ) (cid:39) π ( U G ( )) inducedby the functor F G is just post-composition by the unit of U G ( ). Property (I)asserts that we have an isomorphism U G ( ) (cid:39) triv G (H Z ) of commutative monoidsin SH( G ). The map on endomorphism rings can then be identified with the map π ((triv G ( S ) G ) → π (H Z ∧ triv G ( S ) G )induced by the unit of the Eilenberg–MacLane spectrum H Z . This is an isomor-phism by the Hurewicz theorem since the spectrum triv G ( S ) G is connective (beinga wedge sum of suspension spectra by the tom Dieck splitting theorem). In thisway, we have an identification A ( G ) (cid:39) End
SH( G ) ( ) (cid:39) End
D(H Z G ) ( ) between theBurnside ring and the endomorphism ring of the unit in D(H Z G ).2.21. Corollary.
The comparison map ρ : Spec(D(H Z G ) c ) → Spec( A ( G )) sends P ( H, p ) to p ( H, p ) ∈ Spec( A ( G )) and sends P ( H, to p ( H, ∈ Spec( A ( G )) .Proof. Naturality of the comparison map gives a commutative diagramSpec(D(H Z G ) c ) Spec(SH( G ) c )Spec(End D(H Z G ) ( )) Spec(End SH( G ) ( ))Spec( A ( G )) Spec( A ( G )) ρ Spec( F G ) ρ ∼ = ∼ = ∼ = and the claim follows from Corollary 2.19 and [BS17, Proposition 6.7] (see Re-mark 2.20). The spectrum of the Burnside ring is recalled in [BS17, Section 3]. (cid:3) Theorem.
Let G be a finite group. Every prime ideal of D(H Z G ) c is of theform P ( H, p ) for some H ≤ G and p ∈ Spec Z . Moreover, the prime P ( H, p ) iscompletely determined by the G -conjugacy class of H and the prime ideal p . Thatis, P ( H, p ) = P ( K, q ) if and only if H ∼ G K and p = q .Proof. We will prove the theorem by induction on the order | G | . By construc-tion (F), the geometric fixed points Φ G : D(H Z G ) → D(H Z ) is a finite localization.Hence by the Neeman–Thomason localization theorem [Nee92, Thm. 2.1], the in-duced map Spec(D(H Z ) c ) Spec(Φ G ) −−−−−−→ Spec(D(H Z G ) c )is a homeomorphism onto the subset V ( F G ( G/H + ) | H (cid:12) G ) ⊆ Spec(D(H Z G ) c )consisting of those primes P ∈ Spec(D(H Z G ) c ) which contain F G ( G/H + ) for all H (cid:12) G . In particular, (cid:8) P ( G, p ) (cid:12)(cid:12) p ∈ Spec Z (cid:9) = V ( F G ( G/H + ) | H (cid:12) G ) . The complement is thus given bySpec(D(H Z G ) c ) \ (cid:8) P ( G, p ) (cid:12)(cid:12) p ∈ Spec Z (cid:9) = (cid:91) H (cid:12) G supp( F G ( G/H + ))= (cid:91) H (cid:12) G im(Spec(res GH )) where the last equality is given by Proposition 2.13. By the inductive hypothesis,every prime in D(H Z H ) (for H (cid:12) G ) is of the form P H ( K, p ) for some K ≤ H and p ∈ Spec( Z ). By Lemma 2.12, it gets mapped to P G ( K, p ) under Spec(res GH ).This completes the proof that every prime is of the required form. The uniquenessstatement follows from (H), Corollary 2.19 and [BS17, Theorem 4.14]. (cid:3) Remark.
In other words, Spec(D(H Z G ) c ) is covered by copies of Spec( Z ), onecopy for each conjugacy class of subgroups H ≤ G . These copies are disjoint , so asa set Spec(D(H Z G ) c ) is just the disjoint union of these copies of Spec( Z ). However,the copies of Spec( Z ) are related by the topology of Spec(D(H Z G ) c ). Our nexttask is to determine this topology. This will follow from a series of reductionsculminating in Theorem 2.36.2.24. Remark.
Understanding the topology boils down to understanding the in-clusions among the primes (i.e. understanding the irreducible closed sets) and thecomparison map to the spectrum of the Burnside ring (see Remark 2.20 and Corol-lary 2.21) greatly restricts the possible inclusions (Lemma 2.26 below).2.25.
Remark.
A subgroup H ≤ G is said to be a p -subnormal subgroup if thereexists a subnormal tower from H to G all of whose subquotients have order p . Werefer the reader to [BS17, Section 3] for more details.2.26. Lemma.
Let
K, H ≤ G be two subgroups and p , q ∈ Spec Z . Suppose P G ( K, p ) ⊆ P G ( H, q ) in D(H Z G ) c . Then:(a) If p = (0) then q = (0) and K ∼ G H (in which case the inclusion is anequality).(b) If p = ( p ) then K is G -conjugate to a p -subnormal subgroup of H and q = ( p ) or (0) .Proof. This follows from Corollary 2.19 and [BS17, Proposition 6.9] together withCorollary 2.21 and what is known about the inclusions among the prime ideals ofthe Burnside ring (e.g. from [BS17, Theorem 3.6] or the original [Dre69]). (cid:3)
Remark.
On the other hand, we know that P G ( H, p ) ⊆ P G ( H,
0) for anysubgroup H ≤ G since the map Spec(Φ H ) : Spec(D(H Z ) c ) → Spec(D(H Z G ) c ) pre-serves inclusions and the identification Spec(D(H Z ) c ) ∼ = Spec( Z ) reverses inclu-sions. Armed with Lemma 2.26 and this observation, all that remains to deter-mine the topology of Spec(D(H Z G ) c ) is to understand when we have an inclusion P G ( K, p ) ⊆ P G ( H, p ) when K is G -conjugate to a p -subnormal subgroup of H . Wewill show that this inclusion always holds (see Proposition 2.35 and Theorem 2.36below). To prove this we will use a series of reductions which ultimately reducesthe problem to the case G = C p , the cyclic group of order p .2.28. Remark.
The following result explains how vanishing of the Tate constructionrelates to the geometry of the Balmer spectrum.2.29.
Proposition.
Let T be a rigidly-compactly generated tensor triangulated cat-egory and let K := T c denote its subcategory of compact-rigid objects. For anyThomason subset Y ⊂ Spec( K ) , let K Y = (cid:8) x ∈ K (cid:12)(cid:12) supp( x ) ⊆ Y (cid:9) be the corre-sponding thick tensor-ideal of K , and let e Y → → f Y → Σ e Y be the idempotenttriangle in T for the associated finite localization. For any object x ∈ K , the fol-lowing are equivalent: HE SPECTRUM OF DERIVED MACKEY FUNCTORS 13 (a) The Tate construction t Y ( x ) = 0 vanishes.(b) The exact triangle e Y ⊗ x → x → f Y ⊗ x → Σ e Y ⊗ x splits; that is, f Y ⊗ x → Σ e Y ⊗ x is the zero map.(c) The support of x is a disjoint union of closed sets supp( x ) = Z (cid:116) Z with Z ⊆ Y and Z ∩ Y = ∅ .(d) supp( x ) ∩ (Spec( K ) \ Y ) is Thomason.Proof. Recall that t Y : T → T is defined by t Y := [ f Y , Σ e Y ⊗ − ] where [ − , − ]denotes the internal hom in T (see [BS17, Definition 5.7] or [Gre01]).(a) ⇒ (b): The kernel of t Y : T → T is a thick subcategory of T which is closedunder tensoring with compact-rigid objects. Thus, t Y ( x ) = 0 iff t Y ( Dx ⊗ x ) =[ f Y ⊗ x, Σ e Y ⊗ x ] = 0. This implies (b) since Hom T ( a, b ) ∼ = Hom T ( , [ a, b ]).(b) ⇒ (c): If the exact triangle splits then x (cid:39) ( e Y ⊗ x ) ⊕ ( f Y ⊗ x ). In particular,the objects e Y ⊗ x and f Y ⊗ x are both contained in K (i.e. are compact). Thendefining Z := supp( e Y ⊗ x ) and Z := supp( f Y ⊗ x ), we have supp( x ) = Z (cid:116) Z ,a disjoint union of closed sets. Finally, recall that Loc (cid:104) K Y (cid:105) = e Y ⊗ T . Thus, forany c ∈ K , supp( c ) ⊆ Y is equivalent to f Y ⊗ c = 0. Similarly, if supp( c ) ∩ Y = ∅ then for any d ∈ K Y , d ⊗ c = 0; hence e Y ⊗ c = 0. Conversely, if e Y ⊗ c = 0 then d ⊗ c (cid:39) d ⊗ e Y ⊗ c = 0 for any d ∈ K Y . It follows that supp( c ) ∩ Y = ∅ is equivalentto e Y ⊗ c = 0. This proves (c) by considering c := e Y ⊗ x and c := f Y ⊗ x .(c) ⇒ (d): Observe that Spec( K ) \ Z = Z ∪ (Spec( K ) \ supp( x )) is a unionof two quasi-compact subsets of Spec( K ), and hence is itself quasi-compact. Theclosed set Z = supp( x ) ∩ (Spec( K ) \ Y ) thus has quasi-compact complement, andhence is a Thomason closed subset.(d) ⇒ (a): The hypothesis implies thatsupp( x ) = (supp( x ) ∩ Y ) (cid:116) (supp( x ) ∩ (Spec( K ) \ Y ))is a decomposition into disjoint Thomason sets. Then by the generalized Carlsontheorem [Bal07, Theorem 2.11] we have x (cid:39) a ⊕ b for two objects a, b ∈ K withsupp( a ) ⊆ Y and supp( b ) ∩ Y = ∅ . Then t Y ( x ) = t Y ( a ) ⊕ t Y ( b ) vanishes since f Y ⊗ Da = 0 and e Y ⊗ b = 0. (cid:3) Proposition.
Let e G → → f G → Σ e G be the idempotent triangle in SH( G ) associated to the trivial family of subgroups. (That is, e G = EG + and f G = (cid:101) EG .)For G = C p , this triangle does not split after tensoring with triv G (H F p ) ; that is,the map triv G (H F p ) ⊗ f G → triv G (H F p ) ⊗ Σ e G in SH( G ) is not the zero map.Proof. For notational simplicity, let H := triv G (H F p ). If the map H → H ⊗ f G hasa section in SH( G ), then the map of G -Mackey functors π (H) → π (H ⊗ f G ) wouldhave a section:(2.31) π (H ⊗ f G ) π (H) π (H ⊗ f G ) . ∃ σ id4 IRAKLI PATCHKORIA, BEREN SANDERS, AND CHRISTIAN WIMMER The Mackey functor π (H) can be identified with A ⊗ F p where A denotes theBurnside ring G -Mackey functor G/H (cid:55)→ A ( H ). On the other hand, for G = C p ,the Mackey functor π (H ⊗ f G ) satisfies π (H ⊗ f G )( G/G ) = π G (H ⊗ f G ) ∼ = π (Φ G (H)) ∼ = π (H F p ) ∼ = F p and π (H ⊗ f G )( G/ { } ) = π { } (H ⊗ f G ) ∼ = π (H F p ⊗ res G { } ( f G )) = 0 . Now A ( C p ) is the ring Z [ X ] / ( X − pX ) with restriction A ( C p ) → A ( { } ) = Z given by X (cid:55)→ p and with transfer Z = A ( { } ) → A ( C p ) given by 1 (cid:55)→ X . Asplitting (2.31) of C p -Mackey functors would thus look like F p F p [ X ] / ( X ) F p F p σ where the vertical maps represent restriction and transfer. Since the right-handmap commutes with transfers, it must map X to 0 in F p . Hence, in order for thecomposite to be the identity, the left-hand map σ must map 1 ∈ F p to an element ofthe form 1+ mX ∈ F p [ X ] / ( X ). Since the middle restriction map sends X to 0 ∈ F p ,the element 1 + mX is mapped to 1 ∈ F p . On the other hand, since σ commuteswith restrictions, 1 + mX must be mapped to 0 ∈ F p . This is a contradiction. (cid:3) Proposition.
Consider G = C p . Then P (1 , p ) ⊆ P ( C p , p ) in D(H Z C p ) .Proof. Recall that if F : K → L is a tensor triangulated functor and ϕ :=Spec( F ) : Spec( L ) → Spec( K ) is the induced map on spectra, then for any x ∈ K ,supp L ( F ( x )) = ϕ − (supp K ( x )). Then consider the mod- p Moore spectrum M ( p ).Its support in Spec(SH c ) is precisely { C p, } = { C p,n | ≤ n ≤ ∞} . We can thenpass to the G -equivariant stable homotopy category by giving the mod- p Moorespectrum a trivial G -action. By [BS17, Cor. 4.6], the support of triv G ( M ( p )) inSH( G ) c is (cid:8) P ( H, p, n ) (cid:12)(cid:12) H ≤ G, ≤ n ≤ ∞ (cid:9) ⊂ Spec(SH( G ) c ). Finally, using F G : SH( G ) c → D(H Z G ) c , we can consider F G (triv G ( M ( p ))) in D(H Z G ) c . ByCorollary 2.19, its support is (cid:8) P ( H, p ) (cid:12)(cid:12) H ≤ G (cid:9) . For G = C p this is precisely twopoints: supp( F G (triv G ( M ( p ))) = { P (1 , p ) , P ( C p , p ) } ⊂ Spec(D(H Z G ) c ) . We know from Lemma 2.26 that P (1 , p ) is a closed point and that P ( G, p ) is eithera closed point or else { P ( G, p ) } = { P (1 , p ) , P ( G, p ) } . We claim that the latter holdsi.e. that P (1 , p ) is contained in the closure of P ( G, p ). To this end, let e G → → f G → Σ e G be the idempotent triangle in SH( G ) associated to the Thomason closed subsetsupp( G + ) = (cid:8) P G (1 , p, n ) (cid:12)(cid:12) all p, n (cid:9) (that is, all the primes for the trivial subgroup).By [BS17, Prop. 5.11], F G ( e G ) → → F G ( f G ) → Σ F G ( e G ) is the idempotenttriangle in D(H Z G ) associated to Y := supp( F G ( G + )) = (cid:8) P (1 , p ) (cid:12)(cid:12) all p (cid:9) . Notethat if P (1 , p ) (cid:54)⊆ P ( G, p ) thensupp( F G (triv G ( M ( p )))) = { P (1 , p ) } (cid:116) { P ( G, p ) } = { P (1 , p ) } (cid:116) { P ( G, p ) } is a disjoint union of closed sets Z (cid:116) Z with Z ⊆ Y and Z ∩ Y = ∅ . InvokingProposition 2.29 and letting Z := supp( F G (triv G ( M ( p )))) = { P (1 , p ) , P ( G, p ) } , we HE SPECTRUM OF DERIVED MACKEY FUNCTORS 15 see that P (1 , p ) ⊆ P ( G, p ) if and only if Z = { P ( G, p ) } if and only if Z is irreducibleif and only if Z is connected if and only if the idempotent triangle F G ( e G ) → → F G ( f G ) → Σ F G ( e G )does not split after tensoring with F G (triv G ( M ( p )). Suppose for a contradictionthat it did split. Then passing back to SH( G ) by applying U G and using that U G F G = triv G (H Z ) ⊗ − by (I) and the projection formula [BDS16, Prop. 2.15], itwould follow that the sequence e G → → f G → Σ e G splits after tensoring withtriv G (H Z ) ⊗ triv G ( M ( p )) (cid:39) triv G (H Z ⊗ M ( p )) (cid:39) triv G (H F p ) which contradictsProposition 2.30. (cid:3) Corollary.
Let K be a subgroup of a finite p -group G . Then P G ( K, p ) ⊆ P G ( G, p ) in D(H Z G ) .Proof. As G is a p -group, there is a subnormal tower K = K (cid:1) p · · · (cid:1) p K t = G where each subquotient has order p . By Proposition 2.32, P K i /K i − (1 , p ) is con-tained in P K i /K i − ( K i /K i − , p ) for all i = 1 , . . . , t . Hence by Corollary 2.16, P K i ( K i − , p ) ⊆ P K i ( K i , p ) for all i = 1 , . . . , t . Hence P G ( K i − , p ) ⊆ P G ( K i , p )for all i = 1 , . . . , t by Lemma 2.12, so that P G ( K, p ) ⊆ P G ( G, p ). (cid:3) Proposition. If K is a p -subnormal subgroup of a finite group G , then P G ( K, p ) ⊆ P G ( G, p ) in D(H Z G ) .Proof. The fact that K is p -subnormal in G implies that O p ( G ) ⊆ K (see [BS17,Lem. 3.3]). By Corollary 2.16, P G ( K, p ) ⊆ P ( G, p ) if and only if P G/ O p ( G ) ( K/ O p ( G ) , p ) ⊆ P G/ O p ( G ) ( G/ O p ( G ) , p ) . As G/ O p ( G ) is a p -group, the claim follows from Corollary 2.33. (cid:3) Proposition.
Let
K, H be subgroups of a finite group G . If K is G -conjugateto a p -subnormal subgroup of H then P G ( K, p ) ⊆ P G ( H, p ) in D(H Z G ) .Proof. By assumption K ∼ G K (cid:48) where K (cid:48) ≤ H is p -subnormal. By Proposi-tion 2.34, P H ( K (cid:48) , p ) ⊆ P H ( H, p ) in D(H Z H ). It then follows from Lemma 2.12 that P G ( K, p ) = P G ( K (cid:48) , p ) ⊆ P G ( H, p ) in D(H Z G ). (cid:3) Theorem.
Let G be a finite group, let K, H ≤ G be subgroups and let p , q ∈ Spec Z . Then P G ( K, p ) ⊆ P G ( H, q ) if and only if either(a) p = ( p ) , K is G -conjugate to a p -subnormal subgroup of H , and q = ( p ) or (0) ; or(b) p = (0) , q = (0) and K ∼ G H (in which case the primes are equal).Proof. This follows from Lemma 2.26, Remark 2.27 and Proposition 2.35. (cid:3)
Remark.
The irreducible closed subsets of Spec(D(H Z G ) c ) can thus be com-pletely described as follows: • { P ( H, p ) } = (cid:8) P ( K, p ) (cid:12)(cid:12) K a p -subnormal subgroup of H (cid:9) ; and • { P ( H, } = { P ( H, } ∪ (cid:83) p prime { P ( H, p ) } .These irreducible closed subsets completely determine the topology:2.38. Proposition.
The space
Spec(D(H Z G ) c ) is noetherian. Consequently, theclosed subsets are precisely the finite unions of irreducible closed sets (equivalently,the closures of finite subsets). Moreover, the Thomason subsets are just the special-ization closed subsets, that is, arbitrary unions of closed sets. Proof.
By Theorem 2.22, the space Spec(D(H Z G ) c ) is covered by the images of thecontinuous maps Spec(Φ H,G ) : Spec(D(H Z ) c ) → Spec(D(H Z G ) c ) for H ≤ G . Theclaim that Spec(D(H Z G ) c ) is noetherian follows from the fact that Spec(D(H Z ) c ) ∼ =Spec( Z ) is noetherian, that a continuous image of a noetherian space is noetherian,and that any space covered by finitely many noetherian subspaces is noetherian.For the second claim just note that every subspace of a noetherian space is noe-therian and that a noetherian space has finitely many irreducible components. Thedescription of the Thomason subsets is immediate from the definition as every sub-space of a noetherian space is quasi-compact (cf. [Bal10b, Remark 12]). All of thisis standard: see [Sta20, Section 0050] or [GD71, § (cid:3) Remark.
Theorem 2.22 and Theorem 2.36 together with Proposition 2.38 thusprovide a complete description of the topological space Spec(D(H Z G ) c ) for any finitegroup G . We now explain the precise relationship, alluded to in the introduction,between the three spaces: Spec(D(H Z G ) c ), Spec(SH( G ) c ), and Spec( A ( G )). It maybe helpful to refer to the examples depicted on pages 2 and 3 in the introduction.2.40. Corollary.
For any finite group G , the map Spec( F G ) : Spec(D(H Z G ) c ) (cid:44) → Spec(SH( G ) c ) is a homeomorphism of Spec(D(H Z G ) c ) onto its image, which is the subspace of Spec(SH( G ) c ) consisting of the chromatic height 0 and chromatic height ∞ points.Proof. This follows from Cor. 2.19 and our descriptions of the two spaces; in partic-ular, from Thm. 2.22, Thm. 2.36, [BS17, Thm. 4.14], [BS17, Prop. 6.9], and [BS17,Cor. 8.4]. (cid:3)
Remark.
From the second point of view, both Spec(D(H Z G ) c ) and Spec( A ( G ))consist of a number of copies of Spec( Z ), one for each conjugacy class of subgroupsof G , except that in Spec( A ( G )) the closed points p ( K, p ) and p ( H, p ) are gluedtogether when O p ( K ) ∼ G O p ( H ). Stated differently, each point p ( H, p ) is identifiedwith p (O p ( H ) , p ). For example, if G is a p -group then O p ( H ) is trivial for every H ≤ G , so all the copies of ( p ) — one for each copy of Spec( Z ) — are glued into asingle point. In contrast, if p does not divide | G | then O p ( H ) = H for all H ≤ G ,so no gluing of the copies of ( p ) occurs. The picture given of Spec( A ( C p )) on page 3is indicative of the situation for any p -group.This gluing p ( H, p ) = p (O p ( H ) , p ) in Spec( A ( G )) manifests in Spec(D(H Z G ) c )by the fact that P (O p ( H ) , p ) is contained in the closure of P ( H, p ). Or, rather, thegluing in Spec( A ( G )) is explained by these topological relations in Spec(D(H Z G ) c ).2.42. Corollary.
For any finite group G , the comparison map ρ : Spec(D(H Z G ) c ) → Spec( A ( G )) is a quotient map which identifies points of height ≥ whose closures intersect.In more detail, if P , Q ∈ Spec(D(H Z G ) c ) are distinct points then ρ ( P ) = ρ ( Q ) ifand only if P and Q are points of height ≥ with { P } ∩ { Q } (cid:54) = ∅ if and only if P = P ( H, p ) and Q = P ( K, p ) for some prime number p and subgroups H, K ≤ G such that H ∩ K is a p -subnormal subgroup of both H and K .Proof. The points of height 0 are precisely the points P ( H,
0) while the points ofheight ≥ P ( H, p ). (They can have height greater than 1 because ofthe inclusions among them.) According to Corollary 2.21, [BS17, Theorem 3.6] and
HE SPECTRUM OF DERIVED MACKEY FUNCTORS 17
Theorem 2.22, ρ ( P ( H, (cid:54) = ρ ( P ( K, p )) for any p , while ρ ( P ( H, ρ ( P ( K, H ∼ G K iff P ( H,
0) = P ( K, ρ ( P ( H, p )) (cid:54) = ρ ( P ( K, q )) for p (cid:54) = q , while ρ ( P ( H, p )) = ρ ( P ( K, p )) iff O p ( H ) ∼ G O p ( K ). Thus the only identifications madeby ρ are for primes P ( H, p ) and P ( K, p ) corresponding to the same prime number p ,precisely when O p ( H ) ∼ G O p ( K ). By Remark 2.37, the closure of P ( H, p ) consistsof those P ( K, p ) for K a p -subnormal subgroup of H . So if O p ( H ) ∼ G O p ( K ) then P (O p ( H ) , p ) = P (O p ( K ) , p ) is a point in the intersection { P ( H, p ) } ∩ { P ( K, p ) } .Conversely, if P = P ( H, p ) and Q = P ( K, q ) are height ≥ { P }∩{ Q } (cid:54) = ∅ then p = q and there exists a subgroup L which is G -conjugate to a p -subnormalsubgroup of H and also G -conjugate to a p -subnormal subgroup of K . It followsthat O p ( H ) ∼ G O p ( L ) ∼ G O p ( K ) and hence that ρ ( P ) = ρ ( Q ).Also note that if O p ( H ) ∼ G O p ( K ) then O p ( H ) = O p ( K g ) for some g ∈ G andhence H ∩ K g is a p -subnormal subgroup of both H and K g . The converse alsoholds. We conclude that ρ ( P ) = ρ ( Q ) if and only if P = P ( H, p ) and Q = P ( K, p )for some prime number p and subgroups H, K ≤ G satisfying O p ( H ) = O p ( K )(equivalently, with H ∩ K a p -subnormal subgroup of both H and K ).Finally, one readily checks using Remark 2.37 that the surjective continuousmap ρ is a closed map, hence a quotient map. (cid:3) Example.
Consider G = D the dihedral group of order 8. Its lattice ofconjugacy classes of subgroups is C V D C C and the following diagram depicts the comparison map for the spectrum of D(H Z D )localized at the prime 2:Spec(D(H Z D ) c (2) ) ρ (cid:47) (cid:47) Spec( A ( D ) (2) ) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:57) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:53) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) •◦ •◦•◦ •◦•◦ •◦•◦ •◦•◦ •◦•◦ •◦ •◦ •◦•◦ •◦•◦ •◦•◦ Closure goes up and to the left. For example, the closure of the point P ( C , { P ( C , } = { P ( C , , P ( C , , P ( C , , P ( { } , } . Similarly, the closure of P ( V ,
0) consists of five points. Observe that there is aunique closed point in Spec( A ( D ) (2) ) and that the fiber over this point is a copyof the lattice of conjugacy classes of subgroups of D . The reason is that for p = 2,a subgroup K is a p -subnormal subgroup of H if and only if K is a subgroup of H .On the other hand, at a prime p (cid:54) = 2, there is no gluing in the Burnside ring and the comparison map is the simple bijection of six copies of Spec( Z ( p ) ):Spec(D(H Z D ) c ( p ) ) ρ (cid:47) (cid:47) Spec( A ( D ) ( p ) ) •◦ •◦•◦ •◦•◦ •◦•◦ •◦•◦ •◦•◦ •◦ •◦ •◦•◦ •◦•◦ •◦•◦ •◦•◦ •◦•◦ •◦ Remark.
The above example for G = D is illustrative of the situation forany p -group G . Localized at a prime q (cid:54) = p nothing interesting happens: bothspaces are just a disjoint union of copies of Spec( Z ( q ) ). Localized at the prime p ,on the other hand, the spectrum of the Burnside ring has a single closed pointand the fiber over that unique closed point is a copy of the lattice of conjugacyclasses of subgroups of G . Again, the reason is that the relation “is conjugate to a p -subnormal subgroup of” reduces to “is conjugate to a subgroup of”. For example,a diligent reader can immediately write down the 2-local comparison map for thequaternion group G = Q once they recall the lattice of (conjugacy classes of)subgroups of Q .2.45. Example.
Take G = S the symmetric group on three symbols. Its lattice ofconjugacy classes of subgroups is C S C and there are two primes of interest: p = 2 ,
3. Here is the 2-local comparison mapSpec(D(H Z S ) c (2) ) ρ (cid:47) (cid:47) Spec( A ( S ) (2) ) •◦ •◦•◦ •◦•◦ •◦•◦ •◦ •◦ •◦•◦ •◦•◦•◦ and here is the 3-local comparison mapSpec(D(H Z S ) c (3) ) ρ (cid:47) (cid:47) Spec( A ( S ) (3) ) •◦ •◦•◦ •◦•◦ •◦•◦ •◦ •◦ •◦•◦ •◦•◦ •◦•◦ HE SPECTRUM OF DERIVED MACKEY FUNCTORS 19
Note that although C has index 3 in S , it is not a 3-subnormal subgroup (sinceit is not a normal subgroup). The reader might find it interesting to compare withExample 8.14 in [BS17].2.46. Remark.
Finally, we can translate our computation of Spec(D(H Z G ) c ) into aclassification of the thick tensor-ideal subcategories of D(H Z G ) c :2.47. Theorem.
Let G be a finite group. Say that a subset Y ⊆ Conj( G ) × Spec( Z ) is admissible if it satisfies the following closure properties:(a) If ( H, ∈ Y then ( H, p ) ∈ Y for all prime numbers p .(b) If ( H, p ) ∈ Y then ( K, p ) ∈ Y for all (conjugacy classes of ) p -subnormalsubgroups K of H .There is an inclusion-preserving bijection between the set of admissible subsets of Conj( G ) × Spec( Z ) and the collection of thick tensor-ideal subcategories of D(H Z G ) c given by Y (cid:55)−→ (cid:8) X ∈ D(H Z G ) c (cid:12)(cid:12) Φ H ( X ) ∈ p if ( H, p ) (cid:54)∈ Y (cid:9) and C (cid:55)−→ (cid:8) ( H, p ) ∈ Conj( G ) × Spec( Z ) (cid:12)(cid:12) Φ H ( X ) (cid:54)∈ p for some X ∈ C (cid:9) . Proof.
We have an identification of sets Spec(D(H Z G ) c ) ∼ = Conj( G ) × Spec( Z ) byTheorem 2.22. By Remark 2.37, the “admissible” subsets Y ⊆ Conj( G ) × Spec( Z )are precisely those corresponding to specialization-closed subsets of Spec(D(H Z G ) c ).Moreover, these are precisely the Thomason subsets by Proposition 2.38. Finally,all thick tensor-ideals of D(H Z G ) c are radical by [Bal07, Prop. 2.4] since all objectsare dualizable (A). In this way, the theorem is just a translation of the abstractthick subcategory classification theorem of [Bal05, Theorem 4.10]. (cid:3) Construction of
D(H Z G ) and equivariant spectra Our next goal is to construct the tensor triangulated category D(H Z G ) andestablish properties (A)–(I) from Section 2.3.1. Definition.
For a finite group G , let Sp G denote the ∞ -category of genuine G -spectra as constructed in [GM20, Appendix C]. It is presentable, stable, has asymmetric monoidal structure and comes equipped with a symmetric monoidal leftadjoint(3.2) Σ ∞ : S G ∗ → Sp G where S G ∗ denotes the symmetric monoidal ∞ -category of pointed G -spaces. ByElmendorf’s theorem, S G ∗ is equivalent as a symmetric monoidal ∞ -category toFun( O ( G ) op , S ∗ ) with its pointwise monoidal structure, where O ( G ) denotes theusual orbit category of transitive G -sets and G -equivariant maps.3.3. Remark.
The symmetric monoidal ∞ -category Sp G can be constructed inseveral different ways. For example, it is equivalent to the underlying symmet-ric monoidal ∞ -category [Lur17, § § G -spectra Sp OG [MM02]. In particular, its homotopy categorySH( G ) := Ho(Sp G ) is equivalent as a tensor triangulated category to the usualequivariant stable homotopy category of [LMS86] and [HHR16, Appendix B]. Theapproach taken by [GM20, Appendix C] instead defines Sp G as the colimit of a di-agram of copies of S G ∗ parametrized by a certain poset of G -representations. That this is equivalent to the other construction follows from [GM20, Prop. C.4 andProp. C.9]. The advantage of this definition is that it leads (following work ofRobalo [Rob15]) to a very convenient method for constructing symmetric monoidalfunctors on Sp G . More precisely, as established in [GM20, Corollary C.7] and[Rob15, Corollary 2.22], the functor (3.2) enjoys the following universal property:3.4. Theorem (Robalo, Gepner–Meier) . Let G be a finite group and let ρ G de-note the regular representation of G . Given a presentably symmetric monoidal ∞ -category D and a symmetric monoidal left adjoint F : S G ∗ → D with the propertythat F ( S ρ G ) is invertible, there exists an essentially unique symmetric monoidal leftadjoint F : Sp G → D such that F ◦ Σ ∞ (cid:39) F as symmetric monoidal functors. Remark.
Here and in the sequel, when we say that a functor is essentiallyunique, we mean that the collection of such functors is parametrized by a con-tractible Kan complex. Any two such choices will be equivalent (in a suitable ∞ -category of functors) and will induce naturally isomorphic functors at the levelof homotopy categories.3.6. Remark.
We now recall the universal property of the stable ∞ -category ofnonequivariant spectra Sp. As explained in [Lur17, § ∞ -category P r st ofpresentable stable ∞ -categories and colimit preserving functors has a symmetricmonoidal stucture whose commutative algebra objects are the presentably sym-metric monoidal stable ∞ -categories, that is, symmetric monoidal ∞ -categories C ⊗ whose underlying ∞ -category C is presentable and stable and has the property thatthe bifunctor −⊗− : C × C → C commutes with small colimits in each variable. The ∞ -category of spectra Sp is the unit of P r st and consequently is the initial com-mutative algebra object in P r st . In other words, given any presentably symmetricmonoidal stable ∞ -category D ⊗ ∈ CAlg( P r st ), there is an essentially unique sym-metric monoidal functor Sp ⊗ → D ⊗ whose underlying functor Sp → D commuteswith colimits [Lur17, Cor. 4.8.2.19].3.7. Example.
For any finite group G , there is an essentially unique symmetricmonoidal functor triv G : Sp → Sp G which commutes with colimits.3.8. Remark. If C ⊗ is a presentably symmetric monoidal stable ∞ -category thenfor any commutative algebra A ∈ CAlg( C ⊗ ), the category of A -modules A - Mod ⊗ C is also a presentably symmetric monoidal stable ∞ -category [Lur17, Thm. 3.4.4.2].Moreover, the forgetful functor U A : A - Mod C → C has a left adjoint F A : C → A - Mod C which can be equipped with a symmetric monoidal structure, and thecomposite U A F A is the functor A ⊗ − : C → C (cf. [Lur17, Cor. 4.2.3.7, Cor. 4.2.4.8,Thm. 4.5.2.1, § θ : C ⊗ → D ⊗ induces a functor CAlg( C ⊗ ) → CAlg( D ⊗ ) between the ∞ -categories of commuta-tive algebra objects. Moreover, for any A ∈ CAlg( C ⊗ ) there is an induced symmet-ric monoidal functor A - Mod ⊗ C → θ ( A ) - Mod ⊗ D for which both squares in C D A - Mod C θ ( A ) - Mod D θF FU θ U HE SPECTRUM OF DERIVED MACKEY FUNCTORS 21 commute up to equivalence. Moreover, if θ : C → D commutes with limits orcolimits then so does the induced functor θ : A - Mod C → θ ( A ) - Mod D (cf. [Lur17,Cor. 4.2.3.3 and Cor. 4.2.3.5]).3.9. Definition.
Let H Z ∈ CAlg(Sp) denote the Eilenberg–MacLane spectrum of theintegers. For any finite group G , let H Z G := triv G (H Z ) ∈ CAlg(Sp G ) and considerits ∞ -category of modules H Z G - Mod := H Z G - Mod Sp G . We defineD(H Z G ) := Ho(H Z G - Mod)to be its homotopy category.3.10. Remark.
By construction, we have a free-forgetful adjunctionSp G H Z G - Mod F G U G and U G ( ) ∼ = triv G (H Z ) as commutative algebras in Sp G .3.11. Remark. If C ⊗ ∈ CAlg( P r st ) is a presentably symmetric monoidal stable ∞ -category then Ho( C ) has the structure of a triangulated category as well as aclosed symmetric monoidal structure that is compatible with the triangulation (inthe sense of [HPS97, Def. A.2.1]). Moreover, if C is compactly generated by a set ofobjects G then for any commutative algebra A ∈ CAlg( C ⊗ ), the stable ∞ -category A - Mod C is compactly generated by the set F A ( G ). Note that an object of a stable ∞ -category D is compact if and only if it is compact as an object of Ho( D ) in theusual triangulated category sense (see [Lur17, Prop. 1.4.4.1]). Moreover, a set G ofcompact objects generates D under colimits if and only if G generates Ho( D ) in theusual triangulated category sense (see the proof of [Lur17, Cor. 1.4.4.2]).3.12. Example.
The stable ∞ -category H Z G - Mod is compactly generated by { F G ( G/H + ) | H ≤ G } . Moreover, since these compact generators are dualizable and the unit is com-pact, it follows that an object of D(H Z G ) is dualizable if and only if it is compact(cf. [HPS97, Theorem A.2.5]). In other words, D(H Z G ) is a rigidly-compactly gen-erated tensor triangulated category.3.13. Remark.
Any homomorphism of groups α : G → G (cid:48) induces a left adjoint(3.14) α ∗ : S G (cid:48) ∗ → S G ∗ which can be constructed as follows. Restriction along α provides a functor fromthe category of G (cid:48) -sets to the category of G -sets. This functor always has a leftadjoint which sends transitive G -sets to transitive G (cid:48) -sets and hence restricts to afunctor α ! : O ( G ) → O ( G (cid:48) ) on the orbit categories. Restricting along α ! providesthe functor (3.14). Note that it preserves colimits hence is a left adjoint (since S ∗ is presentable).By Theorem 3.4, there is an essentially unique symmetric monoidal left adjoint α ∗ : Sp G (cid:48) → Sp G such that the diagram S G (cid:48) ∗ S G ∗ Sp G (cid:48) Sp G Σ ∞ α ∗ Σ ∞ α ∗ commutes up to an equivalence of symmetric monoidal functors.By the universal property of Remark 3.6, the compositeSp triv G (cid:48) −−−−→ Sp G (cid:48) α ∗ −−→ Sp G is equivalent as a symmetric monoidal functor to triv G : Sp → Sp G . In particular, α ∗ (H Z G (cid:48) ) ∼ = H Z G as commutative algebras in Sp G . Hence by Remark 3.8, there isa symmetric monoidal left adjoint α ∗ : H Z G (cid:48) - Mod → H Z G - Mod such that bothsquares in Sp G (cid:48) Sp G H Z G (cid:48) - Mod H Z G - Mod F G (cid:48) α ∗ F G U G (cid:48) α ∗ U G commute up to equivalence.3.15. Remark. If G α −→ G (cid:48) β −→ G (cid:48)(cid:48) is a composite of group homomorphisms, itfollows from the constructions that the functor ( β ◦ α ) ∗ is equivalent to α ∗ ◦ β ∗ (inthe relevant ∞ -category of symmetric monoidal functors) at the level of spaces S G ∗ ,spectra Sp G , and H Z G - Mod.3.16. Remark.
As usual, for a quotient α : G → G/N , we call infl
GG/N := α ∗ theinflation functor and for an inclusion α : H (cid:44) → G , we call res GH := α ∗ the restrictionfunctor. Since the functor res GH : Sp G → Sp H preserves limits, the induced functorres GH : H Z G - Mod → H Z H - Mod also preserves limits (Rem. 3.8) and hence has aleft adjoint (see [Lur09, Cor. 5.5.2.9 and Cor. 5.4.7.7]).3.17. Remark.
The smashing Bousfield localizations of a presentably symmetricmonoidal stable ∞ -category C correspond to the smashing Bousfield localizationsof the tensor triangulated category Ho( C ). More precisely, smashing Bousfield lo-calizations correspond to idempotent commutative algebras (in C or Ho( C ) respec-tively) and the ∞ -category of idempotent commutative algebras in C is equivalent tothe ordinary category (in fact poset) of idempotent commutative algebras in Ho( C )(see [Lur17, § Proposition.
Let C ⊗ ∈ CAlg( P r st ) be a presentably symmetric monoidalstable ∞ -category and let L : C → C be a smashing Bousfield localization. For anycommutative algebra A ∈ CAlg( C ⊗ ) , the functor (3.19) A - Mod C → LA - Mod L C is a smashing Bousfield localization. Moreover, if C is rigidly-compactly gener-ated and L is the finite localization associated to a set G of compact objects in C then (3.19) is the finite localization associated to the set F A ( G ) of compact objectsin A - Mod C . HE SPECTRUM OF DERIVED MACKEY FUNCTORS 23
Proof.
Given two algebras
A, B ∈ CAlg( C ), the extension-of-scalars F B : C → B - Mod C induces a functor A - Mod C → F B ( A ) - Mod B - Mod C by Remark 3.8 whichunder the equivalences F B ( A ) - Mod B - Mod C (cid:39) U B F B ( A ) - Mod C (cid:39) ( B ⊗ A ) - Mod C (cid:39) ( A ⊗ B ) - Mod C (cid:39) U A F A ( B ) - Mod C (cid:39) F A ( B ) - Mod A - Mod C is the extension-of-scalars A - Mod C → F A ( B ) - Mod A - Mod C associated to F A ( B ) ∈ CAlg( A - Mod C ). (Nesting of module categories behaves as expected for symmetricmonoidal ∞ -categories: see [Lur17, § § L : C → L C is nothing but extension-of-scalars C → L - Mod C withrespect to the idempotent algebra L ∈ CAlg( C ). Taking B = L above, wefind that the the functor A - Mod C → LA - Mod L C induced by L : C → L C is thesmashing Bousfield localization associated to the idempotent algebra F A ( L ) ∈ CAlg( A - Mod C ).To prove the second part, it suffices to check (at the level of homotopy cate-gories) that F A ( L ) is the smashing idempotent associated to the indicated finitelocalization. This is the content of [BS17, Proposition 5.11]. (cid:3) Remark.
The following useful proposition is a variation on ideas that haveappeared in a few different places (e.g. [MNN17, § Proposition.
Let F : C → D be a symmetric monoidal functor between pre-sentably symmetric monoidal stable ∞ -categories which admits a right adjoint G .If C is generated by a set of compact-rigid objects then the functor F is an equiva-lence if and only if(a) the functor F sends a set of compact generators of C to a set of compactgenerators of D , and(b) the commutative algebra G ( D ) is equivalent to the commutative algebra C .Proof. The ( ⇒ ) direction is immediate once we recognize that the unit η : C → GF ( C ) (cid:39) G ( D ) is a map of algebras. On the other hand, hypothesis (a) im-plies that the right adjoint G preserves colimits and is, moreover, conservative.Hence the adjunction F (cid:97) G is monadic by the Barr–Beck–Lurie Theorem [Lur17,Theorem 4.7.3.5]. Now the projection formula G ( x ) ⊗ y → G ( x ⊗ F ( y )) is an equiva-lence under our assumptions (as can be checked at the level of homotopy categories[BDS16, Prop. 2.15]) and the natural equivalence G ( ) ⊗ y (cid:39) GF ( y ) provides anisomorphism between the monad associated to the algebra object G ( ) ∈ CAlg( C )and the monad GF of the adjunction (see the proof of [BDS15, Lemma 2.8]). Thefunctor F is thus, up to equivalence, just extension of scalars C → G ( ) - Mod C withrespect to the algebra G ( ). This is an equivalence by the second hypothesis. (cid:3) Proposition.
Let N (cid:2) G be a normal subgroup and let L : H Z G - Mod → H Z G - Mod be the finite localization associated to the set (cid:8) F G ( G/H + ) (cid:12)(cid:12) H (cid:54)⊇ N (cid:9) . The compos-ite H Z G/N - Mod infl
GG/N −−−−−→ H Z G - Mod L −→ L (H Z G - Mod) is an equivalence of symmetric monoidal ∞ -categories.Proof. By Proposition 3.18, the result follows from the analogous statement forSp
G/N namely that if L : Sp G → Sp G is the finite localization associated to the set (cid:8) G/H + (cid:12)(cid:12) H (cid:54)⊇ N (cid:9) then the composite(3.23) Sp G/N infl
GG/N −−−−−→ Sp G L −→ L (Sp G )is an equivalence of symmetric monoidal ∞ -categories. This is well-known at thelevel of homotopy categories (see [LMS86, Cor. II.9.6]) but we will provide a proofthat holds at the level of ∞ -categories. For simplicity of notation, let f ∗ denote thecomposite (3.23) and let f ∗ denote a right adjoint. The functor f ∗ has a symmetricmonoidal structure (being a composite of such) and we just need to establish that itis an equivalence. First note (e.g. by Remark 3.11) that the smashing localization L maps the set of compact generators (cid:8) G/H + (cid:12)(cid:12) H ≤ G (cid:9) of Sp G to a set of compactgenerators for L (Sp G ) which is thus compactly generated by (cid:8) L ( G/H + ) (cid:12)(cid:12) H ⊇ N (cid:9) .Note that these are precisely the images under f ∗ of the generators of Sp G/N . Wecan thus invoke Proposition 3.21 if we prove that the homomorphism of algebras S G/N = → f ∗ f ∗ = ( (cid:103) E F [ (cid:54)⊇ N ]) N is an equivalence. This can be checked in thehomotopy category, where it is well-known (see [LMS86, Prop. II.9.10.(ii)] or theproof of [MNN17, Thm. 6.11]). (cid:3) Remark.
For any g ∈ G , consider the inner automorphism α := c g = ( − ) g : G ∼ −→ G . The induced automorphism of the category of G -sets is naturally isomor-phic to the identity functor. Consequently, the induced functor α ∗ : S G ∗ → S G ∗ ofRemark 3.13 is equivalent (as a symmetric monoidal functor) to the identity func-tor. It follows that the same is true for the induced functors c ∗ g : Sp G → Sp G and c ∗ g : H Z G - Mod → H Z G - Mod.3.25. Remark.
To summarize, we have established (A) through (I) as follows: (A)by Definition 3.9, Remark 3.10 and Example 3.12; (B) by construction (Definition3.9); (C) by Remark 3.13; (D) by Remark 3.15; (E) by Remark 3.16; (F) by Propo-sition 3.22; (G) is just a definition; (H) by Remark 3.24; and (I) by Remark 3.10.3.26.
Remark.
The approach we have taken is not the only way to construct thetensor triangulated category D(H Z G ). An alternative approach is to consider theinflation functor (cid:15) ∗ G : Sp O → Sp OG in the context of orthogonal G -spectra. Wecan take a cofibrant replacement H Z c → H Z of associative algebras with respectto the model structure given in [MM02, Section III.7]. The homotopy categoryHo( (cid:15) ∗ G H Z c - Mod) is then a model for D(H Z G ). However, there are subtleties con-cerning the commutative structures on (cid:15) ∗ G H Z c . One can show that (cid:15) ∗ G H Z c is nota genuine G - E ∞ -ring spectrum (see Example 3.27 below). It does however have anaive E ∞ -structure (see [BH15]) and this is good enough to produce the symmetricmonoidal structure on D(H Z G ). Alternatively one can do the same thing using thestable model category Sp G Σ of G -equivariant symmetric spectra of [Hau17] basedon simplicial sets. The latter is monoidally Quillen equivalent to Sp OG but has theadvantage that it is combinatorial. It turns out however that making this all work HE SPECTRUM OF DERIVED MACKEY FUNCTORS 25 brings forth a number of point-set level technicalities and some of the delicate issuesrelated to model categories of modules over naive equivariant E ∞ -ring spectra thatare still not covered in the literature. Instead we have chosen to construct D(H Z G )using simple universal properties and stable ∞ -categories.3.27. Example.
We show that (cid:15) ∗ G H Z c for G = C cannot be modelled by a genuine C - E ∞ -ring spectrum. Indeed, if this were the case, then (cid:15) ∗ C H Z c would admit astrictly commutative orthogonal spectrum model as a C -equivariant ring spectrum.We would then have a commutative diagram in the homotopy category of spectraH Z Φ C ( N (cid:15) ∗ C H Z c ) Φ C ( (cid:15) ∗ C H Z c ) ∼ = H Z H Z ( N (cid:15) ∗ C H Z c ) tC H Z tC ∆∆ where N is the Hill–Hopkins–Ravenel norm, the top ∆ is the Hill–Hopkins–Raveneldiagonal [HHR16, Proposition B.209] and ( − ) tC is the Tate construction [GM95,NS18]. The lower ∆ is the Tate diagonal of [NS18]. It follows from [NS18, TheoremIV.1.15] that the lower horizontal composite (which is the Tate valued Frobenius)splits as a sum containing all even Steenrod squares if we use the splitting of H Z tC coming from the canonical H Z -module structure. On the other hand, with respectto the same splitting, the right-hand vertical map is the inclusion of a summand andhence does not contain any non-trivial Steenrod squares. This gives a contradiction.4. Equivalence with the categories of Kaledin and Barwick
Our goal in this section is to prove that the category D(H Z G ) constructed inSection 3 and whose spectrum was computed in Section 2 is equivalent to Kaledin’scategory of derived Mackey functors [Kal11]. We will achieve this in two steps bypassing first through the category of spectral Mackey functors [Bar17, BGS20]. Spectral Mackey Functors.
Recall the effective Burnside ∞ -category A eff (Fin G )introduced by [Bar17]. Its objects are finite G -sets and its n -simplices are n -foldspans of finite G -sets (see Definition 4.16 below). It is a semi-additive ∞ -categorywith biproduct given by the disjoint union and with a symmetric monoidal structureprovided by [BGS20, Section 2]. A spectral Mackey functor is an additive functor A eff (Fin G ) → Sp. The ∞ -category of spectral Mackey functorsFun add ( A eff (Fin G ) , Sp)is a smashing localization of the functor category Fun( A eff (Fin G ) , Sp). As such, it isstable, presentable and can be equipped with the localized Day convolution product(see [Gla16] and [BGS20, Lemma 3.7]). It also comes equipped with a symmetricmonoidal left adjoint Σ ∞ : S G ∗ → Fun add ( A eff (Fin G ) , Sp)which inverts representation spheres (see [Nar17, Appendix A]) and whose rightadjoint Ω ∞ is induced by the inclusion O ( G ) op → A eff (Fin G ) . Proposition.
For any finite group G , the essentially unique colimit-preservingsymmetric monoidal functor (4.2) F : Sp G → Fun add ( A eff (Fin G ) , Sp) which commutes with Σ ∞ is an equivalence of symmetric monoidal ∞ -categories.Proof. The existence and essential uniqueness of the functor F follows from theuniversal property of Theorem 3.4. To prove the claim it suffices to check that F is an equivalence of underlying ∞ -categories. Moreover, since F is an exact func-tor between stable ∞ -categories, it suffices to check that the induced functor onhomotopy categories F : Ho(Sp G ) → Ho(Fun add ( A eff (Fin G ) , Sp))is an equivalence. Note that both homotopy categories are equivalent as triangu-lated categories to the homotopy category of orthogonal G -spectra. For Sp G thisis established in [GM20, Appendix C] while for spectral Mackey functors it goesback to [GM17]. We do not need the full strength of these results (just certainfacts about the homotopy categories, such as the fact that they are both generatedby suspension spectra of orbits) but morally what the following proof really doesis establish that any system of endofunctors on the equivariant stable homotopycategories satisfying certain compatibility properties must be an equivalence. It isa rigidity result. With these comments in mind, let us continue.Note that the tensor triangulated category Ho(Sp G ) is rigidly-compactly gener-ated by { Σ ∞ ( G/H + ) | H ≤ G } and, by construction, the universal functor F commutes with suspension spectra: F (Σ ∞ ( G/H + )) (cid:39) Σ ∞ ( G/H + ) . Moreover, these suspension spectra rigidly-compactly generate the tensor triangu-lated category of spectral Mackey functors Ho(Fun add ( A eff (Fin G ) , Sp)) by [Nar17,Lemma A.8]. Since F preserves coproducts, a thick subcategory argument reducesthe problem of showing that F is an equivalence to the problem of showing that(4.3) F : [Σ ∞ ( G/H + ) , Σ ∞ ( G/K + )] G ∗ → [ F (Σ ∞ ( G/H + )) , F (Σ ∞ ( G/K + ))] G ∗ is an isomorphism for all H, K ≤ G . Here we use the notation [ − , − ] G ∗ to denotegraded morphisms in the homotopy categories.The degree zero morphisms [Σ ∞ ( G/H + ) , Σ ∞ ( G/K + )] G are in both cases gener-ated by conjugations, restrictions and transfers subject to the same relations. Thefunctor F preserves conjugations and restrictions since they come from morphismsof G -spaces and F Σ ∞ (cid:39) Σ ∞ . Transfers are preserved since transfers are duals ofrestrictions and F commutes with duality (being a symmetric monoidal functor).Hence we conclude that F : [Σ ∞ ( G/H + ) , Σ ∞ ( G/K + )] G → [ F (Σ ∞ ( G/H + )) , F (Σ ∞ ( G/K + ))] G is an isomorphism for any two subgroups H and K .Now, to prove that (4.3) is an isomorphism in general, it suffices by [Pat16,Proposition 5.1.1] to establish the H = K case. For this, we use induction on the HE SPECTRUM OF DERIVED MACKEY FUNCTORS 27 order of H . The fact that Sp is the free stable ∞ -category on one generator [Lur17,Corollary 1.4.4.6] implies that the diagramSp Fun add ( A eff (Fin ) , Sp)Sp G Fun add ( A eff (Fin G ) , Sp) F G + ∧− G + ∧− F commutes up to equivalence and, moreover, that the top arrow is an equivalence.Furthermore, for both homotopy categories we have an isomorphism[Σ ∞ G + , Σ ∞ G + ] G ∗ ∼ = [ S , S ] ∗ ⊗ [Σ ∞ G + , Σ ∞ G + ] G induced by the functor G + ∧ − . Combining the latter two results we conclude that F : [Σ ∞ G + , Σ ∞ G + ] G ∗ → [ F (Σ ∞ G + ) , F (Σ ∞ G + )] G ∗ is an isomorphism.To do the induction step we need the geometric fixed point functors. For Sp G they are defined in the proof of Proposition 3.22 while for the category of spectralMackey functors Fun add ( A eff (Fin G ) , Sp) they are defined in [Nar16, Example A.20].Both are smashing localizations, hence are symmetric monoidal, and the essentialuniqueness of Theorem 3.4 implies that F (cid:101) Φ N,G (cid:39) (cid:101) Φ N,G F . For a subgroup H ≤ G with normalizer N ( H ) and Weyl group W ( H ) := N ( H ) /H , let (cid:101) Φ H,G denote thecomposite (cid:101) Φ H,N ( H ) ◦ res GN ( H ) . We will use the notation E P ( H ) for the classifyingspace of the family of proper subgroups of H . Then consider the following diagram(in which we have suppressed the Σ ∞ symbols): [ G/H + , G × H E P ( H ) + ] G ∗ [ G/H + , G/H + ] G ∗ [ W ( H ) + , W ( H ) + ] W ( H ) ∗ [ F ( G/H + ) , F ( G × H E P ( H ) + )] G ∗ [ F ( G/H + ) , F ( G/H + )] G ∗ [ F ( W ( H ) + ) , F ( W ( H ) + )] W ( H ) ∗ . F proj ∗ F (cid:101) Φ H,G FF (proj) ∗ (cid:101) Φ H,G
It follows from [Pat16, Proposition 6.3.2] that the top row is a short exact sequence.Since F commutes with suspension spectra, so is the bottom row (as it is isomorphicto the analog of the top row in the homotopy category of spectral Mackey functors).The left square commutes by the functoriality of F and the right square commutesby the fact that F commutes with geometric fixed points. Since G × H E P ( H )has smaller isotropy than H (see [Pat16, Proof of Lemma 7.2.2]), the inductionhypothesis implies that the left-hand map is an isomorphism. On the other hand,the right-hand map is an isomorphism by the base case of the induction. It followsthat the middle map is also an isomorphism and this completes the proof. (cid:3) Remark.
Combined with [GM20, Proposition C.9], Proposition 4.1 provides asymmetric monoidal equivalence between the symmetric monoidal ∞ -category of or-thogonal G -spectra and the symmetric monoidal ∞ -category of spectral G -Mackeyfunctors. Such an equivalence is also established by different methods in [CMNN20].4.5. Remark.
Evaluation at the object
G/G ∈ A eff (Fin G ) admits a symmetricmonoidal left adjoint F : Sp → Fun( A eff (Fin G ) , Sp)whose composition with the localization L : Fun( A eff (Fin G ) , Sp) → Fun add ( A eff (Fin G ) , Sp) is the essentially unique symmetric monoidal left adjoint LF making the diagram S ∗ Sp S G ∗ Fun add ( A eff (Fin G ) , Sp) triv G Σ ∞ LF Σ ∞ commute up to an equivalence of symmetric monoidal functors. This follows bytaking right adjoints and noting that the inclusion functor { G/G } → A eff (Fin G )factors through the nerve of O ( G ) op . Moreover the universal property also impliesthat LF coincides with the compositeSp triv G −−−→ Sp G (cid:39) −→ Fun add ( A eff (Fin G ) , Sp)up to equivalence of symmetric monoidal functors. An immediate consequence is:4.6.
Corollary.
There is an equivalence of symmetric monoidal ∞ -categories H Z G - Mod (cid:39) LF H Z - Mod . Remark.
Our next task is to compare LF H Z - Mod with H Z -valued spectralMackey functors: Fun add ( A eff (Fin G ) , H Z - Mod).4.8. Proposition.
Let C ⊗ and D ⊗ be symmetric monoidal ∞ -categories with C small, D presentable and − ⊗ − : D × D → D preserving colimits in each variable.Equip Fun( C , D ) with the Day convolution.(a) The left adjoint F : D → Fun( C , D ) to evaluation at the unit ∈ D has asymmetric monoidal structure.(b) Let A ∈ CAlg( D ) be a commutative algebra. There exists an essentiallyunique symmetric monoidal equivalence Fun( C , A - Mod D ) (cid:39) F A - Mod
Fun( C , D ) where the left-hand side is equipped with the Day convolution. Moreover,this equivalence commutes with the free-forgetful adjunctions to Fun( C , D ) .Proof. Part (a) is established by [Nik16, Corollary 3.8]. To prove part (b) notethat the free-forgetful adjunction F : D (cid:29) A - Mod D : U induces an adjunction onfunctor categories F ∗ : Fun( C , D ) (cid:29) Fun( C , A - Mod D ) : U ∗ by post-composition. We then apply the monadic machinery used in the proof ofProposition 3.21 (cf. [Lur17, Corollary 4.8.5.21]). The Day convolution productpreserves colimits in each variable by [Gla16, Lemma 2.13] and F ∗ is symmetricmonoidal by [Nik16, Corollary 3.7]. Since equivalences and colimits in functorcategories are detected pointwise, it follows that the right adjoint U ∗ preservescolimits and is conservative. The projection formula can also be verified pointwiseby using the equivalenceMap Fun( C , D ) ( F ⊗ U ∗ ( G ) , H ) (cid:39) Map
Fun( C × C , D ) ( ⊗ D ◦ ( F × U ∗ ( G )) , H ◦ ⊗ C )of mapping spaces (see e.g. [Nik16, Corollary 3.6]) and the fact that U ∗ is also aleft adjoint. Thus the adjunction F ∗ (cid:97) U ∗ is monadic. HE SPECTRUM OF DERIVED MACKEY FUNCTORS 29
Finally, to see that F A is equivalent as a commutative algebra to the imageunder U ∗ of the unit of the Day convolution on Fun( C , A - Mod) just observe thatin the following commutative diagram D Fun( C , D ) Fun( C , A - Mod D ) D Fun( C , D ) Fun( C , D ) UF F U ∗ F ∗ F ∗ U ∗ F the top row is symmetric monoidal and hence, in particular, preserves units. (cid:3) As a consequence we obtain4.9.
Proposition.
There is an equivalence of symmetric monoidal ∞ -categories LF H Z - Mod (cid:39)
Fun add ( A eff (Fin G ) , H Z - Mod) . Proof.
Proposition 4.8 provides an equivalence of symmetric monoidal ∞ -categories(4.10) F H Z - Mod (cid:39) Fun( A eff (Fin G ) , H Z - Mod)which commutes with the forgetful functors to Fun( A eff (Fin G ) , Sp). Then considerFun( A eff (Fin G ) , Sp) Fun add ( A eff (Fin G ) , Sp)Fun( A eff (Fin G ) , H Z - Mod) F H Z - Mod LF H Z - Mod L (cid:39) where L denotes the smashing localization onto the full subcategory of additivefunctors. By Proposition 3.18, the induced functor F H Z - Mod → LF H Z - Mod isalso a smashing localization. Moreover, since the forgetful functor is conservative,an application of the projection formula shows that the local objects of the inducedlocalization are precisely those that are sent under the forgetful functor to a localobject of the original localization (that is, to an additive functor). As additivity isdetected by the forgetful functors, we conclude that the bottom row is the smashinglocalization of Fun( A eff (Fin G ) , H Z - Mod) whose local objects are the additive func-tors. In particular, Fun add ( A eff (Fin G , H Z - Mod) with its localized Day convolutionis equivalent to LF H Z - Mod as a symmetric monoidal ∞ -category. (cid:3) Corollary.
There is an equivalence of symmetric monoidal ∞ -categories H Z G - Mod (cid:39)
Fun add ( A eff (Fin G ) , H Z - Mod) and, consequently, an equivalence of tensor triangulated categories
D(H Z G ) (cid:39) Ho(Fun add ( A eff (Fin G ) , H Z - Mod)) for any finite group G .Proof. This follows from Corollary 4.6 and Proposition 4.9. (cid:3) ∗ ∗ ∗
Comparison to Kaledin’s derived Mackey functors.
Our next task is to showthat the tensor triangulated categoryHo(Fun add ( A eff (Fin G ) , H Z - Mod))is equivalent to Kaledin’s category of derived Mackey functors from [Kal11]. Thissection is based on the arguments of [Kal11, Sections 3–5] modified as appropriate tothe ∞ -categorical context to facilitate comparison with Barwick’s spectral Mackeyfunctors.4.12. Remark.
Kaledin [Kal11] provides two different constructions of the triangu-lated category of derived Mackey functors DMack( G ). His first definition is in termsof an A ∞ -category associated to the (2 , G -sets, whilea second approach uses a Waldhausen type construction on the category of finite G -sets. We will take the latter construction as our starting point (see Definition 4.38below). Nevertheless, as the monoidal structure [Kal11] constructs on DMack( G )uses the A ∞ -approach, we will ultimately need to recall this construction as well(see Definition 4.46).4.13. Definition.
For a small category C with pullbacks, Kaledin [Kal11, Section 4]defines a category S C as a subcategory of the Grothendieck fibration associatedwith Fun( − , C ) op : ∆ op → Cat . In more detail, S C is the category whose objects are the pairs([ n ] , X −→ X −→ · · · −→ X n )where X −→ X −→ · · · −→ X n is a diagram in C for n ≥
0. A morphism ( α, f ) :([ n ] , X • ) → ([ m ] , Y • ) consists of a map α : [ n ] → [ m ] in ∆ and f : α ∗ ( Y • ) → X • such that for each i ≥ j the commutative square Y α ( j ) X j Y α ( i ) X if j f i is cartesian. The morphism is called special if α is the inclusion of an end-segmentand all the maps f i are isomorphisms. We denote the set of special morphismsby I .4.14. Remark.
The case of interest is C = Fin G the category of finite G -sets. Ourfirst task is to relate Kaledin’s category S Fin G to Barwick’s ∞ -category A eff (Fin G ).This will be achieved in Theorem 4.19 which will recognize A eff (Fin G ) as the infinity-categorical localization of S Fin G with respect to the class of special morphisms.We begin with some general notation and then recall the definition of A eff (Fin G ).4.15. Notation.
Given an ∞ -category D , we use the notationMap D ( x, y ) = { x } × D Fun(∆ , D ) × D { y } for the mapping space between two objects x, y ∈ D (see [Lur09, § | D | for the Kan complex replacement (which isequivalent to inverting all arrows or to geometric realization) and D ∼ ⊂ D willdenote the ∞ -groupoid core (the maximal Kan subcomplex). HE SPECTRUM OF DERIVED MACKEY FUNCTORS 31
Definition.
Let C be a small category with pullbacks. The ∞ -category A eff ( C )is defined in simplicial degree n to be the subset of cartesian diagrams A eff ( C ) n ⊂ Fun(Tw([ n ]) op , C )where Tw([ n ]) denotes the twisted arrow category of [ n ] with objects ( i, j ) for 0 ≤ i ≤ j ≤ n and exactly one morphism ( i , j ) → ( i , j ) for i ≤ i and j ≤ j .Cartesian here means that a diagram X is contained in A eff ( C ) n if and only if thesquare X ij (cid:47) (cid:47) (cid:15) (cid:15) X kj (cid:15) (cid:15) X il (cid:47) (cid:47) X kl is cartesian for all integers 0 ≤ i ≤ k ≤ l ≤ j ≤ n .4.17. Remark.
The mapping spaces in A eff ( C ) can be concretely identified as spansin the following way. Given objects X and Y of C , we denote by Span C ( X, Y ) thecategory with objects the spans X ← Z → Y and morphisms Z → Z (cid:48) lying over X and Y . There is a natural equivalence of Kan complexesMap A eff ( C ) ( X, Y ) (cid:39) −→ N Span C ( X, Y ) ∼ (see [Bar17, 3.7]) which sends an n -simplex f : ∆ × ∆ n → A eff ( C ) to the n -tupleof isomorphisms f | ∆ ×{ } f −→ f | ∆ ×{ } f −→ · · · f n − −→ f | ∆ ×{ n } in Span C ( X, Y ). Here the map f i is the vertical composition in the commutativediagram X Z i +1 YX W i YX Z i Y ∼ = ∼ = in C encoded by the two 2-simplices of f | ∆ ×{ i,i +1 } in the Burnside category (think-ing of a square subdivided with a diagonal into two triangles), where the bottomand top row are the spans associated with f | ∆ ×{ i } and f | ∆ ×{ i +1 } .4.18. Construction.
We define a functor N S C → A eff ( C ) as follows: On objects itsends ([ n ] , X • ) to X n and on morphisms it sends ( α, f ) : ([ n ] , X • ) → ([ m ] , Y • ) tothe span X n f n ←− Y α ( n ) −→ Y m .It sends a 2-simplex([ n ] , X , • ) α (cid:47) (cid:47) ([ n ] , X , • ) α (cid:47) (cid:47) ([ n ] , X , • )to X ,α ( α ( n )) X ,α ( n ) X ,α ( n ) X ,n X ,n X ,n . A general k -simplex([ n ] , X , • ) α (cid:47) (cid:47) ([ n ] , X , • ) α (cid:47) (cid:47) . . . α k − (cid:47) (cid:47) ([ n k ] , X k, • )is sent to the obvious diagram Y with Y j,i + j = X i + j,α j + i − ( α j + i − ( ··· α j ( n j ))) for 0 ≤ i ≤ k − j . The cartesian condition on morphisms in S C gives the cartesiancondition for A eff ( C ). The functor N S C → A eff ( C ) sends special morphisms toequivalences.4.19. Theorem.
The functor N S C → A eff ( C ) exhibits the effective Burnside ∞ -category as the ∞ -categorical localization N S C [ I − ] (cid:39) A eff ( C ) of S C with respect tothe class of special morphisms I . In order to prove Theorem 4.19, we consider the following more general situation:4.20.
Definition ([Kal11, Definition 4.3]) . Let Φ be a small category with distin-guished classes of morphisms I and P . Then (cid:104) P, I (cid:105) forms a complementary pair ifthe following axioms are satisfied:(a) The classes P and I are closed under composition and contain all isomor-phisms.(b) For every object c ∈ Φ, the full subcategory Φ
I/b ⊂ Φ /b of the slice categoryconsisting of the maps in I admits an initial object i b : ι ( b ) → b .(c) Every map f in Φ factors uniquely (up to unique isomorphism) as f = i ( f ) ◦ p ( f ) with i ( f ) ∈ I and p ( f ) ∈ P .(d) For every pair of morphisms b p ← b i → b in Φ with p ∈ P and i ∈ I , thereexists a pushout square b b b b pi i (cid:48) p (cid:48) with i (cid:48) ∈ I and p (cid:48) ∈ P .4.21. Example.
Take Φ = S C , I the class of special morphisms, and P the collectionof morphisms ( α, f ) : ([ n ] , X • ) → ([ m ] , Y • )) such that α (0) = 0. This pair (cid:104) P, I (cid:105) iscomplementary by [Kal11, Lemma 4.8].4.22.
Construction.
Let R Φ be the category of diagrams b i ←− b i −→ b such that i , i ∈ I . Using the canonical projections π , π : R Φ → Φ we construct for everypresentable ∞ -category D an endofunctorSpcl : Fun(Φ , D ) → Fun(Φ , D ) HE SPECTRUM OF DERIVED MACKEY FUNCTORS 33 as the composition Spcl = ( π ) ! ◦ ( π ) ∗ of restriction along π with left Kan extensionalong π . The common section δ : Φ → R Φ of the projections π , π provides anatural transformation τ : Id → Spcl which is defined asId = δ ∗ ◦ π ∗ → δ ∗ ◦ π ∗ ◦ ( π ) ! ◦ π ∗ = ( π ) ! ◦ π ∗ . Lemma.
The value of
Spcl on a functor F ∈ Fun(Φ , D ) can be computed at b ∈ Φ as the colimit Spcl( F )( b ) (cid:39) colim Φ b π ∗ F where Φ b is the full subcategory of the under category Φ ιb/ with objects the maps ιb → ˜ b ∈ I and π : Φ b → Φ is the projection to Φ . Under this identification thenatural transformation τ corresponds to the canonical map ( η F )( b ) : F b → colim Φ b π ∗ F into the colimit system at the initial map ιb → b .Proof. We use the pointwise formula for Kan extensions [Lur09, Lemma 4.3.2.13and Definition 4.3.3.2]. It follows from the axioms that the projection π : R Φ → Φis a weak op-fibration (cofibered functor). Hence the left Kan extension can becomputed at b ∈ Φ as the colimit (using cofinality, see e.g. [Lur09, Theorem 4.1.3.1])Spcl( F )( b ) = ( π ) ! π ∗ ( F )( b ) (cid:39) colim π − ( b ) π ∗ F over the fiber π − ( b ). Moreover, there is an adjunction π − ( b ) (cid:29) Φ b where the right adjoint sends b ← ˜ b → ˜˜ b to the composite ιb → ˜ b → ˜˜ b with theunique arrow ιb → ˜ b lying over b . (cid:3) Proposition.
Let Φ be a small category together with a complementary pair (cid:104) P, I (cid:105) of classes of maps and let D be a presentable ∞ -category. Then Spcl is alocalization functor with local objects the functors F ∈ Fun I (Φ , D ) that invert themaps in I .Proof. We first claim that if F ∈ Fun(Φ , D ) inverts morphisms in I , then the map τ F : F −→ Spcl( F )is an equivalence. This follows from the object-wise description of Lemma 4.23together with the observation that the natural transformation from the constantfunctor is an equivalence F ( ιc ) (cid:39) π ∗ F of functors Φ c → D whose source has aninitial object [Lur09, Corollary 4.4.4.10]. Moreover, for any F ∈ Fun(Φ , D ) thefunctor Spcl( F ) inverts morphism in I , since for i : c → c (cid:48) ∈ I there is a uniqueisomorphism ιc ∼ = ιc (cid:48) lying over i , which induces an equivalence of the relevant slicecategories [Kal11, Section 4.3]. This shows that the essential image of Spcl is givenby the I -inverting functors and that τ Spcl( F ) : Spcl( F ) → Spcl(Spcl( F ))is an equivalence for all F : Φ → D . It remains to show that the mapSpcl( τ F ) : Spcl( F ) → Spcl(Spcl( F ))is an equivalence. Under the zigzag of equivalencesSpcl( G ) = ( π ) ! ( π ) ∗ G (cid:39) ( π ) ! ( π ) ∗ G for I -inverting G it is given by the unit map( π ) ! π ∗ F ( π ) ! η π ∗ F −−−−−−→ ( π ) ! π ∗ ( π ) ! π ∗ F, which can be seen pointwise by evaluating η π ∗ F at the zig-zag of (horizontal) mor-phisms c c c c c c c c c in R Φ. The counit (cid:15)
Spcl( F ) is a left inverse and so it suffices to show that it isan equivalence. Furthermore, for any I -inverting functor G , under the zigzag ofequivalences Spcl( G ) (cid:39) ( π ) ! π ∗ G , the natural map τ G corresponds to G = δ ∗ π ∗ G δ ∗ η π ∗ G −−−−−→ δ ∗ π ∗ ( π ) ! π ∗ G = ( π ) ! π ∗ G. This is an equivalence (using the first paragraph of the proof) with a left inverse (cid:15) G = δ ∗ π ∗ (cid:15) G , which is thus also an equivalence. (cid:3) Remark.
A formal consequence of the Yoneda Lemma [Lur09, 5.1.3] is thatthe mapping spaces in the ∞ -categorical localization (N Φ)[ I − ] can be identifiedas Map (N Φ)[ I − ] ( c , c ) (cid:39) Spcl(Φ( c , − ))( c ) . Definition.
Let Q I ( b , b ) be the category of cospans b p → b i ← b with p ∈ P and i ∈ I . Forgetting the map p and composing with the map ιb → b defines afunctor j : Q I ( b , b ) → Φ b . Recall that we also have the functor π : Φ b → Φ. Anobject c ∈ C is called simple if the map ιc → c is an isomorphism.4.27. Lemma.
For a simple b , the natural transformation ∆ → j ∗ π ∗ Φ( b , − ) , ( p, i ) (cid:55)→ p of functors Q I ( b , b ) → S exhibits the functor π ∗ Φ( b , − ) as a left Kan extension π ∗ Φ( b , − ) (cid:39) j ! (∆ ) along j . Here ∆ is the constant functor associated with the point and S the ∞ -category of spaces.Proof. We fix a map i (cid:48) : ιb → b (cid:48) ∈ I and consider the slice category j /i (cid:48) withobjects the diagrams b b b ιb b (cid:48) p i i (cid:48) Sending such a diagram to the composite b → b (cid:48) defines a functor to the discretecategory Φ( b , b (cid:48) ) = π ∗ Φ( b , − )( i (cid:48) ), which is a left adjoint by unique factorization(Property (c) of Definition 4.20). In fact the right adjoint is given by the uniquefactorization and using that b is simple. The claim now follows from the pointwise HE SPECTRUM OF DERIVED MACKEY FUNCTORS 35 formula for Kan extensions [Lur09, Lemma 4.3.2.13 and Definition 4.3.3.2] andcofinality [Lur09, Theorem 4.1.3.1]. (cid:3)
Corollary.
The localization of the corepresented functor Φ( c , − ) evaluatedat a simple object c is given by Spcl(Φ( c , − ))( c ) (cid:39) | Q I ( c , c ) | and the universal arrow is induced by the functor Φ( c , c ) → Q I ( c , c ) that sends f = i ◦ p : c → c to the cospan c p −→ c ← ιc ∼ = c . Example.
In the case of the complementary pair from Example 4.21 (whereΦ = S C ) the map from the above corollary is given as follows: S C ( X, Y ) = C ( Y, X ) −→ Q I ( X, Y )( Y → X ) = (([0] , X ) → ([0] , Y )) (cid:55)−→ ([0] , X ) → ([0] , Y ) = ([0] , Y )for simple objects ([0] , X ) and ([0] , Y ), i.e. X, Y ∈ C .We can now show that the effective Burnside category A eff ( C ) is obtained from S C by inverting the special maps: Proof of Theorem 4.19.
The functor N S C → A eff ( C ) is clearly essentially surjectiveand so it suffices to consider its effect on mapping spaces. By inspection of thedefinitions one sees that the compositeMap N S C ( X, Y ) −→ Map A eff ( C ) ( X, Y ) (cid:39) −→ N Span C ( X, Y ) ∼ is induced by the map S C ( X, Y ) → Span C ( X, Y ) that sends X f ←− Y to the span X f ←− Y = Y . It factors as the composition S C ( X, Y ) −→ Q I ( X, Y ) −→ Span C ( X, Y ) ∼ of the map from Corollary 4.28 with the second functor defined by sending a cospan([0] , X ) → ([ n ] , Z • ) ← ([0] , Y )to the span X f ←− Z → Z n ∼ = Y .This is a right adjoint and so induces an equivalence on classifying spaces. Hencethe above composite is a universal arrow. It follows from Corollary 4.28 thatMap (N S C )[ I − ] ( X, Y ) (cid:39) Spcl( S C ( X, − ))( Y ) (cid:39) −→ Map A eff ( C ) ( X, Y )is a weak equivalence. Finally, we note that any object in N S C [ I − ] is equivalentto an object of the form ([0] , X ). (cid:3) Remark.
Let C be a small category with pullbacks and a terminal object ∗ .The cartesian product of C does not induce a symmetric monoidal structure on S C .It only yields a functor S C × ∆ S C ∼ = S ( C × C ) → S C . However it passes to a product S ( C × C ) S C ( S C )[ I − ] × ( S C )[ I − ] ( S C )[ I − ]on the localization. Here the left vertical arrow exhibits the target as the localization S ( C × C )[ I − ] which follows from the description of the mapping spaces in the proofof Theorem 4.19. We now promote this to a full symmetric monoidal structure.4.31. Construction.
We use notation from [Lur17, Section 2.1.1]. The symbol (cid:104) n (cid:105) stands for the pointed set {∗ , , , . . . , n } and (cid:104) n (cid:105) ◦ for the set { , , . . . , n } . Wewrite Fin ∗ for the category of finite pointed sets.The product on C can be rectified to a functor C × : Fin ∗ −→ Cat , (cid:104) n (cid:105) (cid:55)→ Fun Π ( P ( (cid:104) n (cid:105) ◦ ) , C )where P ( − ) denotes the poset of all subsets and Fun Π ( P ( (cid:104) n (cid:105) ◦ ) , C ) is the full sub-category of those functors F : P ( (cid:104) n (cid:105) ◦ ) → C such that for every S ⊆ (cid:104) n (cid:105) ◦ restrictingto the elements of S exhibits F ( S ) ∼ = (cid:81) s ∈ S F ( { s } ) as a product ( F ( ∅ ) = ∗ ). Inparticular, evaluating at the elements of (cid:104) n (cid:105) ◦ induces an equivalenceFun Π ( P ( (cid:104) n (cid:105) ◦ ) , C ) (cid:39) −→ C n . The construction S ( − )[ I − ] preserves products and equivalences. Applying it ob-jectwise, we thus obtain a functor S C × [ I − ] : N Fin ∗ → Cat ∞ that still satisfies the Segal conditions S C × [ I − ]( (cid:104) n (cid:105) ◦ ) (cid:39) ( S C )[ I − ] n and thus en-codes a symmetric monoidal structure on S C [ I − ]. Similarly, we obtain a ‘pointwise’symmetric monoidal structure A eff × ( C ) = A eff ◦ C × : N Fin ∗ → Cat ∞ using the fact that A eff preserves products and sends equivalences of ordinary cat-egories to equivalences of ∞ -categories.4.32. Corollary.
The comparison functor N S C → A eff ( C ) of Theorem 4.19 inducesan equivalence S C × [ I − ] ∼ −→ A eff × ( C ) of symmetric monoidal ∞ -categories. Remark.
The symmetric monoidal structure we have constructed on A eff ( C )agrees with the one constructed in [BGS20, Section 2]. To explain this, we need torecall the relative nerve construction (see [Lur09, Section 3.2.5]).4.34. Definition.
Let f : D → Set ∆ be a functor from a small category D to thecategory of simplicial sets. The relative nerve of f is the simplicial set N f with an n -simplex consisting of a chain d φ −→ d φ −→ · · · φ n − −→ d n in D together with a collection of diagrams { τ J : ∆ J → f ( d j ) } J ⊆ [ n ] indexed bythe non-empty subsets of [ n ], where j ∈ J denotes the maximal element of J .These are compatible in the following sense: For any inclusion of non-empty subsets HE SPECTRUM OF DERIVED MACKEY FUNCTORS 37 J (cid:48) ⊂ J ⊂ [ n ], with maximal elements j (cid:48) ≤ j , the diagram∆ J (cid:48) (cid:127) (cid:95) (cid:15) (cid:15) τ J (cid:48) (cid:47) (cid:47) f ( d j (cid:48) ) (cid:15) (cid:15) ∆ J τ J (cid:47) (cid:47) f ( d j )commutes.4.35. Remark.
Let D be a small category and F : D → Cat a functor such that f ( d ) has pullbacks for any object d and for any φ : d → d (cid:48) , the functor F ( φ )preserves pullbacks. Then we get a functor A eff ◦ F : D → Set ∆ . It follows from[Lur09, 3.2.5.21] that the canonical map N A eff F → N D is a cocartesian fibrationclassified by the functor A eff ◦ F : N D → Cat ∞ . Now suppose that C is a smallcategory with pullbacks and a terminal object. Consider the special case of thelatter D = Fin ∗ and F = Fun Π ( P ( (cid:104)−(cid:105) ◦ ) , C ). A direct but tedious inspection showsthat N A eff F agrees with the total space A eff ( C ) ⊗ of the cocartesian fibration of[BGS20, Proposition 2.14 and Notation 2.6]. As a consequence we obtain:4.36. Corollary.
The cocartesian fibration A eff ( C ) ⊗ → N Fin ∗ is classified by thefunctor A eff × ( C ) = A eff ◦ C × : N Fin ∗ → Cat ∞ . Consequently, the symmetric monoidal structure on A eff ( C ) defined in [BGS20] agrees with the symmetric monoidal structure provided by Construction 4.31. Corollary.
For any finite group G , there is an equivalence of symmetricmonoidal ∞ -categories Fun add ( A eff (Fin G ) , H Z - Mod) (cid:39)
Fun add ( S (Fin G )[ I − ] , H Z - Mod) where both sides are equipped with localized Day convolutions.Proof.
Corollary 4.36 provides an equivalence of symmetric monoidal ∞ -categoriesFun( A eff (Fin G ) , H Z - Mod) (cid:39) Fun( S (Fin G )[ I − ] , H Z - Mod)where both sides are equipped with Day convolution products. Additive functorscorrespond under this equivalence and after localizing we get the desired compati-bility of localized Day convolution products [BGS20, Lemma 3.7]. (cid:3) Definition.
Consider the derived category D(Fun( S Fin G , Ab)) of the abeliancategory of all functors S Fin G → Ab. Note that the objects of the derived categorycan be regarded as functors S Fin G → Ch( Z ). The triangulated category of derivedMackey functors in the sense of Kaledin can be defined as the full subcategoryDMack( G ) ⊂ D(Fun( S Fin G , Ab))consisting of all functors S Fin G → Ch( Z ) which send the special morphisms of S Fin G to quasi-isomorphisms and which are “additive” in the sense that theirrestriction along the embedding Fin op G (cid:44) → S Fin G results in a functor Fin op G → Ch( Z ) that preserves products up to quasi-isomorphism. In Kaledin’s notation thiscategory is denoted DS add (Fin G , Ab) and defined in [Kal11, Def. 4.1 and Def. 4.11].It follows from [Kal11, Thm. 4.2 and Prop. 4.7] that it is equivalent to the categorydefined in [Kal11, Def. 3.3] (cf. Remark 4.47 below).
Remark.
For any small category C , the abelian categoryCh(Fun( C , Ab)) ≡ Fun( C , Ch( Z ))can be equipped with the projective model structure. The underlying ∞ -categoryD ∞ (Fun( C , Ab)) is equivalent to the functor ∞ -category Fun(N C , D ∞ ( Z )), whereD ∞ ( Z ) denotes the underlying ∞ -category of D( Z ). In particular, we have anequivalence of triangulated categories(4.40) D(Fun( C , Ab)) ∼ = Ho(Fun(N C , D ∞ ( Z ))) . If I ⊂ Mor( C ) is a class of morphisms, then under the equivalence (4.40), thefull subcategory of D(Fun( C , Ab)) consisting of functors that send morphisms in I to quasi-isomorphisms is equivalent to the full subcategory of the right-hand sideconsisting of functors of ∞ -categories N C → D ∞ ( Z ) which send morphisms in I to equivalences in D ∞ ( Z ). Similarly, if C is semi-additive, the full subcategory ofD(Fun( C , Ab)) consisting of functors which are additive up to quasi-isomorphismis equivalent to the full subcategory of additive functors N C → D ∞ ( Z ). Also notethat there is a symmetric monoidal equivalence D ∞ ( Z ) ∼ = H Z - Mod by [Lur17,Thm. 7.1.2.13] which is an ∞ -categorical version of a theorem of Schwede andShipley [SS03].4.41. Proposition.
There is an equivalence of triangulated categories
Ho(Fun add ( A eff (Fin G ) , H Z - Mod)) ∼ = DMack( G ) for any finite group G .Proof. According to Definition 4.38 and Remark 4.39, DMack( G ) is equivalent tothe full subcategory of Ho(Fun(N S Fin G , H Z - Mod)) consisting of those functorswhich send the special morphisms in S Fin G to equivalences in H Z - Mod and whoserestriction along NFin op G (cid:44) → N S Fin G sends products to direct sums. By the univer-sal property of localization we have an equivalence of ∞ -categoriesFun I (N S Fin G , H Z - Mod) (cid:39) Fun(N S (Fin G )[ I − ] , H Z - Mod)and since N S Fin G [ I − ] (cid:39) A eff (Fin G ) by Theorem 4.19, it remains to understandthe additivity condition. To this end note that the compositeFin op G (cid:44) → S Fin G (cid:16) S Fin G [ I − ] (cid:39) A eff (Fin G )is the (opposite) of the usual embedding. This embedding Fin op G (cid:44) → A eff (Fin G ) sendsproducts to direct sums (see the proof of [Bar17, Prop. 4.3]) and, consequently, sincethe embedding is surjective on objects, a functor A eff (Fin G ) → H Z - Mod preservesdirect sums if and only if the composite Fin op G (cid:44) → A eff (Fin G ) → H Z - Mod sendsproducts to direct sums. The diagramN S Fin G H Z - ModNFin op G A eff (Fin G ) F F then shows that for a functor F : N S Fin G → H Z - Mod which maps special mor-phisms to equivalences, the induced functor F is additive if and only if the restric-tion of F to Fin op G preserves products. Putting everything together, we concludethat the homotopy category ofFun add (N S Fin G [ I − ] , H Z - Mod) (cid:39) Fun add ( A eff (Fin G ) , H Z - Mod) HE SPECTRUM OF DERIVED MACKEY FUNCTORS 39 is equivalent to the triangulated category DMack( G ) of Definition 4.38. (cid:3) Corollary.
For any finite group G , there is an equivalence of triangulatedcategories D(H Z G ) ∼ = DMack( G ) Proof.
This follows from Corollary 4.11, Corollary 4.37 and Proposition 4.41. (cid:3)
Remark.
The last thing that remains is to compare the monoidal structure ofD(H Z G ) with the monoidal structure of DMack( G ) defined in [Kal11, Section 5.2].This monoidal structure is actually constructed on an equivalent triangulated cat-egory (denoted DMack Q ( G ) below) which Kaledin constructs using A ∞ -categories.We briefly recall the construction.4.44. Remark.
Every 2-category C gives rise to an associated A ∞ -category B ( C )which has the same objects and whose complex of morphisms (for a pair of objects X and Y ) is the simplicial chain complex C • ( C ( X, Y )) of the nerve of the category C ( X, Y ). See [Kal11, Sections 1.5–1.6] for details. We can then consider its derivedcategory of A ∞ -modules D( B ( C )), that is, the derived category of A ∞ -functors from B ( C ) to the dg-category Ch( Z ) (see [Kel01] for instance).4.45. Example.
Applied to an ordinary category C , the resulting A ∞ -category B ( C )is quasi-isomorphic to the additive category Z [ C ] (obtained by linearizing the homsets) regarded as a dg-category whose morphism complexes are concentrated indegree 0. In this case, an A ∞ -module is just an ordinary functor C → Ch( Z ) andthe derived category D( B ( C )) is simply the derived functor category D(Fun( C , Ab)).4.46.
Definition.
Let Q (Fin G ) denote the (2 , G -setswhose 2-morphisms are the isomorphisms of spans. The usual embedding Fin op G (cid:44) → Q (Fin G ) can be regarded as a 2-functor and it induces a map between the associated A ∞ -categories Z [Fin op G ] → B ( Q (Fin G )). Kaledin considers the full subcategoryDMack Q ( G ) ⊂ D( B ( Q (Fin G )))of the derived category of A ∞ -modules consisting of those A ∞ -functors B ( Q (Fin G )) → Ch( Z )that are additive in the sense that their restriction (to an ordinary functor) Fin op G → Ch( Z ) preserves products up to quasi-isomorphism (see [Kal11, Definition 3.2]).4.47. Remark.
The category DMack Q ( G ) is equivalent to the triangulated categoryDMack( G ) defined in Definition 4.38. This is established in [Kal11, Section 4] butwe reformulate the result in a way that suits our purposes. Regarding S (Fin G ) asa discrete 2-category, there is a 2-functor φ : S (Fin G ) → Q (Fin G ) given on objectsand morphisms exactly like the functor N S (Fin G ) → A eff (Fin G ) from Construc-tion 4.18 (see [Kal11, Section 4.2]). This functor φ preserves products (disjointunions) when restricted to Fin op G and sends morphisms in I to equivalences. Fur-ther, it induces an A ∞ -functor Z S (Fin G ) → B ( Q (Fin G )) where Z S (Fin G ) is thelinearization of the category S (Fin G ) (see Example 4.45). Restricting along this A ∞ -functor provides a triangulated functor(4.48) χ : DMack Q ( G ) → DMack( G )which one proves is an equivalence by combining [Kal11, Theorem 4.2] and [Kal11,Proposition 4.7]. Construction.
We now recall the symmetric monoidal structure that [Kal11,Section 5.2] constructs on DMack Q ( G ). The Cartesian product of finite G -setsinduces a 2-functor m : Q (Fin G × Fin G ) → Q (Fin G )which provides a functor m ∗ : D( B ( Q (Fin G ))) → D( B ( Q (Fin G × Fin G ))) . Let m add! denote the left adjoint of the composite functorDMack Q ( G ) (cid:44) → D( B ( Q (Fin G ))) m ∗ → D( B ( Q (Fin G × Fin G ))) . Using the Alexander–Whitney map one can define an external productD( B ( Q (Fin G ))) × D( B ( Q (Fin G ))) (cid:2) → D( B ( Q (Fin G × Fin G ))))and the symmetric monoidal product on DMack Q ( G ) is defined to be the composite m add! ◦ (cid:2) ◦ ( i × i ) where i : DMack Q ( G ) (cid:44) → D( B ( Q (Fin G ))) denotes the inclusion.The original category DMack( G ) then obtains a symmetric monoidal structure viathe equivalence 4.48.4.50. Theorem.
There is an equivalence of tensor triangulated categories
D(H Z G ) ∼ = DMack( G ) for any finite group G .Proof. We have already established a symmetric monoidal equivalenceH Z G - Mod (cid:39) Fun add ( S (Fin G )[ I − ] , H Z - Mod)which provides a triangulated equivalence D(H Z G ) ∼ = DMack( G ) at the level ofhomotopy categories (Cor. 4.42). It remains to show that the triangulated equiva-lence χ : DMack Q ( G ) ∼ −→ DMack( G ) of Remark 4.47 is symmetric monoidal whenDMack( G ) is equipped with the symmetric monoidal structure induced by the lo-calized Day convolution product on Fun add ( S (Fin G )[ I − ] , H Z - Mod).To this end, note that we have a strictly commutative diagram of 2-functors(4.51) S (Fin G ) × S (Fin G ) Q (Fin G ) × Q (Fin G ) S (Fin G × Fin G ) Q (Fin G × Fin G ) S (Fin G ) Q (Fin G ) φ × φm φ ∼ mφ HE SPECTRUM OF DERIVED MACKEY FUNCTORS 41 where m : S (Fin G × Fin G ) → S (Fin G ) is induced from the Cartesian product.We can apply the B ( − ) construction (Rem. 4.44) to obtain a corresponding di-agram of A ∞ -categories. For example, the bottom arrow of the diagram be-comes the A ∞ -functor Z S (Fin G ) → B Q (Fin G ) which in turn provides a func-tor D( B Q (Fin G )) → D( Z S (Fin G )) ∼ = D(Fun( S (Fin G ) , Ab)) by restriction (Exam-ple 4.45). Moreover, we can add the following piece to the top of the diagram(4.52) Z S (Fin G ) ⊗ Z S (Fin G ) B Q (Fin G ) ⊗ B Q (Fin G ) Z [ S (Fin G ) × S (Fin G )] B ( Q (Fin G ) × Q (Fin G )) B ( φ ) ⊗B ( φ ) ∼ B ( φ × φ ) where ⊗ denotes tensor product of A ∞ -categories and where the right-hand verticalfunctor is induced by the Alexander–Whitney map. That the diagram commutessimply follows from the observation that the Alexander–Whitney map is an isomor-phism in degree zero (given by the Cartesian product of basis sets). Finally, let’sarm ourselves with the following commutative diagram of ∞ -categories(4.53) S (Fin G ) × S (Fin G ) S (Fin G )[ I − ] × S (Fin G )[ I − ] S (Fin G × Fin G ) S (Fin G × Fin G )[ I − ] S (Fin G ) S (Fin G )[ I − ] q × qm q ∼ mq where we have omitted nerves in the notation (cf. Rem. 4.30). We are now preparedto compare the monoidal structures. WritingD( D ) := D(Fun( D , Ab)) ∼ = Ho(Fun(N D , H Z - Mod))for an ordinary category D and then using the abbreviationsDM := DMack( G ) , DM Q := DMack Q ( G ) , and C := Fin G , consider the following diagram DM Q × DM Q D( B Q C ) × D( B Q C ) D( B Q ( C × C )) DM Q D( S ( C × C )[ I − ]) DMDM × DM D( S C [ I − ]) × D( S C [ I − ]) D( S C [ I − ] × S C [ I − ]) D( S C [ I − ]) . i × i (1) ∼ χ × χ ∼ φ ∗ × φ ∗ (cid:2) ∼ m add! (3) ∼ χ (2) ∼ (cid:101) m add! (4) i × i (cid:2) t ! add Here (cid:101) m add! denotes the left adjoint of restriction alongDMack( G ) (cid:44) → D( S (Fin G )[ I − ]) m ∗ −−→ D( S (Fin G × Fin G )[ I − ])(cf. Cons. 4.49) and t ! denotes the left adjoint of restriction along S (Fin G )[ I − ] × S (Fin G )[ I − ] → S (Fin G )[ I − ] . The lower (cid:2) is the external product Fun( A , B ) × Fun( A , B ) → Fun( A × A , B ) for thesymmetric monoidal ∞ -categories A = S (Fin G )[ I − ] and B = H Z - Mod. The com-posite t ! ◦ (cid:2) is the Day convolution on Fun( S (Fin G )[ I − ] , H Z - Mod) (see [Nik16]). The commutativity of the diagram (up to isomorphism) can be established asfollows. Region (1) comes directly from the definition of the comparison functor χ (Rem. 4.47) while the more involved region (2) can be checked using the commu-tative diagrams (4.51), (4.52) and (4.53). For the commutativity of (3) note thatDMack Q ( G ) D( B Q (Fin G )) D( B Q (Fin G × Fin G ))DMack( G ) D( S (Fin G )[ I − ]) D( S (Fin G × Fin G )[ I − ]) χ φ ∗ m ∗ φ ∗ m ∗ commutes. Kaledin [Kal11, Section 4] establishes that all three vertical functorsare equivalences. Hence, we can replace the top and bottom rows with their leftadjoints and the diagram still commutes. Finally, the commutativity of region (4)is immediate from the definitions. This completes the proof that the localizedDay convolution on DMack( G ) coincides with the symmetric monoidal structureon DMack( G ) ∼ = DMack Q ( G ) constructed by Kaledin. (cid:3) Modules over the Burnside ring Mackey functor
Instead of considering modules over the equivariant ring spectrum H Z G :=triv G (H Z ), a natural alternative is to consider modules over the equivariant ringspectrum H A G , that is, the Eilenberg-MacLane G -spectrum associated to the Burn-side G -Mackey functor A G . As observed by Greenlees–Shipley [GS14, Section 5],the derived category of H A G -modules is equivalent to the ordinary derived cate-gory of G -Mackey functors. We will begin by providing a new proof of this fact —one which takes the monoidal structures into account — and then explain how thestory changes with H Z G replaced by H A G (i.e. with the category of derived Mackeyfunctors replaced with the ordinary derived category of Mackey functors).5.1. Remark.
The ordinary abelian category of G -Mackey functors M ack ( G ) isequivalent to the heart of the standard t -structure on Sp G . Under this equiva-lence, every Mackey functor M ∈ M ack ( G ) is associated to its Eilenberg-MacLane G -spectrum H M ∈ Sp G . The inclusion N M ack ( G ) ∼ = (Sp G ) ♥ (cid:44) → Sp G is lax sym-metric monoidal and hence induces a functorNCAlg( M ack ( G )) = CAlg(N M ack ( G )) → CAlg(Sp G ) . Thus the Eilenberg-MacLane G -spectrum of a commutative Green functor has thestructure of a commutative algebra in the symmetric monoidal ∞ -category Sp G .5.2. Remark.
The homotopy category Ho( A eff (Fin G )) of the effective Burnside ∞ -category (Def. 4.16) is the ordinary effective Burnside category B eff G whose ob-jects are finite G -sets and whose morphisms are isomorphism classes of spans. Itis a semi-additive category whose group completion is the usual Burnside cate-gory B G . Since the category of abelian groups is additive, there is an equivalenceFun add ( B G , Ab) ∼ = Fun add ( B eff G , Ab). In other words, although G -Mackey functorsare usually defined to be additive functors B G → Ab, they can equivalently bedefined as functors B eff G → Ab which are “additive” in the sense that they preservebiproducts (equivalently, preserve products).5.3.
Remark.
For each finite G -set T ∈ B G , we have the evaluation functorev T : M ack ( G ) → Ab M (cid:55)→ M ( T ) HE SPECTRUM OF DERIVED MACKEY FUNCTORS 43 which, by the Yoneda lemma, is representable: ev T ∼ = Hom M ack ( G ) ( M T , − ) where M T := Hom B G ( T, − ) ∈ M ack ( G ) is the Mackey functor represented by T ∈ B G .5.4. Example.
The Burnside G -Mackey functor A G is the representable Mackeyfunctor M G/G . It is the unit for the Day convolution product on M ack ( G ) (seeExample 5.9 below).5.5. Definition.
For any finite group G , let H A G ∈ CAlg(Sp G ) denote the Eilenberg–MacLane G -spectrum associated to the Burnside G -Mackey functor A G (see Re-mark 5.1 and Example 5.4). We letD(H A G ) := Ho(H A G - Mod Sp G )denote the homotopy category of the ∞ -category of H A G -modules.5.6. Lemma.
The triangulated category D( M ack ( G )) is compactly generated bythe set of Mackey functors (cid:8) M G/H (cid:12)(cid:12) H ≤ G (cid:9) regarded as complexes concentratedin degree 0.Proof. Since every finite G -set is a finite coproduct of orbits, a Mackey functor N is zero in M ack ( G ) if and only if ev G/H ( N ) = 0 for all H ≤ G . Since the functorev G/H : M ack ( G ) → Ab is exact, the representing object M G/H is projective, andwe have(5.7) Hom D( M ack ( G )) ( M G/H [ n ] , N • ) ∼ = H n (ev G/H ( N • )) ∼ = ev G/H ( H n ( N • ))for any complex of Mackey functors N • . A complex is thus zero in D( M ack ( G ))if and only if (5.7) vanishes for all H ≤ G and n ∈ Z . Morever, (5.7) also showsthat the M G/H [0] are compact objects of D( M ack ( G )) since the right-hand sidecommutes with coproducts. (cid:3) Remark. If C is a small symmetric monoidal additive category, then the categoryof additive functors Fun add ( C , Ab) is closed symmetric monoidal with respect to theadditive Day convolution. The tensor product is given by the coend( F ⊗ add G )( c ) = (cid:90) ( c ,c ) F ( c ) ⊗ G ( c ) ⊗ C ( c ⊗ c , c )which implicitly uses that the target category Ab is copowered over the enrichingcategory (Ab itself). This is not the same as the Day convolution on Fun( C , Ab)that does not use the Ab-enrichment. For example, the unit of the additive Dayconvolution on Fun add ( C , Ab) is the functor which maps c ∈ C to the abelian group C ( , c ), while the unit of the non-enriched Day convolution on Fun( C , Ab) is thefunctor which maps c ∈ C to the free abelian group generated by the set C ( , c ).5.9. Example.
The category of Mackey functors M ack ( G ) = Fun add ( B G , Ab) isclosed symmetric monoidal under the additive Day convolution (with respect to thesymmetric monoidal structure on B G induced from the cartesian product of finite G -sets). This symmetric monoidal structure is sometimes called the “box product”of Mackey functors. The unit is the Mackey functor corepresented by the unit ofthe monoidal structure on B G , that is, the Burnside Mackey functor A G = M G/G .We equip the derived category D( M ack ( G )) with the derived symmetric monoidalstructure. Theorem.
There is an equivalence of tensor triangulated categories
D(H A G ) ∼ = D( M ack ( G )) for any finite group G .Proof. For any pointed simplicial presheaf Y ∈ Fun( O ( G ) op , sSet ∗ ), the relativenormalized simplicial chain complex (cid:101) C • ( Y ) := C • ( Y ( − ) , ∗ ) can be regarded as acomplex of coefficient systems. This provides a functor(5.11) (cid:101) C • ( − ) : Fun( O ( G ) op , sSet ∗ ) → Ch(Fun( O ( G ) op , Ab))which, using the Eilenberg–Zilber map, is lax symmetric monoidal with respect tothe pointwise smash product onFun( O ( G ) op , sSet ∗ )and the pointwise monoidal structure onCh(Fun( O ( G ) op , Ab)) = Fun( O ( G ) op , Ch( Z )) . To pass from coefficient systems to Mackey functors, we use the induction functor i ! : Fun( O ( G ) op , Ab) → M ack ( G ), which is left adjoint to the restriction functoralong i : O ( G ) op (cid:44) → B G . The induction functor i ! is symmetric monoidal withrespect to the pointwise monoidal structure on the category of coefficient systemsand the box product monoidal structure on the category of Mackey functors (seeExample 5.9). It then induces a symmetric monoidal functor(5.12) i ! : Ch(Fun( O ( G ) op , Ab)) → Ch( M ack ( G ))on the categories of chain complexes.These categories are symmetric monoidal model categories when equipped withthe projective model structures, and the functors (5.11) and (5.12) are left Quillenfunctors. We denote the composite i ! ( (cid:101) C • ( − )) by (cid:101) C • ( − , A G ). This choice of notationcan be explained as follows: If X is a pointed G -simplicial set then G/H (cid:55)→ X H gives a cofibrant pointed simplicial presheaf on O ( G ) and a straightforward calcu-lation (using [MPN06, Section 3], for example) shows that the associated complexof Mackey functors i ! ( (cid:101) C • ( X ( − ) )) is nothing but the Mackey-valued Bredon chaincomplex of X with coefficients in A G (see [tD87, Section II.9]).All told, we have a lax symmetric monoidal left Quillen functor (cid:101) C • ( − , A G ) : Fun( O ( G ) op , sSet ∗ ) → Ch( M ack ( G ))whose lax monoidal structure maps are weak equivalences. By [NS18, Theorem A.7](see also [Lur17, Example 4.1.7.6]), we obtain a symmetric monoidal left adjoint(5.13) (cid:101) C • ( − , A G ) : S G ∗ → D ∞ ( M ack ( G ))between the underlying symmetric monoidal ∞ -categories. We would like to showthat (5.13) induces a symmetric monoidal functor Sp G → D ∞ ( M ack ( G )). To in-voke Theorem 3.4, we need to show that (cid:101) C • ( S ρ G , A G ) is invertible. This can bechecked at the level of homotopy categories. Indeed, given a finite G -CW spec-trum X (i.e. a finite G -spectrum with a preferred cellular decomposition in thehomotopy category), we can define the cellular chain complex C cell • ( X, A G ) usingthe relative equivariant stable homotopy groups (see [BDP17, Section 4]) and it isstraightforward to check that the following equivalences hold in D( M ack ( G )): C cell • (Σ ∞ ( G/G + ) , A G ) (cid:39) M G/G [0] = A G [0] HE SPECTRUM OF DERIVED MACKEY FUNCTORS 45 and C cell • ( X ∧ Y, A G ) (cid:39) C cell • ( X, A G ) ⊗ C cell • ( Y, A G ) . Moreover, it follows from [BDP17, Section 4] that (cid:101) C • ( S ρ G , A G ) is quasi-isomorphicto C cell • ( S ρ G , A G ). Since orbits are self-dual, a cellular structure on S ρ G inducesa cellular structure on S − ρ G and the cellular chain complex C cell • ( S − ρ G , A G ) is aninverse of (cid:101) C • ( S ρ G , A G ) in D( M ack ( G )). We can thus invoke Theorem 3.4 and assertthat there is an essentially unique symmetric monoidal left adjoint L : Sp G → D ∞ ( M ack ( G ))such that L ◦ Σ ∞ (cid:39) (cid:101) C • ( − , A G ). If R denotes a right adjoint to L then(5.14) π H ∗ R ( N • ) ∼ = H ∗ ( N • ( G/H ))for any complex of Mackey functors N • since L (Σ ∞ G/H + ) (cid:39) M G/H [0]. More-over, since L sends a set of compact generators to a set of compact generators(see Lemma 5.6), the right adjoint R commutes with colimits and is conserva-tive. Hence the adjunction is monadic by the Barr–Beck–Lurie Theorem [Lur17,Theorem 4.7.3.5]. Since the projection formula holds, we conclude that there is asymmetric monoidal equivalenceD ∞ ( M ack ( G )) (cid:39) R ( ) - Mod Sp G and it remains to show that R ( ) is equivalent as a commutative algebra to H A G .The unit map S → R ( ) = R ( A G [0]) is a morphism of commutative algebras andtruncates to an isomorphism π ( S ) ∼ −→ π ( R ( A G [0])) in the heart (see Rem. 5.1and [Lur17, Exa. 2.2.1.10]), since both sides are abstractly isomorphic to A G andthe latter is the initial commutative Green functor. This provides an isomorphismH A G ∼ −→ R ( ) since R ( ) has non-trivial homotopy groups only in degree 0. (cid:3) Remark.
Many of the basic features of the category D(H Z G ) developed inSection 3 hold just as well for D(H A G ) by simply replacing all instances of H Z G with H A G . Indeed, properties (A)–(E) all hold for D(H A G ). The crucial propertythat does not hold is property (F) which says that we obtain D(H Z ) if we kill off allthe generators of D(H Z G ) associated to proper subgroups. From the authors’ pointof view, the geometric fixed point functor Φ G of an equivariant category is thislocalization killing off all the generators for proper subgroups (morally, killing ev-erything that comes by induction from proper subgroups). For SH( G ) the result ofthis localization is the category associated with the trivial subgroup: the nonequiv-ariant stable homotopy category SH. Property (F) asserts that the same is true forthe category of derived Mackey functors D(H Z G ): the result of the localization isD(H Z ) ∼ = D(H Z ). We will show below that property (F) fails for D(H A G ) even forthe smallest nontrivial group G = C . Nevertheless, Proposition 3.18 does providea general description of the localization, as follows:5.16. Corollary.
For any finite group G , the finite localization of H A G - Mod Sp G associated to the set (cid:8) F G ( G/H + ) (cid:12)(cid:12) H (cid:12) G (cid:9) is, up to equivalence, the functor onmodule categories H A G - Mod Sp G → Φ G (H A G ) - Mod Sp induced by the geometric fixed point functor Φ G : Sp G → Sp . In particular, D(H A G ) / Loc ⊗ (cid:104) F G ( G/H + ) | H (cid:12) G (cid:105) ∼ = D(Φ G (H A G )) . Remark.
In other words, the target of the “geometric fixed point” functor Φ G associated to the category D(H A G ) is the derived category D(Φ G (H A G )) of the ringspectrum Φ G (H A G ) ∈ CAlg(Sp). More generally, for any subgroup H ≤ G , thetarget of the geometric fixed point functor Φ H on D(H A G ) is the derived categoryof the ring spectrum Φ H (H A G ) ∼ = Φ H (H A H ) ∈ CAlg(Sp). The heart of the issueis that the equivariant Eilenberg-MacLane spectra H A G do not behave well withrespect to geometric fixed points. For categorical fixed points, it is immediate fromthe definition that the non-equivariant spectrum (H A G ) H is the Eilenberg-MacLanespectrum H A ( H ) associated to the Burnside ring A ( H ). In contrast, the geometricfixed points Φ H (H A G ) seem more mysterious and more complicated.5.18. Proposition.
The homotopy ring π ∗ Φ C H A C is isomorphic to the gradedring Z [ x ] / (2 x ) where x has degree two. In particular, Φ C H A C is not equivalent to H Z .Proof. Let H Z denote the C -equivariant Eilenberg-MacLane spectrum associatedto the constant Mackey functor Z . It follows from the Tate square of [GM95] and theTate cohomology of C that π ∗ Φ C H Z is the polynomial algebra F [ x ] where x hasdegree two. Comparing the isotropy separation sequences for H Z and H A C , we con-clude that the canonical map Φ C H A C → Φ C H Z induces an isomorphism on homo-topy groups in positive degrees. On the other hand, again using the isotropy separa-tion sequence, we know that π of geometric fixed points is isomorphic to π of gen-uine fixed points modulo proper transfers. Hence π Φ C H A C ∼ = A ( C ) / [ C ] ∼ = Z .This completes the proof. (cid:3) Corollary.
The derived category of the integers
D(H Z ) is not equivalent as atriangulated category to the homotopy category of modules over Φ C H A C .Proof. If P is any compact generator in D(H Z ) then it is a perfect complex andhence quasi-isomorphic to a direct sum of finitely generated abelian groups. Thisimplies that the endomorphism ring spectrum of P has bounded homotopy groups.But the homotopy groups of Φ C H A C are not bounded by Proposition 5.18. (cid:3) Remark.
This demonstrates that for the ordinary derived category of Mackeyfunctors D( M ack ( G )) ∼ = D(H A G ), the target category of the geometric fixed pointfunctor Φ H varies with the subgroup H . For example, D(H Z ) is always the targetof the geometric fixed point functor Φ ∼ = res G associated to the trivial subgroup,but if G contains a subgroup H = C then the target of Φ H is D(Φ C (H A C )) whichis not equivalent to D(H Z ). This is in stark contrast to examples like SH( G ) orthe category of derived Mackey functors D(H Z G ) where the geometric fixed pointfunctors Φ H always land in the same category, namely the category associated tothe trivial group. References [Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories.
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