Descent and vanishing in chromatic algebraic K -theory via group actions
aa r X i v : . [ m a t h . K T ] N ov DESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIAGROUP ACTIONS DUSTIN CLAUSEN, AKHIL MATHEW, NIKO NAUMANN, AND JUSTIN NOEL
Abstract.
We prove some K -theoretic descent results for finite group actions on stable ∞ -categories, including the p -group case of the Galois descent conjecture of Ausoni–Rognes. We alsoprove vanishing results in accordance with Ausoni–Rognes’s redshift philosophy: in particular,we show that if R is an E ∞ -ring spectrum with L T ( n ) R = 0, then L T ( n +1) K ( R ) = 0. Our keyobservation is that descent and vanishing are logically interrelated, permitting to establish themsimultaneously by induction on the height. Contents
1. Introduction 12. Mackey functors and equivariant algebraic K -theory 73. Review of nilpotence 134. Descent for p -groups; proof of Theorem A and Theorem B 175. Descent by normal bases; proof of Theorem D 236. Swan induction and applications; proofs of Theorem F and G 257. Swan induction theorems; proof of Theorem E 29Appendix A. Mackey functors and orthogonal G -spectra 39References 451. Introduction
In this paper, we prove some results concerning the algebraic K -theory of ring spectra and stable ∞ -categories after (telescopic) T ( n )-localization. Our starting point is the following two resultsconcerning classical commutative rings R : Theorem 1.1 ([Mit90]) . For n ≥ , we have L T ( n ) K ( R ) = 0 . Theorem 1.2 ([Tho85], [TT90], [CMNN20]) . For G a finite group and R → R ′ a G -Galois exten-sion, the natural comparison map L T (1) K ( R ) → ( L T (1) K ( R ′ )) hG is an equivalence. Date : November 18, 2020. Here our telescopes T ( n ) are taken at a fixed implicit prime p and height n ≥
0; we adopt the convention T (0) = S [1 /p ]. Following [LMMT20], we will also denote the Bousfield localization L T (0) ⊕ ... ⊕ T ( n ) by L p,fn . In otherwords, L p,fn is the localization away from a finite complex of type n + 1 at the prime p . Thus, the K -theory of an ordinary commutative ring has no chromatic information beyond heightone, and the localization to height one is well-behaved in its descent properties. Moving from ordinary rings to more general ring spectra, Ausoni–Rognes suggested that theabove two theorems should fit into a broader “redshift” philosophy in algebraic K -theory, [AR02,AR08]. For an E -ring spectrum R , one expects that taking algebraic K -theory increases the “chro-matic complexity” of R by one. In the setting of Theorem 1.1, the Eilenberg–MacLane spectrum HR has no chromatic information at heights ≥
1, while the result states that K ( R ) = K ( HR )has no chromatic information at heights ≥
2; furthermore, Theorem 1.2 and its refinement tohyperdescent control the height one information very precisely.The notion of chromatic complexity is especially well-behaved when R is an E ∞ -ring, thanks toa theorem of Hahn [Hah16]: if L T ( n ) R = 0, then L T ( m ) R = 0 also for all m > n . If R is an E ∞ -ring,then so is K ( R ), and in this setting one possible expression of the redshift philosophy would bethat L T ( n ) R = 0 ⇔ L T ( n +1) K ( R ) = 0. Here we prove half of this statement:
Theorem A.
Let R be an E ∞ -ring and n ≥ . If L T ( n ) R = 0 , then L T ( n +1) K ( R ) = 0 . This clearly generalizes Mitchell’s vanishing Theorem 1.1. We note that there is a more gen-eral statement which applies also to E -rings A : if both L T ( n ) A = 0 and L T ( n +1) A = 0, then L T ( n +1) K ( A ) = 0; see Corollary 4.9, which is also explored in [LMMT20]. Even ignoring this, onecan directly use Theorem A to prove vanishing of L T ( n +1) K ( A ) provided A is an R -algebra forsome E ∞ -ring R with L T ( n ) R = 0, as then K ( A ) is a module over K ( R ). Variants of this basictrick, that a module over the zero ring is necessarily zero, play a recurring role in this paper.As for Thomason’s descent Theorem 1.2, we have a similar statement, but the hypothesis on T ( n )-local vanishing of R is replaced by the weaker one of T ( n )-local vanishing of the C p -Tateconstruction R tC p (taken with respect to the trivial action). Theorem B.
Let R be an E ∞ -ring and n ≥ . Suppose L T ( n ) ( R tC p ) = 0 . Then for C any R -linear stable ∞ -category equipped with an action of a finite p -group G , the homotopy fixed pointcomparison map for T ( n + 1) -local K -theory is an equivalence: L T ( n +1) K ( C hG ) ∼ −→ ( L T ( n +1) K ( C )) hG . If R → R ′ is a G -Galois extension of commutative rings, then by Galois descent we havePerf( R ) ∼ → Perf( R ′ ) hG ; thus, when n = 0 this theorem recovers the p -group case of Theorem 1.2.But in fact the case of general G in Theorem 1.2 reduces to the p -group case by a simple transferargument, as already pointed out and exploited by Thomason. However, the more general Theo-rem B does not hold true for an arbitrary finite group G , essentially because the G -action is allowedto be arbitrary. In fact, for the trivial action of G on Perf( C ) and n = 0, one can calculate both In fact, the height one theory is even better-behaved than suggested by Theorem 1.2: under mild finitenesshypotheses, the Galois descent can be upgraded to an ´etale hyperdescent result, which leads to a descent spectralsequence from ´etale cohomology to T (1)-local K -theory as produced by [Tho85, TT90]. Furthermore, one knows thatunder such conditions, the map K ( R ; Z p ) → L T (1) K ( R ) is an equivalence in high enough degrees, i.e., one has theLichtenbaum–Quillen conjecture, thanks to the work of Voevodsky–Rost, cf. [RØ06, CM19] for accounts. Howeverwe will not touch on these more advanced aspects in this paper. For clarity, we note that this was not formulated by Ausoni–Rognes; we will discuss the relation with the con-jectures formulated by Ausoni–Rognes later on. Also, for the reverse implication, when n = 0 it may be necessary torestrict to connective R (where the statement does indeed hold by mapping to a separably closed field of characteristic = p and invoking Gabber–Suslin rigidity). For convenience, we actually assume all our stable ∞ -categories are idempotent-complete, though this shouldbe irrelevant. ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 3 sides using Suslin’s equivalence [Sus84] between topological and algebraic K -theory, and for G oforder prime-to- p the result is evidently false, while for G a p -group it amounts to the p -completeAtiyah–Segal completion theorem. On the other hand, there is a generalization of Theorem B toarbitrary finite groups which shows that the descent question for arbitrary G reduces to that forcyclic subgroups of order prime to p ; see Theorem 6.4.We remark that these theorems also hold with K -theory replaced by an arbitrary additive invari-ant, and one also has “co-descent” or “assembly map” equivalences dual to the descent statementsof Theorem B; see Proposition 4.1 for more details.Theorem A clearly applies to R = L p,fn − S ; and recalling Kuhn’s “blueshift” theorem [Kuh04]that if X is L p,fn -local then X tC p is L p,fn − -local, we see that Theorem B applies to R = L p,fn S . Thuswe deduce the following: Theorem C.
Let n ≥ , and let C be an L p,fn -local stable ∞ -category. Then L T ( m ) K ( C ) = 0 forall m ≥ n + 2 , and for any finite p -group G acting on C we have L T ( n +1) K ( C hG ) ∼ → ( L T ( n +1) K ( C )) hG . In fact, for the proofs of Theorem A and Theorem B we proceed by first proving this special case,Theorem C. Then we combine with a recent result of Land–Mathew–Meier–Tamme [LMMT20] tothe effect that L T ( n ) K ( R ) ∼ → L T ( n ) K ( L p,fn R ) (for n ≥
1) which lets us deduce the general case.(Actually, we also use the result of [LMMT20] in the proof of Theorem C, but in a more indirectway.)An interesting aspect of our arguments is that we show a logical connection between the vanishingand the descent theorems. This is expressed in the following result.
Theorem 1.3 (Inductive vanishing, Lemma 4.7) . Let R be an E ∞ -ring spectrum and n ≥ . Thenfor the following conditions, we have the implications (1) ⇒ (2) ⇒ (3): (1) L T ( n ) R = 0 and L T ( n ) K ( R tC p ) = 0 . (2) For any action of a finite p -group G on an R -linear stable ∞ -category C , the comparisonmap L T ( n ) K ( C hG ) ∼ → ( L T ( n ) K ( C )) hG is an equivalence. (3) L T ( i ) K ( R ) = 0 for i ≥ n + 1 . Thus, a vanishing result for R tC p implies a descent result for R which implies a vanishing resultfor R at the next height. For the proof of (2), we place the descent question in the context ofequivariant algebraic K -theory, in essentially the form developed by Barwick, [Bar17]. Then wemix equivariant and chromatic technology to reduce to proving a statement about C p -assemblymaps in algebraic K -theory. There we first handle the case of ordinary rings using essentially acounting argument. By the Dundas–Goodwillie–McCarthy theorem [DGM13], this reduces the caseof an arbitrary connective E ∞ -ring to the analogous question in topological cyclic homology. Andthat is amenable to an easy analysis thanks to a recent theorem of Hesselholt–Nikolaus [HN19].Finally, we deduce the general case from the connective case using results of [LMMT20]. The proofof part (3) also uses essentially the theorem of Hahn [Hah16] referenced above. In fact we use thefollowing a priori stronger version of Hahn’s theorem, which we however deduce from it: if R is an E ∞ -ring with L T ( n ) ( R tC p ) = 0, then L T ( m ) R = 0 for m ≥ n + 1, see Lemma 4.5.Theorem 1.3 allows an inductive approach to simultaneously proving vanishing and descentstatements. In fact, Theorem C follows immediately from it by inductively taking R = L p,fn S , viaKuhn’s blueshift theorem. DUSTIN CLAUSEN, AKHIL MATHEW, NIKO NAUMANN, AND JUSTIN NOEL
Many special cases of our vanishing results have previously appeared in the literature. The caseof BP h n i for n = 1 or n = 2 and p = 2 , BP h n i is known to admit an E ∞ -structure) has been proved by Angelini-Knoll–Salch [AKS20], using methods of topological cyclichomology. At height 1 and for p ≥
5, the result for ku (or equivalently the Adams summand)is a consequence of the calculations of Ausoni–Rognes [AR02] and Ausoni [Aus10]. Using ourresults, one can remove the restrictions on n and p in these previous results. For example, sinceTheorem A applies to R = ( L p,fn S ) ≥ and BP h n i is an ( L p,fn S ) ≥ -algebra, we deduce that the result L T ( n +2) K ( BP h n i ) = 0 of [AKS20] holds for all primes p and heights n . (One could also use the“purity” statement Corollary 4.9 for this deduction.)Concerning the general descent result Theorem B, we have already mentioned that it recov-ers Galois descent for K (1)-local K-theory and the p -completed p -group case of the Atiyah-Segalcompletion theorem. We also use it to obtain the following p -group case of a conjecture of Ausoni–Rognes [AR08, Conj. 4.2]: Corollary (Corollary 4.13 below) . Let A → B be a T ( n ) -local G -Galois extension of E ∞ -rings, inthe sense of Rognes [Rog08] , for G a finite p -group. Then the maps L T ( n +1) K ( A ) → L T ( n +1) ( K ( B ) hG ) → ( L T ( n +1) K ( B )) hG are equivalences. Besides the above thread of results, we also prove some other descent results of a slightly differentnature with different techniques. Like the results of our previous paper [CMNN20], these workuniformly for all chromatic heights, including height zero, and do not assume G to be a p -group;but on the other hand they make more restrictive assumptions on the action of the group G . Theorem D.
Let C be a monoidal, stable ∞ -category with biexact tensor product equipped withan action of a finite group G . Let tr : C → C hG denote the G -equivariant biadjoint to the for-getful functor C hG → C . Suppose the G -equivariant object tr( ) ∈ Fun(
BG, C hG ) has class in K (Fun( BG, C hG )) equal to that of the induced G -object L G C hG ∈ Fun(
BG, C hG ) . Then thecomparison map K ( C hG ) → K ( C ) hG induces an equivalence after T ( n ) -localization for any n ≥ at any prime p . Theorem D states that a type of normal basis property (e.g., for a Galois extension of fields,the condition follows from the normal basis theorem) for the G -equivariant object tr( ) ∈ C hG implies that the homotopy fixed point comparison map for K -theory is an equivalence after T ( n )-localization. The argument is based on the vanishing [Kuh04] of Tate spectra in telescopic homotopytheory. In fact, Theorem D yields another new proof of Theorem 1.2 avoiding the use of E ∞ -structures, see Remark 5.6.Next, we use [MNN15] to prove a third descent result (Theorem F below), which applies in moregeneral situations, albeit with a weaker conclusion. For this, we first formulate a generalization ofthe homotopy fixed point comparison maps with respect to a family of subgroups of G . Construction 1.4 (Comparison maps for families of subgroups) . Let C be a stable ∞ -categorywith an action of a finite group G . Let F be a family of subgroups of G , i.e., F is nonempty andclosed under subconjugation, and let O F ( G ) be the category of G -sets of the form G/H, H ∈ F .We obtain a comparison map(1.1) K ( C hG ) → lim ←− G/H ∈O F ( G ) op K ( C hH );this generalizes the homotopy fixed point comparison map, which is the case where F = { (1) } . ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 5 The map (1.1) is dual to the type of assembly maps which (for infinite groups) play a rolein the Farrell–Jones conjecture and its variants. In the rest of this paper, we will introduce abasic condition on an E ∞ -ring that guarantees the maps (1.1) are equivalences after telescopiclocalization, and implies a bound on the chromatic complexity of the algebraic K -theory. Definition 1.5 (Swan K -theory, Malkiewich [Mal17]) . Let R be an E ∞ -ring spectrum, and let G be a finite group. We define the ring Rep( G, R ), called the
Swan K -theory of R , viaRep( G, R ) = K (Fun( BG,
Perf( R ))) . Using induction and restriction functors, one makes Rep( − , R ) into a Green functor, cf. Defini-tion 6.1.The ∞ -category Fun( BG,
Perf( R )) is an analog of the category of complex representations of thefinite group G ; studying this in analogy with complex or modular representation theory for R = KU has been proposed by Treumann [Tre15]. In this analogy, the ring Rep( G, R ) is an analog of theclassical representation ring of G (to which it reduces when R = H C ). In general, the calculationof the rings Rep( G, R ) seems to be an interesting problem (e.g., for R = KU ), although we knowvery few examples. Definition 1.6 ( R -based Swan induction) . Fix a finite group G and an E ∞ -ring R . If the Greenfunctor Rep( − , R ) ⊗ Q (for subgroups of G ) is induced from a family F of subgroups of G , thenwe say that R -based Swan induction holds for the family F .In [Swa60], Swan shows that H Z -based Swan induction holds for the family of cyclic groups forany finite group; see also [Swa70] for a detailed treatment of Swan K -theory for a discrete ring.For H C , this is Artin’s classical induction theorem for the representation ring. Via some explicitgeometric arguments, we prove the following instances of Swan induction for E ∞ -ring spectra. Theorem E. If R is an E ∞ -ring, then for every finite group, R -based Swan induction holds for: (1) the family of abelian subgroups if R is complex orientable as an E -ring. (2) the family of abelian subgroups of rank ≤ if R = KU . (3) the family of abelian subgroups of p -rank ≤ n + 1 and ℓ -rank ≤ for primes ℓ = p if R = E n is Morava E -thory of height n at the prime p = 2 . This statement subsumes Theorem 7.4, Theorem 7.13, and parts of Theorem 7.12. Informally,(2) states that while a complex representation of a finite group G is determined by its character(i.e., its restriction to cyclic subgroups), the class in Rep( G, KU ) of a “representation” of G with KU -coefficients should be determined by a sort of “2-character,” defined on pairs of commutingelements of G ; moreover, there should be generalizations to higher heights. We conjecture that (3)should be true for odd primes too (Conjecture 7.22).We show that the Swan induction condition guarantees that the maps (1.1) become equivalencesafter telescopic localization, for every R -linear ∞ -category. This relies on similar techniques as in[CMNN20]. Theorem F (Descent via Swan induction, see Theorem 6.3) . Let R be an E ∞ -ring spectrum.Suppose that R -based Swan induction holds for the family F of subgroups of the finite group G .Then, for every R -linear stable ∞ -category C with an action of G , the natural map (1.1) , namely K ( C hG ) → lim ←− G/H ∈O F ( G ) op K ( C hH ); DUSTIN CLAUSEN, AKHIL MATHEW, NIKO NAUMANN, AND JUSTIN NOEL becomes an equivalence after T ( i ) -localization, for any i and any implicit prime p . Furthermore,the limit in (1.1) commutes with L T ( i ) . In fact, Theorem F can be combined with Theorem B, yielding the following result (Theo-rem 6.4): if L T ( n ) ( R tC p ) = 0, then the comparison map (1.1) becomes an equivalence after T ( n +1)-localization, for F the family of cyclic subgroups of order prime to p .Our final main result, which is inspired by the character theory of Hopkins–Kuhn–Ravenel[HKR00], is that a certain case of Swan induction implies the vanishing of the T ( i )-localizations ofalgebraic K -theory for large i . Theorem G.
Let R be an E ∞ -ring, p a prime, and n > . Suppose that R -based Swan inductionholds for the family of proper subgroups of C × np . Then L T ( i ) K ( R ) = 0 for i ≥ n at the implicitprime p . As a consequence, we obtain a new proof of Mitchell’s theorem (Theorem 1.1), and we recoverseveral chromatic bounds, e.g., that if p = 2 or n = 1, then we have L T ( i ) K ( E n ) = 0 for i ≥ n + 2.These bounds are special cases of Theorem A above, although the method is different and could beuseful in other settings as well; for instance, in Theorem 7.12 we prove 2-primary Swan inductionresults for M O h n i which therefore implies bounds on the chromatic complexity of K ( M O h n i ). Conventions.
We let S denote the ∞ -category of spaces, Sp denote the ∞ -category of spectra,and Sp G the ∞ -category of (genuine) G -spectra. We denote by S the unit of either of these (i.e.,the sphere spectrum). We write D X for the Spanier–Whitehead dual of X .We let L p,fn denote the finite localization [Mil92] on Sp away from a finite type n + 1 spectrum(at the implicit prime p ). In particular, for n = 0, we have L p,f ( X ) = X [1 /p ]. This conventionfollows [LMMT20]; for p -local spectra, it agrees with what is usually denoted L fn . Equivalently, if T ( i ) denotes the telescope of a v i -self map of a finite type i complex (so by convention T (0) can betaken to be S [1 /p ]), then L p,fn = L T (0) ⊕···⊕ T ( n ) .We write K for connective algebraic K -theory. Most of our results are new only after telescopiclocalization, after which there is no difference between connective and nonconnective K -theory, andwe will anyway state them in the generality of additive invariants.We write Cat perf ∞ for the ∞ -category of small, idempotent-complete stable ∞ -categories and exactfunctors between them, cf. [BGT13]. More generally, given an E ∞ -ring R , we write Cat perf R, ∞ forthe ∞ -category of small, idempotent-complete R -linear stable ∞ -categories and R -linear functorsbetween them, so Cat perf R, ∞ is Perf( R )-modules in Cat perf ∞ . Compare [HSS17] for a treatment.An ∞ -category C is called preadditive (also called semiadditive in the literature) if it is pointed,admits finite coproducts, and finite coproducts are (canonically) identified with finite products, see[GGN15, Sec. 2]. Given a preadditive ∞ -category C , we say that C is additive if the E ∞ -spacesHom C ( X, Y ) for
X, Y ∈ C are grouplike.
Acknowledgments.
We thank Clark Barwick, Jesper Grodal, Rune Haugseng, Michael Hopkins,Marc Hoyois, Markus Land, Jacob Lurie, Lennart Meier, John Rognes, Georg Tamme, and CraigWesterland for helpful discussions. The second author thanks the University of Copenhagen and theUniversity of Regensburg for hospitality. The second author was supported by the NSF GraduateFellowship under grant DGE-114415 as this work was begun and was a Clay Research Fellow whilethis work was completed. The third and fourth authors were partially supported by the SFB 1085,Regensburg.
ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 7 Mackey functors and equivariant algebraic K -theory In this section, we review the setup of equivariant algebraic K -theory which plays an integralrole in our approach to the present descent theorems. The use of equivariant algebraic K -theoryrefines the use of the transfer, which is central to all such descent results, going back to [Tho85]. In studying the comparison map K ( C hG ) → K ( C ) hG , one observes that there is also a map(2.1) K ( C ) hG → K ( C hG ) , arising from the G -equivariant functor C → C hG biadjoint to the forgetful functor C hG → C , suchthat the composition with the comparison map is the norm map K ( C ) hG → K ( C ) hG . In the caseof a G -Galois extension R → R ′ , equation (2.1) is the map K ( R ′ ) hG → K ( R )which arises from restriction of scalars from R ′ -modules to R -modules. These types of transfermaps and their functorialities can be encoded using the language of (genuine) G -spectra, and someof the techniques for proving descent results can be expressed using the language of F -nilpotence[MNN17, MNN19].Several authors have considered the setup of equivariant algebraic K -theory, including Merling[Mer17], Barwick [Bar17], Malkiewich–Merling [MM19], and Barwick–Glasman–Shah [BGS20]. Wewill follow the setup of [Bar17, BGS20], but will try to keep the exposition mostly self-contained.In particular, we will use the theory of spectral Mackey functors, which is equivalent to the theoryof G -spectra by work of Guillou–May [GM11] and Nardin [Nar16]; we will also give another proofof this equivalence in the appendix. Definition 2.1 (The effective Burnside ∞ -category, [Bar17, Sec. 3]) . For a finite group G , letBurn eff G denote the effective Burnside ∞ -category of the category of finite (left) G -sets and G -maps;informally, Burn eff G is the nerve of the (weak) (2 , • The objects of Burn eff G are finite G -sets S , • Given finite G -sets S and T , Hom Burn eff G ( S, T ) is the nerve of the groupoid of spans of finite G -sets U (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ S T and isomorphisms of spans. • Composition is given by pullback of spans.The ∞ -category Burn eff G is preadditive, and the direct sum comes from the disjoint union of finite G -sets.One then obtains the following definition [Bar17, Sec. 6] of a Mackey functor; this reduces to theclassical notion when C is the category of abelian groups. Definition 2.2 (Mackey and semiMackey functors) . Given any presentable, preadditive ∞ -category C , we define a C -valued semiMackey functor (for the finite group G ) to be a C -valued presheafon Burn eff G which takes finite coproducts of G -sets to products in C . We let sMack G ( C ) denotethe ∞ -category of C -valued semiMackey functors. When C is actually additive rather than merelypreadditive, we will write Mack G ( C ) rather than sMack G ( C ) and call these C -valued Mackey functors . A toy example of this argument is the Galois descent for rationalized algebraic K -theory, cf. [Tho85, Th. 2.15]. DUSTIN CLAUSEN, AKHIL MATHEW, NIKO NAUMANN, AND JUSTIN NOEL
Let M ∈ sMack G ( C ). Given a subgroup H ⊆ G , we write M H def = M ( G/H ) , and call this the H -fixed points of M . Remark 2.3 (Comparison with the P Σ -construction) . Consider the nonabelian derived ∞ -category P Σ (Burn eff G ) of Burn eff G , in the sense of [Lur09, Sec. 5.5.8], i.e., P Σ (Burn eff G ) is the ∞ -category ofpresheaves on Burn eff G which preserve finite products, or equivalently P Σ (Burn eff G ) is obtained byfreely adding sifted colimits to Burn eff G . Then sMack G ( C ) can be described as the Lurie tensorproduct P Σ (Burn eff G ) ⊗ C , cf. [Lur17, Sec. 4.8.1]. Remark 2.4 (Spectral Mackey functors and G -spectra) . Suppose C = Sp is the ∞ -category ofspectra. Then by [GM11, Nar16], we have an equivalence between Mack G (Sp) and the ∞ -categorySp G of (genuine) G -spectra, cf. also the appendix for an independent account of this equivalence.The target of equivariant algebraic K -theory will naturally be Mack G (Sp), and so we can equallyregard equivariant algebraic K -theory as a G -spectrum. Example 2.5 (The case of the trivial group) . Suppose G = (1) is the trivial group. Then Burn eff(1) isthe category of finite sets and correspondences between them. This is the free preadditive categoryon a single generator, a result due to Cranch [Cra10], cf. [BH17, Prop. C.1] for another account.It follows that sMack (1) ( C ) = C . In particular, it follows that P Σ (Burn eff(1) ) is the ∞ -category of E ∞ -spaces, since this is the free presentable preadditive ∞ -category on one object, cf. [GGN15]. Construction 2.6 (Relation to the orbit category) . Let O ( G ) be the orbit category of G , i.e., thecategory of nonempty transitive G -sets. We have a natural functor O ( G ) → Burn eff G which sendsthe G -set S to S ∈ Burn eff G and the G -map f : S → T to the span S id (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ f ❅❅❅❅❅❅❅ S T.
We also obtain a natural functor O ( G ) op → Burn eff G in a similar (dual) manner. Suppose f : G/H → G/H ′ is a morphism in O ( G ). Given a C -valued semiMackey functor M , we then obtain morphismsin C f ∗ : M H ′ → M H , f ∗ : M H → M H ′ . Thus, given the semiMackey functor M , we obtain two functors O ( G ) op → C , O ( G ) → C , which both send G/H M ( G/H ) = M H , and such that the functoriality is via f ∗ in the first caseand via f ∗ in the second case. Construction 2.7 (The symmetric monoidal structure on semiMackey functors) . Suppose now C is a presentably symmetric monoidal ∞ -category which is preadditive. Then there is a canonicalstructure of a presentably symmetric monoidal structure on sMack G ( C ), obtained (implicitly by Dayconvolution) as follows. We consider the symmetric monoidal structure on Burn eff G obtained fromthe cartesian product on finite G -sets (and products of spans). This symmetric monoidal structurecommutes with finite coproducts in each variable. Applying P Σ , we obtain a canonical presentablysymmetric monoidal structure on P Σ (Burn eff G ) such that the Yoneda functor is symmetric monoidal, ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 9 [Lur17, Prop. 4.8.1.10]. Now via the Lurie tensor product, sMack G ( C ) = P Σ (Burn eff G ) ⊗ C thenacquires the structure of a presentably symmetric monoidal ∞ -category.Next, we discuss the most basic source of Mackey functors: the Borel-equivariant ones. We beginwith the case where C is given by E ∞ -monoids in spaces. Proposition 2.8.
There is a symmetric monoidal Bousfield localization functor P Σ (Burn eff G ) → Fun(
BG, P Σ (Burn eff(1) )) such that the essential image of its right adjoint inclusion consists of thoseproduct-preserving presheaves F : (Burn eff G ) op → S such that for each finite G -set S , the naturalmap F ( S ) → F ( G × S ) hG is an equivalence. Here G acts on G (in the category of finite G -sets,and hence in Burn eff G ) by right multiplication.Proof. Let y : Burn eff G → P Σ (Burn eff G ) be the Yoneda embedding, and consider the Bousfield local-ization functor L I on P Σ (Burn eff G ) with respect to the maps I = { y ( G × S ) hG → y ( S ) } , for eachfinite G -set S . Here we use the map of G -sets G × S → S given by projection onto the secondfactor, and the G -action on the source (in the category of G -sets) by right multiplication on thefirst factor.Since y is symmetric monoidal and the tensor product on P Σ (Burn eff G ) commutes with colimitsin each variable, the class I is preserved by tensoring with objects in the image of y , and wesee that this Bousfield localization L I respects the symmetric monoidal structure. Unwinding thedefinitions, we see that the image of L I is precisely those product-preserving presheaves F as inthe statement because Hom P Σ (Burn eff G ) ( y ( G × S ) hG , F ) = F ( G × S ) hG . In particular, for any finite G -set S which is G -free, y ( S ) is I -local, as one sees by unwinding the definition of mapping spacesin Burn eff G .We claim that the { y ( S ) } for S finite and G -free form a set of compact projective generators for L I P Σ (Burn eff G ). Compactness and projectivity follow because for a finite free G -set S , the functor F 7→ F ( S ) (with values in S ) commutes with sifted colimits on P Σ (Burn eff G ) and carries the mapsin I to equivalences. Moreover, the y ( T ) for T ∈ Burn eff G can be expressed up to I -equivalence ascolimits of the y ( S ) for S finite G -free by construction; therefore, the { y ( S ) } generate. This verifiesthe claim about L I P Σ (Burn eff G ).The symmetric monoidal functor Burn eff G → Fun(
BG,
Burn eff(1) ) which remembers an underly-ing set, or correspondence, with G -action extends to a cocontinuous symmetric monoidal func-tor P Σ (Burn eff G ) → Fun(
BG, P Σ (Burn eff(1) )). Evidently, this functor carries the class of maps I to equivalences, and factors symmetric monoidally through the Bousfield localization L I . It re-mains to show that the induced functor L I P Σ (Burn eff G ) → Fun(
BG, P Σ (Burn eff(1) )) is an equiv-alence. The compact projective generators on both sides are given by y ( S ) for S a finite free G -set, so it suffices to compare maps between them. Equivalently, it suffices to show that the mapHom Burn eff G ( S, T ) → Hom
Fun(
BG,
Burn eff(1) ) ( S, T ) is an equivalence for
S, T finite free G -sets (in fact, itsuffices for the G -action on one of them to be free). By decomposing S and T and using duality, itsuffices to prove that this map is an equivalence when S = ∗ and T = G ; then one checks directlythat both sides are the free E ∞ -space on a generator and the map is an equivalence. (cid:3) Construction 2.9 (Borel-equivariant objects) . Let C be a presentably symmetric monoidal, pread-ditive ∞ -category. Tensoring the Bousfield localization of Proposition 2.8 with C , we obtain asymmetric monoidal Bousfield localization functorsMack G ( C ) → Fun(
BG, C ) , with a fully faithful lax symmetric monoidal right adjoint functor of “Borelification”,( − ) Bor : Fun(
BG, C ) → sMack G ( C ) . The essential image of ( − ) Bor (called
Borel-equivariant objects) is given by those product-preservingpresheaves F : (Burn eff G ) op → C such that for any finite G -set S , we have F ( S ) ∼ −→ F ( G × S ) hG . Inother words, the restriction of F to O ( G ) op should be right Kan extended from the full subcategoryspanned by the G -set G .We will be interested in the above construction when C = Cat perf ∞ . For this, recall that Cat perf ∞ is preadditive (cf. [Bar16, Prop. 4.7] for this result in the closely related context of Waldhausen ∞ -categories). Moreover, Cat perf ∞ is presentable [BGT13, Cor. 4.25], and symmetric monoidal under theLurie tensor product [BGT13, Th. 3.1]. Informally, for an idempotent-complete stable ∞ -category A with G -action, we obtain a Cat perf ∞ -valued semiMackey functor M A such that M A ( G/H ) = A hH .For a map of finite G -sets f : G/H → G/K , then f ∗ is the natural pullback map A hK → A hH . Wewill need to know that in this case, f ∗ can also be described explicitly. Proposition 2.10.
Let M ∈ sMack G (Cat perf ∞ ) be Borel-equivariant. Then for any map f : S → T of finite G -sets, the functors ( f ∗ , f ∗ ) : M ( T ) ⇄ M ( S ) form an adjoint pair (in either direction).Proof. We will verify this by invoking from [Bar17] a construction of a Cat perf ∞ -valued semiMackeyfunctor which does have the desired adjointness property, which is Borel, and whose underlyingobject of Fun( BG,
Cat perf ∞ ) agrees with that of M .Let A = M ( G ) ∈ Fun(
BG,
Cat perf ∞ ). Note first that for any map of finite sets f : S → T , thepullback functor f ∗ : Fun( T , A ) → Fun( S , A ) admits a right (and left) adjoint f ∗ : Fun( S , A ) → Fun( T , A ) given by summing over the fibers. Moreover, one has the base-change property: givena pullback square of finite sets U (cid:15) (cid:15) / / V (cid:15) (cid:15) S / / T , the induced square in Cat perf ∞ obtained by applying pullback everywhere is left and right adjointable.Now for every G -set S , we consider Fun G ( S, A ) ∈ Cat perf ∞ ; this is also Fun( S, A ) hG for thediagonal G -action (with G acting on both S and A ). Given a map of G -sets f : S → T , we havea pullback functor f ∗ : Fun G ( T, A ) → Fun G ( S, A ); we obtain a Cat perf ∞ -valued presheaf on thecategory of finite G -sets. We claim that for any map f : S → T , the functor f ∗ : Fun G ( T, A ) → Fun G ( S, A ) admits an adjoint (in either direction), and furthermore that for any pullback squareof finite G -sets U (cid:15) (cid:15) / / V (cid:15) (cid:15) S / / T, the induced square in Cat perf ∞ obtained by applying pullback everywhere is adjointable (in eitherdirection). This follows from the above verification in the case of finite sets, and then takinghomotopy fixed points in view of [Lur17, Cor. 4.7.4.18]. Indeed, the result of loc. cit. shows that ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 11 given a square in Fun( BG,
Cat ∞ ) which is left (or right) adjointable, the square in Cat ∞ obtainedby taking G -homotopy fixed points remains left (or right) adjointable.We thus have a Cat perf ∞ -valued presheaf on the category of finite G -sets, S Fun G ( S, A ), andwe have verified the adjointability conditions needed to apply the unfurling construction of [Bar17,Sec. 11], which produces a Cat perf ∞ -valued semiMackey functor M ′ extending the above presheafwhose restriction to O ( G ) is given by the adjoints f ∗ . In particular, M ′ satisfies the condition ofthe proposition. The semiMackey functor M ′ is clearly Borel-equivariant (since this condition onlydepends on the restriction to the category of finite G -sets) and must therefore agree with M , sincethe restrictions of M, M ′ in Fun( BG,
Cat perf ∞ ) agree; it follows now that M has the desired propertyin the proposition. (cid:3) Now we describe the fundamental construction for our purposes, the equivariant algebraic K -theory of group actions, in the form constructed by Barwick–Glasman–Shah, cf. [BGS20, Sec. 8].First we need a technical remark. Remark 2.11 (Functoriality of sMack G ( − )) . Let C , D be presentable, preadditive ∞ -categories.Let F : C → D be an accessible functor which commutes with finite coproducts (but not necessarilyall colimits). Then we can still define a natural functor sMack G ( C ) → sMack G ( D ) induced by F by sending a C -valued Mackey functor to the corresponding D -valued one (i.e., composing with F ).However, this is slightly awkward to formulate in our setup where sMack G ( C ) = C ⊗ P Σ (Burn eff G ),since this tensor product takes place in the world of presentable ∞ -categories. We can modifythis by fixing a suitable cardinal κ , considering the κ -compact objects C κ ⊆ C , then defining thecocontinuous functor Ind( C κ ) → D and applying sMack G ( − ) = ( − ) ⊗ P Σ (Burn eff G ) to it. Varying κ ,we obtain a functor out of C . In particular, this also shows that if C , D are symmetric monoidal andif F has a lax symmetric monoidal structure, then sMack G ( C ) → sMack G ( D ) has a lax symmetricmonoidal structure; alternatively one could see this using Day convolution, cf. [Gla16] or [Lur17,Sec. 2.2.6]. Construction 2.12 (Equivariant K -theory of group actions [BGS20, Sec. 8]) . It follows fromConstruction 2.9 that we have a lax symmetric monoidal functor of “Borelification”( − ) Bor : Fun(
BG,
Cat perf ∞ ) → sMack G (Cat perf ∞ ) , and composing it with the lax symmetric monoidal algebraic K -theory functor as in Remark 2.11,we obtain a lax symmetric monoidal functor(2.2) K G : Fun( BG,
Cat perf ∞ ) → Mack G (Sp) , such that K G ( A ) H = K ( A hH ) whenever A ∈
Fun(
BG,
Cat perf ∞ ) and H ⊆ G . Example 2.13 (Equivariant algebraic K -theory of E ∞ -rings) . Let R be an E ∞ -ring with G -action.Then we write K G ( R ) for K G (Perf( R )).We will actually need a slight generalization of the above, in order to handle invariants otherthan algebraic K -theory. Given a base E ∞ -ring R , we consider the presentably symmetric monoidal ∞ -category Mot R of R -linear noncommutative motives, cf. [BGT13, HSS17]. We have a symmetricmonoidal functor U : Cat perf R, ∞ → Mot R which is an additive invariant, i.e., it preserves filteredcolimits and carries semiorthogonal decompositions to direct sums in Mot R ; moreover, U is initialfor these data and conditions. Construction 2.14 (Mot R -valued Mackey functors) . Composing the Borel-equivariant functor( − ) Bor with U , we obtain a lax symmetric monoidal functor U G : Fun( BG,
Cat perf R, ∞ ) → Mack G (Mot R ) ≃ Mack G (Sp) ⊗ Mot R , i.e., U G takes values in Mackey functors in noncommutative motives. (Here we use Remark 2.11,since U does not preserve all colimits.) Since for any A ∈
Cat perf R, ∞ , the algebraic K -theory K ( A ) canbe recovered as Hom Mot R ( , U ( A )) by [BGT13] and [HSS17], it follows that the equivariant algebraic K -theory functor K G is the composition of U G and the functor id ⊗ Hom
Mot R ( , − ) : Mack G (Sp) ⊗ Mot R → Mack G (Sp).Finally, in order to treat assembly-type maps for group rings, we will need to discuss the coBorelvariant of the above. Construction 2.15 (coBorel semiMackey functors) . Let M be a C -valued semiMackey functor,for C a presentable preadditive ∞ -category. Note that sMack G ( C ) = P Σ (Burn eff G ) ⊗ C is naturallytensored over P Σ (Burn eff G ). Let y : Burn eff G → P Σ (Burn eff G ) denote the Yoneda embedding.We will say that M is coBorel if the natural map ( M ⊗ y ( G )) hG → M is an equivalence insMack G ( C ). Any M ∈ sMack G ( C ) admits its coBorelification M coBor = ( M ⊗ y ( G )) hG , which is theuniversal coBorel semiMackey functor mapping to M ; this follows because the object y ( G ) hG in P Σ (Burn eff G ) is an idempotent object for the tensor structure. The coBorelification only dependson the underlying object of Fun(
BG, C ) (since M ⊗ y ( G ) does), so we can also consider this as afunctor ( − ) coBor : Fun( BG, C ) → sMack G ( C ) . Dually as in Construction 2.9, ( − ) coBor is fully faithful; the essential image consists of those M ∈ sMack G ( C ) such that the dual comparison maps (cid:0) M { } (cid:1) hH → M H are equivalences for all H ⊆ G .By the universal property, we obtain a natural map ( − ) coBor → ( − ) Bor which is an equivalence onunderlying objects of Fun(
BG, C ).We now describe the coBorelification of the Cat perf ∞ -valued semiMackey functors constructedabove. This involves controlling homotopy orbits in Cat perf ∞ . To begin with, we need some factsabout limits and colimits of presentable, stable ∞ -categories, cf. [Lur09, Sec. 5.5.3]. Let P r L st denotethe ∞ -category of presentable, stable ∞ -categories and left adjoint functors between them. Let P r R st denote the ∞ -category of presentable, stable ∞ -categories and right adjoint functors betweenthem, so we have an equivalence in P r L st ≃ ( P r R st ) op . It follows that the underlying ∞ -category of acolimit in P r L st (of some diagram i
7→ C i , i ∈ I ) can be calculated by taking the inverse limit along I op of the right adjoints [Lur09, Th. 5.5.3.18]. Explictly, via the Grothendieck construction, we canexpress the diagram I → P r L st in terms of a presentable fibration e C → I , which is both a cartesianand a cocartesian fibration (cf. [Lur09, Def. 5.5.3.2]); the limit in P r L st is given by the ∞ -categoryof cocartesian sections, whereas the colimit is given by the ∞ -category of cartesian sections.We can use this to describe colimits in Cat perf ∞ . Construction 2.16 (Colimits in Cat perf ∞ ) . Consider the functor Ind : Cat perf ∞ → P r L st ([Lur09,Sec. 5.3.5]). This functor admits a right adjoint sending a presentable, stable ∞ -category to itssubcategory of compact objects; therefore, Ind commutes with all colimits. To compute a colimitin Cat perf ∞ of an I -indexed diagram, i
7→ A i , we therefore form the I op -indexed diagram of Ind( A i )and the right adjoint functors, and then take the compact objects in the limit. In fact, there is a natural symmetric monoidal functor from the ∞ -category of G -spaces to P Σ (Burn eff G ) whichis the identity on G -sets; then y ( G ) hG is the image of the G -space EG . ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 13 Example 2.17 (Homotopy orbits in Cat perf ∞ ) . Let
A ∈
Fun(
BG,
Cat perf ∞ ). We claim that A hG isnaturally described as the full subcategory of compact objects in Ind( A ) hG .To see this, we first describe the homotopy orbits Ind( A ) hG in P r L st . Form the presentablefibration over BG with fiber Ind( A ); as above, the cocartesian sections give Ind( A ) hG (the homotopylimit in P r L st ) while the cartesian sections give Ind( A ) hG . Since BG is an ∞ -groupoid, the cartesianand cocartesian sections are the same and we have a canonical identification Ind( A ) hG = Ind( A ) hG in P r L st . The claim about A hG now follows by passage to compact objects.Equivalently, we find that A hG ∈ Cat perf ∞ is the full subcategory of A hG generated by the imageof the functor A → A hG adjoint to the forgetful functor, since this image forms a set of compactgenerators of Ind( A ) hG . In particular, we have a natural fully faithful embedding A hG ⊆ A hG (which we verify below to be the norm map); it follows that for any limit diagram A j , j ∈ J in Fun( BG,
Cat perf ∞ ), the natural map (lim ←− J A j ) hG → lim ←− J ( A j ) hG is fully faithful, since ( − ) hG commutes with limits. Proposition 2.18.
Let
A ∈
Fun(
BG,
Cat perf ∞ ) . Then the natural map of Cat perf ∞ -valued semi-Mackey functors A coBor → A Bor , is fully faithful on H -fixed points for H ⊆ G . Moreover, ( A coBor ) H ⊆ ( A Bor ) H = A hH is the thicksubcategory generated by the image of the biadjoint A → A hH .Proof. Let H ⊆ G . Then we claim that the natural map (i.e., the norm map) in Cat perf ∞ ,(2.3) A hH = (( A ⊗ G ) hH ) hG → A hH = (( A ⊗ G ) hG ) hH is fully faithful; this map is the H -fixed points of A coBor → A Bor . But this follows from theobservation in the previous example: we saw that if T : Fun( BG,
Cat perf ∞ ) → Cat perf ∞ is the functor B 7→ B hG , then if B has a G × H -action, then T ( B hH ) → T ( B ) hH is fully faithful. It followsthat A coBor → A Bor is fully faithful on each fixed points. To see that its essential image is thesubcategory as claimed, we observe that
A → A hH has image generating the target as a thicksubcategory. (cid:3) Example 2.19 (Assembly maps) . Let G act trivially on Perf( R ). Then we find that for each sub-group H ⊆ G , one has Perf( R ) hH ≃ Perf( R [ H ]) ⊆ Fun(
BH,
Perf( R )) is the collection of compactobjects in Fun( BH,
Mod( R )). In particular, the Cat perf ∞ -valued Mackey functor (Perf( R )) coBor isprecisely the one that leads to the theory of assembly maps, cf. [RV18]. Construction 2.20 (Equivariant algebraic K -theory, coBorel version) . Combining the above, weobtain a functor U G, coBor : Fun( BG,
Cat perf R, ∞ ) ( − ) coBor −−−−−→ sMack G (Cat perf R, ∞ ) → Mack G (Mot R ) , which is the coBorel version of Construction 2.14. If A is an algebra object of Fun( BG,
Cat perf R, ∞ ),then A coBor = A Bor ⊗ y ( G ) hG is a module over A Bor in sMack G (Cat perf R, ∞ ). Therefore, U G, coBor ( A )is a module over U G ( A ) (which is an algebra object of Mack G (Mot R )).3. Review of nilpotence
To prove our descent theorems, it will be convenient to use the language of nilpotence, as in[MNN17, MNN19]. For the material in section 5 and further, we also need the variant of ǫ -nilpotence,as used in [CMNN20]. Definition 3.1 (Nilpotence) . Given a finite group G and a family F of subgroups, a G -spectrum X is said to be F -nilpotent if it belongs to the thick subcategory (or equivalently the thick ⊗ -ideal)generated by G -spectra which are induced from subgroups in F . We say that a G -spectrum X is( F , ǫ ) -nilpotent if there exists a finite set of prime numbers Σ such that for every finite spectrum F whose localizations at primes in Σ are nontrivial, then X belongs to the thick ⊗ -ideal of G -spectragenerated by F and the F -nilpotent G -spectra. (This somewhat involved definition in particularimplies that every passage to T ( n )-local coefficients makes X F -nilpotent, and this is an if and onlyif for the endomorphism G -ring spectrum of X . Compare [CMNN20, Prop. 2.19].)Given an F -nilpotent G -spectrum X , the natural comparison maps are equivalences(3.1) lim −→ G/H ∈O F ( G ) X H ∼ −−→ X G ∼ −−→ lim ←− G/H ∈O F ( G ) op X H , cf. [MNN19, Prop. 2.8]. In particular, an F -nilpotent object is F -complete , which is the conditionthat the second map above is an equivalence. (For example, when F consists of just the trivialsubgroup { e } then F -completeness is the same as the Borel-completeness discussed in Construc-tion 2.9.) But the notion of nilpotence gives much stronger permanence properties, for example,if X is F -nilpotent then the T ( n )-localization of X remains F -nilpotent by a thick subcategoryargument. From this, it follows formally that when X is ( F , ǫ )-nilpotent, then the comparisonmaps of (3.1) become equivalences after applying L T ( n ) for any n ; moreover, one can apply L T ( n ) either inside or outside the homotopy limit.We now discuss some criteria for nilpotence, starting with the case of the family T consistingonly of the trivial subgroup. Let EG denote the universal free G -space. Let g EG denote the cofiberof EG + → S in Sp G ; it is naturally an algebra object in Sp G , as the localization of S in Sp G awayfrom the localizing ⊗ -ideal generated by the free G -spectra, cf. [MNN17, Sec. 6.1]. In the following,let R be an associative algebra in Sp G . Then we consider the associative algebra ( R ⊗ g EG ) G in Sp.Since EG = ( G ) hG (in the ∞ -category of G -spaces), we have a cofiber sequence R hG → R G → ( R ⊗ g EG ) G , where R hG = R { } hG and R hG → R G is the transfer for the G -spectrum R . Proposition 3.2 (Criteria for T -nilpotence) . An associative algebra R in Sp G is T -nilpotent (for T = { (1) } ) if and only if ( R ⊗ g EG ) G is contractible.Proof. This follows from [MNN17, Th. 4.19], since R ⊗ g EG is the localization of R (in Sp G ) awayfrom the localizing ⊗ -ideal generated by the free G -spectra. (cid:3) Proposition 3.3 (Criterion for ( T , ǫ )-nilpotence) . Let R be an associative algebra in Sp G . Supposethat ( R ⊗ g EG ) G has trivial T ( n ) -localization for n ≥ and all primes p and trivial rationalization.Then R is ( T , ǫ ) -nilpotent.Proof. Our assumptions imply that there exists a finite set of prime numbers Σ such that for everyfinite complex F with nontrivial localizations at primes in Σ, the associative algebra spectrum( R ⊗ g EG ) G belongs to the thick ⊗ -ideal generated by F , cf. [CMNN20, Prop. 2.7]. Thus, the G -spectrum R ⊗ g EG belongs to the thick ⊗ -ideal generated by F ; here we use the natural adjunction( i ∗ , ( − ) G ) : Sp ⇄ Sp G . Therefore, R belongs to the localizing ⊗ -ideal generated by F and by the G -spectrum G + (using the fiber sequence R ⊗ EG + → R → R ⊗ g EG ), and hence it belongs to thesimilarly generated thick ⊗ -ideal by [MNN17, Th. 4.19] again. The result follows. (cid:3) ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 15 We next include three general results about F -nilpotence for an arbitrary family. The firstresult states that when R is rational (i.e, T (0)-local), F -nilpotence is a purely algebraic conditionon π ; the second (which will only be used with F = T ) gives a generalization of this to T ( n )-localobjects. The third result allows us to transfer rational F -nilpotence to ( F , ǫ )-nilpotence in thepresence of an E ∞ -structure, using the May nilpotence conjecture [MNN15]. For this, we let E F denote the universal G -space for the family F and g E F the cofiber of the map E F + → S in Sp G ,so g E F is the localization of S away from the localizing ⊗ -ideal generated by the { G/H + , H ∈ F } . Proposition 3.4 ([MNN19, Prop. 4.11]) . Suppose that the associative algebra R in Sp G is rational.Then R is F -nilpotent if and only if the Green functor π R has defect base contained in F : inother words, the induction map L H ∈ F π ( R H ) → π ( R G ) is surjective, or equivalently has imagecontaining the unit. (cid:3) Let L T ( i ) Sp G denote the full subcategory of Sp G spanned by the T ( i )-local objects, i.e., those forwhich the H -fixed points for each subgroup H ⊆ G are T ( i )-local spectra (at the implicit prime p );this equivalence follows because the orbits form a set of compact generators for Sp G . We next givea criterion for F -completeness in L T ( i ) Sp G . This will use the vanishing of the Tate constructionsin L T ( i ) Sp, due to [Kuh04], in the following equivalent form:
Lemma 3.5. If C is any presentable stable ∞ -category and X ∈ Fun(
BH, C ) is an H -object in C for some finite group H , then X hH ⊗ T ( i ) ∈ C belongs to the thick subcategory generated by X ⊗ T ( i ) .Proof. An equivalent form of the telescopic Tate vanishing is that, as an object of Fun(
BH,
Sp) (withtrivial H -action), T ( i ) belongs to the thick subcategory of Fun( BH,
Sp) generated by T ( i ) ⊗ H + ,cf. [MNN19, Prop. 5.31]. From this, the result easily follows, since ( X ⊗ T ( i ) ⊗ H + ) hH = X ⊗ T ( i ). (cid:3) Proposition 3.6 (Properties of T ( i )-local G -spectra) . Let G be a finite group, F a family ofsubgroups and i ≥ . Let M ∈ L T ( i ) Sp G . Then the following are equivalent: (1) M is F -complete. (2) For every finite type i complex F , the G -spectrum M ⊗ F is F -nilpotent. (3) We have L T ( i ) (Φ H M ) = 0 for H / ∈ F .Proof. We first claim that for each family G of subgroups of G , the G -spectrum E G + ⊗ T ( i ) is G -nilpotent; we prove this by induction on G . To start with, when G = T , then E G + = ( G + ) hG ;this uses the G -action on the G -space G (by right multiplication, so in the category of G -spaces).It follows from Lemma 3.5 that E T + ⊗ T ( i ) is T -nilpotent. Now we treat the inductive step.Fix a proper family G such that E G + ⊗ T ( i ) is G -nilpotent. Choose a subgroup H ⊆ G which isminimal for the property of not belonging to G ; one forms a new family G ′ obtained by adding theconjugates of H to G . Then there is a cofiber sequence of pointed G -spaces E G + → E G ′ + → E G ′ + ∧ e E G = ( G/H + ) hW H ∧ e E G , where W H is the Weyl group of H ⊆ G . Using this, the inductive assumption, and Lemma 3.5, theinductive step follows and the claim is proved.Now we prove the result. Suppose M ∈ L T ( i ) Sp G is F -complete. By the thick subcategorytheorem, condition (2) is independent of the choice of F and we choose a finite type i complex F such that F admits the structure of a ring spectrum; given a v i -self map v of F which we mayassume central, we can take T ( i ) = F [ v − ]. Then M ⊗ F admits the structure of a T ( i )-module,since M is T ( i )-local. It follows that the F -cellularization E F + ⊗ M ⊗ F of M ⊗ F belongs to the thick ⊗ -ideal of Sp G generated by E F + ⊗ T ( i ) and is therefore F -nilpotent, by our initial claim.Consequently, the F -completion F ( E F + , E F + ⊗ M ⊗ F ) (which is M ⊗ F again since this is F -complete) is also F -nilpotent. Thus, (1) implies (2). Clearly (2) implies (3), again by smashingwith F . If (3) holds, then M ⊗ F = M ⊗ T ( i ) has trivial geometric fixed points Φ H for H / ∈ F ,whence M ⊗ F = E F + ⊗ M ⊗ F = E F + ⊗ M ⊗ T ( i ), which we have seen is F -nilpotent. Thus,(3) implies (2). Finally, (2) implies (1) by writing M as an inverse limit of M ⊗ F for suitable finitetype i complexes F . (cid:3) Corollary 3.7.
Let R ∈ L T ( i ) Sp G be an algebra object which is F -complete. Then any R -module M ∈ L T ( i ) Sp G is F -complete.Proof. This follows from item (3) of Proposition 3.6, since Φ H is a symmetric monoidal functor. (cid:3) Corollary 3.8.
Let R ∈ L T ( i ) Sp G be an E -algebra, and let M ∈ L T ( i ) Sp G be an R -module. Thenthe map M G → M hG admits a splitting as R G -modules. Similarly, the map L T ( i ) ( M hG ) → M G admits a section as R G -modules. If G = C p , then M is Borel-complete if and only if either of thesemaps is an equivlaence.Proof. All of this follows because the composite map L T ( i ) ( M hG ) → M G → M hG is the norm,which is an equivalence since Tate constructions vanish in T ( i )-local homotopy [Kuh04]. (cid:3) Proposition 3.9.
Let R be an E ∞ -ring in Sp G . Suppose that the rationalization R Q is F -nilpotent. Then R is ( F , ǫ ) -nilpotent.Proof. By assumption, the E ∞ -ring ( R ⊗ g E F ) G Q is contractible. Therefore, by the main result of[MNN15], the E ∞ -ring ( R ⊗ g E F ) G is annihilated by L T ( n ) for all n and implicit primes p . Inparticular, by [CMNN20, Prop. 2.7], there exists a finite set Σ of primes such that ( R ⊗ g E F ) G belongs to the thick ⊗ -ideal of spectra generated by any finite spectrum F such that F ( p ) = 0 for p ∈ Σ. This implies that R ⊗ E f F belongs to the thick ⊗ -ideal of Sp G generated by F , whence R belongs to the localizing ⊗ -ideal generated by F and { G/H + , H ∈ F } in view of the cofibersequence R ⊗ E F + → R → R ⊗ g E F . Finally, [MNN17, Th. 4.19] again implies that R belongs tothe thick ⊗ -ideal generated by F and { G/H + , H ∈ F } in Sp G , as desired. (cid:3) Definition 3.10.
Let C be a presentably symmetric monoidal stable ∞ -category. We say thatan object of Mack G ( C ) = Mack G (Sp) ⊗ C ≃ Sp G ⊗ C is F -nilpotent (resp. ( F , ǫ ) -nilpotent ) if itbelongs to the thick ⊗ -ideal of Mack G ( C ) generated by the F -nilpotent (resp. ( F , ǫ )-nilpotent)objects in Sp G .It follows that for any cocontinuous functor C →
Sp, the induced functor Mack G ( C ) → Mack G (Sp) ≃ Sp G carries F -nilpotent (resp. ( F , ǫ )-nilpotent) objects in the source to F -nilpotent (resp. ( F , ǫ )-nilpotent) objects in the target. Using the adjunction Sp ⇄ C where the symmetric monoidal leftadjoint carries S to the unit, we obtain the next result. Proposition 3.11.
Let C be a presentably symmetric monoidal stable ∞ -category, and suppose ∈ C is compact. Let A be an object of Mack G ( C ) which admits a unital multiplication in thehomotopy category. Then A is F -nilpotent (resp. ( F , ǫ ) -nilpotent) in Mack G ( C ) if and only ifit is carried to an F -nilpotent (resp. ( F , ǫ ) -nilpotent) object of Mack G (Sp) under the functor Hom C ( , − ) : Mack G ( C ) → Mack G (Sp) . Here we use E ∞ -algebras in the symmetric monoidal ∞ -category Sp G ; we do not need to assume the existenceof norms as in [HHR16]. ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 17 Proof.
Let i ∗ : Mack G (Sp) → Mack G ( C ) denote the canonical symmetric monoidal functor (ob-tained from Sp → C ), and let i ∗ : Mack G ( C ) → Mack G (Sp) denote its right adjoint (which is equallyobtained by the cocontinuous functor Hom C ( , − ) : C →
Sp). By assumption, i ∗ A is F -nilpotent(resp. ( F , ǫ )-nilpotent); thus, so is i ∗ i ∗ A and hence so is A thanks to the map i ∗ i ∗ A → A . (cid:3) Example 3.12.
Let
A ∈
Fun(
BG,
Cat perf R, ∞ ). Consider U G ( A ) ∈ Mack G (Mot R ), a Mackey functorvalued in Mot R . Suppose A is an algebra object of Cat perf R, ∞ (i.e., is an R -linear monoidal stable ∞ -category). Then U G ( A ) is F -nilpotent (resp. ( F , ǫ )-nilpotent) if and only if the G -spectrum K G ( A ) is F -nilpotent (resp. ( F , ǫ )-nilpotent), using the representability of K -theory.4. Descent for p -groups; proof of Theorem A and Theorem B In this section, we give the proof of Theorems A and B via Theorem 1.3. We start with thefollowing general reduction.
Proposition 4.1.
Let R be an E -ring, and let j ≥ . Then the following are equivalent: (1) L T ( j ) (Φ C p K C p ( R )) = 0 . (2) The C p -spectrum L T ( j ) K C p ( R ) is Borel-complete. (3) For every R -linear stable ∞ -category C equipped with an action of a finite p -group G , andevery additive invariant E with values in T ( j ) -local spectra, we have E ( C hG ) ∼ −→ E ( C ) hG . (4) For every R -linear stable ∞ -category C equipped with an action of a finite p -group G , andevery additive invariant E , we have (4.1) L T ( j ) ( E ( C ) hG ) ∼ −→ L T ( j ) E ( C hG ) ∼ −→ L T ( j ) E ( C hG ) ∼ −→ L T ( j ) ( E ( C ) hG ) ∼ −→ ( L T ( j ) E ( C )) hG . Proof. (1) and (2) are equivalent by Proposition 3.6; (2) is the special case of (3) where E = L T ( j ) K ( − ) and G = C p acts trivially on Perf( R ); and (3) is a special case of (4). Thus let us show(1) implies (3) and (3) implies (4).First, we show (1) implies (3). Since every p -group has a composition series with successivequotients cyclic of order p , we can use d´evissage to reduce to the case when G = C p . Let E C p ( C ) = E ( C Bor ) denote the C p -spectrum obtained by applying E to the C p -semiMackey functor C Bor inCat perf ∞ . By construction, E C p ( C ) is a module in C p -spectra over K C p ( R ). In fact, this followsbecause C Bor is a module over Perf( R ) Bor , and U C p ( C ) ∈ Mack C p (Mot) is therefore a module over K C p ( R ). Since E C p ( C ) is T ( j )-local, we find that E C p ( C ) is a module over L T ( j ) K C p ( R ) and istherefore Borel-complete by Corollary 3.7.Finally, we show (3) implies (4). To this end, we will produce a sequence of G -spectra whichwe will show to be Borel-complete, and which on fixed points realizes the maps in (4.1). In fact,consider the G -semiMackey functors C Bor , C coBor with values in Cat perf ∞ ; we have a natural map C coBor → C Bor . Both are modules over Perf( R ) Bor in sMack G (Cat perf ∞ ). Applying E and then L T ( j ) ,we obtain a sequence of G -spectra(4.2) E ( C coBor ) coBor → E ( C coBor ) → E ( C Bor ) → E ( C Bor ) Bor → ( L T ( j ) E ( C Bor )) Bor . Note that all of these G -spectra are modules over K G ( R ). Therefore, the T ( j )-localization of the G -spectra in (4.2) are modules over the G -ring spectrum L T ( j ) K G ( R ), which is Borel-complete by(3) (applied to the trivial G -action on Perf( R )). Consequently, in view of Corollary 3.7, the T ( j )-localizations of the G -spectra in (4.2) are all Borel-complete. Finally, all the maps of G -spectra in(4.2) induce T ( j )-equivalences on underlying spectra; consequently, the T ( j )-localizations induceequivalences on G -fixed points, whence the equivalences in (4). (cid:3) For future reference, we recall also the following lemma.
Lemma 4.2.
Let E i denote Morava E -theory of height i . For any T ( i ) -local E ∞ -algebra R over E i , we have that E hC p i ⊗ E i R ∼ → R hC p , and this is a free R -module of rank p i . Here we always have C p acting trivially, and the relative tensor product is algebraic, not (a priori) T ( i ) -localized.Proof. As E i is complex oriented and even periodic, and the p -series [ p ]( t ) ∈ ( π E i )[[ t ]] of itsassociated formal group law is a nonzerodivisor, the Gysin sequence for S → BC p → BS showsthat E hC p i = E BC p i is also even periodic, and π E hC p i = ( π E i )[[ t ]] / [ p ]( t ). Since the formal grouphas height i , this is a free module of rank p i over π E i . Since E i is T ( i )-local, Kuhn’s Tate vanishingresult from [Kuh04] (or the earlier [GS96]) shows that this implies that L T ( i ) (( E i ) hC p ) is free ofrank p i , on a dual basis for the basis of E hC p i . Mapping out to an arbitrary T ( i )-local E i -module M , we deduce that E hC p i ⊗ E i M ∼ → M hC p , implying all the desired claims. (cid:3) The case of ordinary rings.
For the proof of Theorem 1.3, we will need to give an inde-pendent treatment of a special case: namely, the case where R is an ordinary ring. Note that for n = 1, T (1) and K (1)-local homotopy coincide [Mah81, Mil81], and for all n we have L T ( n ) A = 0if and only if L K ( n ) A = 0 whenever A is a ring spectrum, thanks to the nilpotence theorem; see[LMMT20, Lem. 2.3]. This will let us consider K ( n ) instead of T ( n ).What we will really need for the main proof is the following. Lemma 4.3.
Let R be a ring. Then the assembly map of Example 2.19 K ( R ) hC p → K ( R [ C p ]) is a T ( n ) -equivalence for all n ≥ .Proof. In fact, we will show that the assembly map is a T ( n )-local equivalence for n = 1, andreprove Mitchell’s theorem L T ( n ) K ( Z ) = 0 for n ≥
2, which implies the statement also holds when n ≥ n = 1. Since K ( A ) → K ( A [1 /p ]) is a K (1)-equivalence for all rings A (see [BCM20,LMMT20, Mat20] for three different proofs), we can reduce to the case where R is a Z [1 /p ]-algebra.By transfer along the degree p − Z [1 /p ] → Z [1 /p, ζ p ], we can even assume R is a Z [1 /p, ζ p ]-algebra. The claim is equivalent to the assertion that the C p -spectrum K (Perf( R ) coBor )(obtained by applying K ( − ) to the C p -semiMackey functor Perf( R ) coBor ) has the property that( KU ⊗ K (Perf( R ) coBor )) /p is Borel-complete. Equivalently, we need to show that the map(4.3) ( KU ⊗ K ( R )) hC p → KU ⊗ K ( R [ C p ]))induces an equivalence upon p -completion.The p -completion of the map (4.3) admits a retraction as ( KU ⊗ K ( R )) ˆ p -modules by Corol-lary 3.8. We will show that the p -completions of both sides are free ( KU ⊗ K ( R )) ˆ p -modules of rank p , which will therefore imply the claim. Indeed, the fact that the p -completion of ( KU ⊗ K ( R )) hC p is free of rank p follows from Lemma 4.2. Moreover, the fact that ( KU ⊗ K ( R [ C p ])) b p is free of rank p follows because the standard idempotents in the group ring give R [ C p ] ≃ R × p as R -algebras. Thisproves the claim and hence the n = 1 case of the lemma.Now let us show L T ( n ) K ( Z ) = 0 for n ≥
2; it suffices to prove L K ( n ) K ( Z ) = 0 for such n . ByQuillen’s localization sequence K ( F p ) → K ( Z ) → K ( Z [1 /p ]) and Quillen’s calculation K ( F p ) ( p ) = For this argument, cf. [Mal17].
ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 19 Z ( p ) , it suffices to show L K ( n ) K ( Z [1 /p ]) = 0, or again by a transfer argument L K ( n ) K ( Z [1 /p, ζ p ]) =0. We now run a similar argument as above. Let R = Z [1 /p, ζ p ]. The map(4.4) L K ( n ) ( E n ⊗ K ( R ) hC p ) → L K ( n ) ( E n ⊗ K ( R [ C p ]))admits a retraction of L K ( n ) ( E n ⊗ K ( R ))-modules by Corollary 3.8. We showed above that K ( R [ C p ])is a free K ( R )-module of rank p , whence the right-hand-side of (4.4) is a free L K ( n ) ( E n ⊗ K ( R ))-module of rank p . By Lemma 4.2, the left-hand-side of (4.4) is a free module of rank p n . Inparticular, we obtain a split injection from a free module of rank p n over L K ( n ) ( E n ⊗ K ( R )) to afree module of rank p . As p n > p , this forces L K ( n ) ( E n ⊗ K ( R )) = 0, whence the claim. (cid:3) Extending to higher heights.
In this subsection, we prove Theorem B and Theorem Atogether and in full generality.We begin by proving a criterion for the vanishing of telescopic localizations (Lemma 4.5). Wethen prove a lemma (Lemma 4.7) which lets us bootstrap inductively up the level of chromaticcomplexity. In the following, we let E i +1 denote a Morava E -theory associated to a height i + 1formal group over a perfect field k of characteristic p , so π ( E i +1 ) = W ( k )[[ v , . . . , v i ]]. Lemma 4.4.
The π ( E i +1 ) -algebra π ( E tC p i +1 ) has the property that π ( E tC p i +1 ) / ( p, v , . . . , v i − ) isfaithfully flat over the field π ( E i +1 ) / ( p, v , . . . , v i − )[ v − i ] = k (( v i )) .Proof. Note that π ( E tC p i +1 ) / ( p, v , . . . , v i − ) is nonzero and has v i invertible, since E tC p i +1 has trivial K ( i +1)-localization but nontrivial K ( i )-localization by [GS96, HS96]. Therefore, the result follows. (cid:3) To obtain the desired bounds on the chromatic complexity on K ( R ) in general, we will use thefollowing lemma, a sort of converse to chromatic blueshift which is essentially a version of the resultsof Hahn [Hah16]. Lemma 4.5.
Let A be an E ∞ -ring. Suppose that L T ( i ) ( A tC p ) = 0 . Then L T ( j ) A = 0 for j ≥ i + 1 .Proof. The vanishing results for the telescopic localizations are equivalent to those for the analogous K ( j )-localizations, i.e., it suffices to show that L K ( j ) A = 0 for j ≥ i + 1 (cf. [LMMT20, Lem. 2.3]).By the main result of [Hah16], it suffices to show that L K ( i +1) A = 0. Therefore, without lossof generality, we may replace A with L K ( i +1) ( E i +1 ⊗ A ) and assume that A is a K ( i + 1)-local E ∞ - E i +1 -algebra such that L K ( i ) ( A tC p ) = 0; we then need to show that A = 0.Now we have(4.5) A tC p = A ⊗ E i +1 E tC p i +1 , by Lemma 4.2. Furthermore, π ∗ ( E tC p i +1 ) is a localization of π ∗ ( E hC p i +1 ) and is therefore flat over π ∗ ( E i +1 ), whence π ( A tC p ) = π ( A ) ⊗ π ( E i +1 ) π ( E tC p i +1 ). It follows from Lemma 4.4 that the map π ( A ) / ( p, v , . . . , v i − )[ v − i ] → π ( A tC p ) / ( p, v , . . . , v i − )[ v − i ]is faithfully flat. Our assumption is that the target vanishes since A tC p is L i − -local [Hah16,Lem. 2.2]. Therefore, the source vanishes. If i >
0, then by [Hah16, Th. 1.2], it follows that π ( A ) / ( p, v , . . . , v i − ) = 0. Since A is K ( i + 1)-local, this forces A = 0. If i = 0, the above showsthat π ( A ) ⊗ Q is trivial, whence the K (1)-local E ∞ -ring A is trivial by [MNN15]. (cid:3) To proceed, we will need to use some of the results of [LMMT20] on the chromatic behavior ofalgebraic K -theory. Theorem 4.6 ( [LMMT20, Th. 3.8]) . Let A be an E -ring and let n ≥ . Then the map K ( A ) → K ( L p,fn A ) is a T ( i ) -local equivalence for ≤ i ≤ n . Now we get into the proofs of Theorem B and Theorem A; the following lemma, equivalent toTheorem 1.3 from the introduction (thanks to Proposition 4.1), will be the key inductive step.
Lemma 4.7.
Let R be an E ∞ -ring and let i ≥ . For the following conditions, we have theimplications (1) ⇒ (2) ⇒ (3): (1) L T ( i ) R = 0 and L T ( i ) K ( R tC p ) = 0 . (2) L T ( i ) Φ C p ( K C p ( R )) = 0 . (3) L T ( j ) K ( R ) = 0 for all j ≥ i + 1 . Proof.
In the following proof, we use the following notation: given a C p -semiMackey functor M withvalues in Cat perf ∞ , we simply write K ( M ) , TC( M ), etc. for the associated C p -spectrum obtained byapplying K, TC, etc. With this notation, we have that K C p ( R ) = K (Perf( R ) Bor ); note that ourhypotheses imply that this is an E ∞ -algebra in C p -spectra.We start by showing (1) implies (2). First, we reduce to the case where R is connective. Indeed,given a coconnective spectrum X , the C p -Tate construction X tC p is annihilated by L p,fn for any n ≥
0; this follows by d´evissage and filtered colimits (note that ( − ) tC p commutes with filteredcolimits on coconnective spectra) from the case where X is an Eilenberg–MacLane spectrum in asingle degree. Therefore, the map ( τ ≥ R ) tC p → R tC p induces an equivalence on L p,fn -localizationsfor any n ≥
0. It follows that L T ( i ) K (( τ ≥ R ) tC p ) ∼ −→ L T ( i ) K ( R tC p ) by Theorem 4.6. Therefore,the hypotheses of (1) are preserved by passage from R to τ ≥ R , so we may assume R is connective;note also that the conclusion of (2) for τ ≥ R implies it for R .Now we have the categorical semiMackey subfunctor Perf( R ) coBor ⊆ Perf( R ) Bor . By definition,this map of categorical C p -semiMackey functors is an equivalence on underlying objects (both haveunderlying object of Cat perf ∞ given by Perf( R )), and on C p -fixed points it is given by the inclusionPerf( R [ C p ]) ⊆ Fun( BC p , Perf( R )), see Example 2.19. The Verdier quotient of categorical semi-Mackey functors Perf( R ) Bor / Perf( R ) coBor is therefore a Cat perf ∞ -valued C p -semiMackey functor withtrivial underlying object and C p -fixed points given by the Verdier quotient Fun( BC p , Perf( R )) / Perf( R [ C p ]),which is linear over the E ∞ -ring R tC p (cf. [NS18, Sec. I.3]). Applying K -theory, we obtain a cofibersequence of C p -spectra K (Perf( R ) coBor ) → K C p ( R ) → K (Perf( R ) Bor / Perf( R ) coBor ) . If we suppose that L T ( i ) K ( R tC p ) = 0, then by the above, it follows that L T ( i ) Φ C p ( − ) annihilates thelast term. Therefore, in order to prove (2), it suffices to show that L T ( i ) Φ C p K (Perf( R ) coBor ) = 0.Now K (Perf( R ) coBor ) of any E -algebra is group-ring K -theory, cf. Example 2.19. Thus, since R is now assumed connective, we may apply the Dundas–Goodwillie–McCarthy theorem [DGM13]to obtain a pullback square of C p -spectra, K (Perf( R ) coBor ) (cid:15) (cid:15) / / TC(Perf( R ) coBor ) (cid:15) (cid:15) K (Perf( π R ) coBor ) / / TC(Perf( π R ) coBor ) . Here K (Perf( π R ) coBor ) has trivial T ( i )-localized geometric fixed points by Lemma 4.3. Thus,to prove (2), it suffices to prove that L T ( i ) Φ C p TC(Perf( A ) coBor ) = 0 whenever A is a connective E -ring with L T ( i ) A = 0 (as this holds for both A = R by hypothesis and A = π R trivially). ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 21 Now we use an expression for the p -completion of Φ C p TC(Perf( A ) coBor ) given in the work ofHesselholt–Nikolaus, [HN19, Th. 1.4.1]. Indeed, Φ C p TC(Perf( A ) coBor ) is the cofiber of the assemblymap TC( A ) ⊗ BC p + → TC( A [ C p ]) and loc. cit. shows that after p -completion, this cofiber becomesTHH( A ; Z p ) h T p [1] ⊗ C p for T p the p -fold cover of the circle T . In particular, our assumption that L T ( i ) A = 0 thus implies that L T ( i ) (cid:0) Φ C p TC(Perf( A ) coBor ) (cid:1) = 0 as desired. This shows (1) implies(2).Finally, we show (2) implies (3). The Borel-completion of the C p -spectrum K C p ( R ) is the Borel-complete C p -spectrum associated to the trivial C p -action on K ( R ); in particular, we have a mapof E ∞ -rings Φ C p K C p ( R ) → K ( R ) tC p . It follows from (2) that L T ( i ) ( K ( R ) tC p ) = 0, whence (3) byLemma 4.5. (cid:3) We now prove Theorem A and Theorem B from the introduction, by starting with their specialcase Theorem C, which we restate here:
Theorem 4.8.
Let n ≥ , and let C be an L p,fn -local stable ∞ -category. Then L T ( m ) K ( C ) = 0 forall m ≥ n + 2 , and for any finite p -group G acting on C we have L T ( n +1) K ( C hG ) ∼ → ( L T ( n +1) K ( C )) hG . Proof.
Taking R = L p,fn S , by applying Lemma 4.7 and Proposition 4.1 it suffices to show that L T ( n +1) R = 0 and L T ( n +1) K ( R tC p ) = 0. The first vanishing follows from the definition. As for thesecond vanishing, we use induction on n . When n = 0 we have R = S [1 /p ] so R tC p = 0. When n >
0, Kuhn’s blueshift theorem [Kuh04] shows that R tC p is L p,fn − -local, whence K ( R tC p ) is amodule over K ( L p,fn − S ) and we conclude by induction. (cid:3) As a corollary of combining this theorem with the results of [LMMT20] (in particular, The-orem 4.6), one obtains the following purity result in T ( n )-local K -theory; this also appears in[LMMT20] and is explored further there. Corollary 4.9.
Let A be an E -ring, and let n ≥ . Then the map A → L T ( n − ⊕ T ( n ) A inducesan equivalence on L T ( n ) K ( − ) .Proof. By Theorem 4.6, we may assume that A is already L p,fn -local. We have a pullback square K ( A ) (cid:15) (cid:15) / / K ( L T ( n − ⊕ T ( n ) A ) (cid:15) (cid:15) K ( L p,fn − A ) / / K ( L p,fn − ( L T ( n − ⊕ T ( n ) A )) , since both vertical homotopy fibers are given by the (non-connective) K -theory of the thick sub-category of Perf( A ) generated by A ⊗ F , for F a finite type n − T ( n )-acyclic. (cid:3) Now we can input this back in to our arguments and obtain Theorem A and Theorem B, whichwe combine and restate here.
Theorem 4.10.
Let R be an E ∞ -ring. (1) Suppose L T ( n ) ( R tC p ) = 0 for some n ≥ . Let C be an R -linear stable ∞ -category equippedwith an action of a finite p -group G . Let E be an additive invariant of R -linear stable ∞ -categories (e.g., E could be K ( − ) ). Then the natural maps induce equivalences L T ( n +1) ( E ( C ) hG ) ∼ −→ L T ( n +1) E ( C hG ) ∼ −→ L T ( n +1) E ( C hG ) ∼ −→ L T ( n +1) ( E ( C ) hG ) ∼ −→ ( L T ( n +1) E ( C )) hG . (2) Suppose L T ( n +1) R = 0 for some n ≥ − . Then L T ( j ) K ( R ) = 0 for j ≥ n + 2 .Proof. For (1), by Lemma 4.7 and Proposition 4.1 it suffices to show that L T ( n +1) R = 0 and L T ( n +1) K ( R tC p ) = 0. The first follows from Lemma 4.5; for the second, we also get the weakervanishing L T ( n +1) ( R tC p ) = 0 (Hahn’s theorem, [Hah16]) so this follows from the purity resultCorollary 4.9. For (2), Hahn’s theorem shows L T ( n +2) R = 0 as well, so this follows from Corol-lary 4.9. (cid:3) Comparison with the redshift conjectures.
Finally, we discuss the relationship of ourresults to redshift. Conjecture 4.2 of [AR08] predicts that if A → B is a K ( n )-local G -Galoisextension of E ∞ -rings in the sense of [Rog08], then L T ( n +1) K ( A ) ≃ L T ( n +1) ( K ( B ) hG ). Here wewill prove this conjecture in the case where G is a p -group. In fact, we will allow the (a priori moregeneral) case of a T ( n )-local G -Galois extension.We recall that the condition of being a T ( n )-local G -Galois extension, in which the map B ⊗ A B → Q G B need only be a T ( n )-equivalence, is much weaker than being a G -Galois extension ofunderlying E ∞ -ring spectra (also known as a “global Galois extension”), and fundamental examplessuch as the Galois extensions of the K ( n )-local sphere produced by Devinatz–Hopkins are only T ( n )-locally (or K ( n )-locally) Galois. Thus the descent results in our previous paper [CMNN20] do notapply to them. Moreover, even in the case of underlying G -Galois extensions our previous resultsrequired being able to verify an extra condition: the rational surjectivity of the transfer map.In the global case, we directly obtain from Theorem 4.10 and Galois descent the following. Corollary 4.11.
Let A −→ B be a faithful G -Galois extension of E ∞ -rings with G a finite p -group, and suppose that L T ( n ) ( A tC p ) = 0 . Then the maps L T ( n +1) K ( A ) → L T ( n +1) ( K ( B ) hG ) → ( L T ( n +1) K ( B )) hG are equivalences. (cid:3) Now we consider the T ( n )-local case, where we can also obtain results, but with an additionalargument. Given a T ( n )-local E ∞ -ring A , we write K ′ ( A ) for the K -theory of the small, symmetricmonoidal, stable ∞ -category D ( A ) of dualizable objects in T ( n )-local A -modules. We have a naturalinclusion Perf( A ) ⊆ D ( A ), whence a map of E ∞ -rings K ( A ) → K ′ ( A ). The next result (togetherwith Theorem A) shows that this map is a T ( n + 1)-equivalence and implies that either K or K ′ can be used equivalently in the Ausoni–Rognes conjecture. Proposition 4.12.
Let A be a T ( n ) -local E ∞ -ring. Then the homotopy fiber of K ( A ) → K ′ ( A ) isnaturally a module over K ( L p,fn − S ) .Proof. Indeed, consider the Verdier quotient D ( A ) / Perf( A ). We claim that this stable ∞ -categoryis naturally L p,fn − S -linear. To this end, we need to show that if F is a finite type n complex, thenfor any M ∈ D ( A ), we have M ⊗ F ∈ Perf( A ) (so that this vanishes in the Verdier quotient). Tothis end, we observe that dualizability implies that the functorHom L T ( n ) Mod( A ) ( M, − ) : L T ( n ) Mod( A ) → L T ( n ) Mod( A )commutes with all colimits. Tensoring with F , we find easily that M ⊗ F is a compact object of L T ( n ) Mod( A ); this uses that tensoring with F yields a colimit-preserving functor L T ( n ) Sp → Sp.Since L T ( n ) Mod( A ) is compactly generated by A ⊗ F , it follows that M ⊗ F belongs to the thicksubcategory generated by A ⊗ F and is therefore a perfect A -module, whence the result. (cid:3) ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 23 Given a T ( n )-local G -Galois extension A → B with G a finite p -group, we have D ( A ) ≃ D ( B ) hG ;this follows because L T ( n ) Mod( A ) ≃ (cid:0) L T ( n ) Mod( B ) (cid:1) hG by Galois descent, and using [Lur17,Prop. 4.6.1.11] to commute the formation of dualizable objects over homotopy fixed points. There-fore, the next result follows in a similar manner; this proves [AR08, Conj. 4.2] in the case of a p -group. Corollary 4.13.
Let A → B be a T ( n ) -local G -Galois extension, with G a finite p -group. Then L T ( n +1) K ( A ) ≃ −→ L T ( n +1) ( K ( B ) hG ) ≃ −→ ( L T ( n +1) K ( B )) hG .Proof. In fact, Theorem B yields the analog of this result with K ′ ( − ) replacing K , since D ( − )satisfies T ( n )-local Galois descent. Using Proposition 4.12, we find that the difference betweenthe statements for K ′ ( − ) and K ( − ) is controlled by modules over K ( L p,fn − S ), which have trivial T ( n + 1)-localizations by Theorem A. (cid:3) An important example of a T ( n )-local (pro-)Galois extension is the map L K ( n ) S → E n , wherethe profinite group in question is the extended Morava stabilizer group G n . We have a short exactsequence 1 → S n → G n → Gal( F p / F p ) → , where S n has an open subgroup which is pro- p . For each open subgroup H ⊆ G n , we write E hHn for the (Devinatz–Hopkins) continuous homotopy fixed points, [DH04]. Now, we choose an opensubgroup U of G n such that U ∩ S n is a pro- p -group. Then for any normal inclusion V ′ E V ⊆ U of open subgroups, we have L T ( n +1) K ( E hVn ) ∼ −→ ( L T ( n +1) K ( E hV ′ n )) h ( V/V ′ ) , i.e., we obtain Galoisdescent for the K ( n )-local finite Galois extensions of E hUn , and we therefore obtain a sheaf of T ( n +1)-local spectra on finite continuous U -sets. This follows from the descent for p -groups provedabove as well as the descent for finite ´etale extensions, proved as in [CMNN20]. Our methods donot (to the best of our knowledge) yield hyperdescent for this sheaf.Finally, [AR08, Conjecture 4.3] predicts that for appropriate K ( n )-local E ∞ -ring spectra B (e.g., L K ( n ) S ), and for a finite type n + 1-complex V , the map V ⊗ K ( B ) → L T ( n +1) ( V ⊗ K ( B )) is anequivalence in high enough degrees; this is a higher chromatic analog of the Lichtenbaum–Quillenconjecture, cf. [AR02, Aus10] for instances. Our methods are certainly not strong enough to provesuch statements; however, this conjecture would imply the weaker assertion L T ( n + i ) K ( B ) = 0 for i ≥
2, which we have proved above as Theorem A.5.
Descent by normal bases; proof of Theorem D
In this section, we will give another condition that guarantees T ( n )-local descent, which willwork uniformly for all n (including n = 0). Construction 5.1 (The transfer) . We use the transfer map of the finite group G , which is a mapof spectra tr BG : BG + → S . The adjoint map of spaces BG → Ω ∞ S arises from interpreting the target as the K -theory of thecategory Fin of finite sets (the Barratt–Priddy–Quillen theorem), and considering the G -action onthe G -set G (by right multiplication), so we take the composite map BG → Fin ≃ → Ω ∞ K (Fin) =Ω ∞ S . Our basic tool will be the following observation: Note that A → B is automatically T ( n )-locally faithful. In fact, A ≃ B hG ≃ L T ( n ) B hG , so tensoring with B isconservative on L T ( n ) Mod( A ). Theorem 5.2.
For any n ≥ and implicit prime p , the map L T ( n ) (tr BG ) : L T ( n ) BG + → L T ( n ) S admits a section.Proof. This follows from (and is equivalent to, as explained in [CM17]) the vanishing of Tate spectrain the T ( n )-local category, due to Kuhn [Kuh04]. In fact, this vanishing yields that L T ( n ) BG + ∼ −→ C ∗ ( BG, L T ( n ) S ) via the norm map, and the transfer is the composite of the norm with the projection C ∗ ( BG, L T ( n ) S ) → L T ( n ) S , which clearly admits a section. (cid:3) Proposition 5.3.
Let R be an associative algebra in Sp G . Suppose that there is a factorization of BG + tr BG −−−→ S → R G through the R -transfer R hG → R G . Then R is ( T , ǫ ) -nilpotent.Proof. By Proposition 3.3, it suffices to show that L T ( n ) ( R G /R hG ) = L T ( n ) ( R ⊗ g EG ) G = 0 for any p and n (including p = 0). For this, it suffices to show that the map L T ( n ) ( R hG ) → L T ( n ) ( R G )has image containing the unit. But this follows because we have seen above that L T ( n ) (tr BG ) : L T ( n ) ( BG + ) → L T ( n ) S has image containing the unit. (cid:3) We will apply this below to associative G -ring spectra of a particularly special kind, where onehas a homotopy commutative diagram(5.1) R hG / / R G BG + O O tr BG / / S , O O in which the factorization BG + → R hG required in Proposition 5.3 is obtained as the G -homotopyorbits of the map S → R ; in particular, these satisfy the conditions of Proposition 5.3. Note thatthis now is merely a condition on the algebra R in Sp G , namely that the diagram(5.2) R hG / / R G BG +( η ) hG O O tr BG / / S , η O O should commute, where η denotes the unit map. Such R arise via the following categorical con-struction, namely by taking R = K G ( C ) below.Let ( C , ⊗ , ) be a monoidal, stable ∞ -category equipped with a G -action. Let f : G → ∗ bethe map of G -sets. We use the induction functor f ∗ : C → C hG (biadjoint to the forgetful functor C hG → C ); we note that this is G -equivariant with respect to the trivial action on the target. Since ∈ C is G -invariant, we obtain a G -action on f ∗ ( ) ∈ C hG . Definition 5.4 (The normal basis condition) . We say that the G -action on C as above satisfies the normal basis property if the object f ∗ ( ) ∈ Fun(
BG, C hG ) defines the same K -class as the object C hG ⊗ G + ∈ Fun(
BG, C hG ). Note also that as explained in [CM17], the existence of the section in the essential case G = C p fol-lows via the Bousfield–Kuhn functor [Bou01, Kuh89] from the Kahn–Priddy theorem [KP78], which states thatΩ ∞ +1 (tr BG ): Ω ∞ +1 BG + → Ω ∞ +1 S has a section. ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 25 In other words, the normal basis condition implies that the following diagram, which is not commutative,(5.3) C ≃ hG f ∗ / / ( C hG ) ≃ BG [ ] hG O O tr BG / / Ω ∞ S , C hG O O gives rise to two objects in Fun( BG, C hG ) with the same K -class; the class obtained by going rightand up is the normal basis class C hG ⊗ G + , while the class obtained by going up and right is f ∗ ( ).It follows that we do have a homotopy commutative diagram if we replace the top right in (5.3)with Ω ∞ K ( C hG ).We now prove the following result, which is a slight refinement of Theorem D. Theorem 5.5.
Suppose R is an E ∞ -ring, C is an algebra object of Cat perf R, ∞ equipped with an actionof a finite group G , and the G -action on C satisfies the normal basis property. Then U G ( C ) ∈ Mack G (Mot R ) is ( T , ǫ ) -nilpotent. In particular, for any additive invariant E on Cat perf R, ∞ , the map E ( C hG ) → E ( C ) hG induces an equivalence after T ( i ) -localization for any i and any implicit prime p , including p = 0 .Proof. We show that U G ( C ) ∈ Mack G (Mot R ) is ( T , ǫ )-nilpotent, which also implies the other claims.By Example 3.12, it suffices to show that K G ( C ) is ( T , ǫ )-nilpotent. By Proposition 5.3, it sufficesto show that we have a factorization of the BG + → S through K ( C ) hG → K ( C hG ). However,this follows from the diagram (5.3) (which is not commutative, but which becomes homotopycommutative when we replace the upper right by Ω ∞ K ( C hG ) by our hypotheses). (cid:3) Remark 5.6 (Alternative proof of Theorem 1.2) . Let R → R ′ be a G -Galois extension of commu-tative rings. Then Zariski locally on R , one has the normal basis property (even before passage to K ): the R [ G ]-module R ′ is locally isomorphic to R [ G ]; indeed, this follows because of the usualnormal basis theorem when R is a field, and hence more generally a local ring. Using Zariski descentfor K -theory [TT90], one reduces to this case, whence the result via Theorem 5.5. Remark 5.7.
In fact, the above argument for Theorem 1.2 is valid more generally (with L T ( n ) -localization for any n ) if R → R ′ is a G -Galois extension of E ∞ -ring spectra in the sense of [Rog08]in the case where π ( R ) → π ( R ′ ) is additionally G -Galois, i.e., R → R ′ is ´etale in the sense of[Lur17, Sec. 7.5]. This is a special case of the results of [CMNN20], which assume a weaker conditionon R → R ′ , but which essentially use the E ∞ -structures on the algebras in question.6. Swan induction and applications; proofs of Theorem F and G
In this section, we recall the notion of Swan K -theory, and prove Theorems F and G from theintroduction. In the final section, we will give a number of examples of Swan induction.The Swan K -theory of a ring spectrum with respect to a finite group was introduced by Malkiewichin [Mal17], following ideas of Swan [Swa60, Swa70] who defined it for discrete rings. Throughoutthe subsection, let R be an E ∞ -ring spectrum. Definition 6.1 ([Mal17, Def. 4.11]) . Given a finite group G , we let Rep( G, R ) denote the Grothendieckring of the stable ∞ -category Fun( BG,
Perf( R )). We will call this the Swan K -theory of R withrespect to G . The groups { Rep(
H, R ) } H ⊆ G form a Green functor, as the π of the E ∞ -algebra K G ( R ) in Sp G . For a family F of subgroups of G , we will say that F -based Swan induction holds for R if thereexist classes x H ∈ Rep(
H, R ) ⊗ Q for H ∈ F such that(6.1) 1 = X H ∈ F Ind GH ( x H ) ∈ Rep(
G, R ) ⊗ Q , for Ind GH : Rep( H, R ) → Rep(
G, R ) the map obtained by induction of representations on R -modules. Example 6.2 (Classes in Rep(
G, R )) . Let M be a space with the homotopy type of a finite CWcomplex, equipped with a G -action. Then R ⊗ M + defines an object of Fun( BG,
Perf( R )) andconsequently an element [ R ⊗ M + ] ∈ Rep(
G, R ). If M has the homotopy type of a G -CW complex,then a cell decomposition shows that the class [ R ⊗ M + ] actually belongs to the image of the map A ( G ) → Rep(
G, R ), for A ( G ) the Burnside ring.We now prove the descent statement in the K -theory of R -linear ∞ -categories that Swan induc-tion implies (this is Theorem F); the use of a rational statement to deduce telescopic ones follows[CMNN20]. Theorem 6.3 (Descent via Swan induction) . Let R be an E ∞ -ring and let G be a finite group.Suppose that R -based Swan induction holds for the family F . Then for any R -linear ∞ -category C equipped with a G -action, and for any additive invariant E on Cat perf R, ∞ , the maps (6.2) E ( C hG ) → lim ←− G/H ∈O F ( G ) op E ( C hH ) and (6.3) lim −→ G/H ∈O F ( G ) E ( C hH ) → E ( C hG ) become an equivalence after T ( n ) -localization, for any n and any implicit prime p .Proof. For the first claim, it suffices to show that U G ( C ) ∈ Mack G (Mot R ) is ( F , ǫ )-nilpotent. Bymultiplicativity, this reduces to showing that U G (Perf( R )) is ( F , ǫ )-nilpotent, where the G -actionon Perf( R ) is trivial; for this in turn, it suffices to show that K G ( R ) ∈ Sp G is ( F , ǫ )-nilpotent asin Example 3.12. By Proposition 3.9, it suffices to show that K G ( R ) Q is F -nilpotent, which isprecisely the condition of R -based Swan induction for F .For the second claim, we use the coBorel construction of Construction 2.20. We claim that U G, coBor ( C ) ∈ Mack G (Mot R ) is ( F , ǫ )-nilpotent. But this follows because it is a module over U G (Perf( R )), which we have just seen is ( F , ǫ )-nilpotent. (cid:3) Now we record a variant of Theorem 6.3 specifically in the context where R = L p,fn S , and wherethe localization is precisely at height n + 1; this relies on similar techniques as in section 4, andfollows from combining them with the above. In fact, this yields a slight refinement of the resultsof section 4 to non- p -groups. Theorem 6.4.
Fix n ≥ . Let R be an E ∞ -ring such that L T ( n ) ( R tC p ) = 0 . Let C be an R -linear ∞ -category equipped with an action of a finite group G . Then for any additive invariant E , themaps (6.2) and (6.3) become equivalences after T ( n + 1) -localization, for F the family of cyclicsubgroups of G of prime-to- p -power order. ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 27 Proof.
To begin with, observe that for any finite group H , we have L T ( n ) ( R tH ) = 0. Indeed, wereduce to the case where H is a p -group by restricting to a p -Sylow. Inductively, we have a normalsubgroup C p E H . The norm map R hH → R hH factors as R hH = ( R hC p ) h ( H/C p ) → ( R hC p ) h ( H/C p ) → ( R hC p ) h ( H/C p ) = R hH . By induction on H and the assumption L T ( n ) ( R tC p ) = 0, we see that each map above has T ( n )-acyclic cofiber, whence R tH is T ( n )-acyclic.The rest of the argument will closely follow that of Lemma 4.7. As above, it suffices to showthat L T ( n +1) K G ( R ) is F -complete, for F as in the statement (cf. also Proposition 4.1 and itsproof). Indeed, for any additive invariant E , the G -spectra E ( C Bor ) , E ( C coBor ) , F ( E F + , E ( C Bor ))are modules over K G ( R ), so this claim follows using Corollary 3.7.To see that L T ( n +1) K G ( R ) is F -complete, we let D = Perf( R ) with trivial G -action. We havethe fully faithful inclusion of G -semiMackey functors D coBor → D Bor ; the cofiber takes values in R tH -linear ∞ -categories for various subgroups H ⊆ G ; indeed, this follows because the Verdierquotient Fun( BH,
Perf( R )) / Perf( R [ H ]) is linear over R tH as in [NS18, Sec. I.3]. Therefore, usingTheorem A, we find an equivalence of G -Mackey functors L T ( n +1) K ( D coBor ) ≃ L T ( n +1) K ( D Bor ) = L T ( n +1) K G ( R ). Thus, it suffices to show that L T ( n +1) K ( D coBor ) is F -complete, or equivalentlythat its T ( n + 1)-local geometric fixed points vanish at all subgroups except possibly those whichare cyclic of order prime to p (Proposition 3.6).Now K ( D coBor ) is group-ring K -theory. By Theorem 4.6, we have for any subgroup H ⊆ G , L T ( n +1) K ( R [ H ]) ≃ L T ( n +1) K (( τ ≥ R )[ H ]). In particular, L T ( n +1) K ( D coBor ) = L T ( n +1) K ((Perf( τ ≥ R ) coBor ) . We obtain from the Dundas–Goodwillie–McCarthy theorem [DGM13] a pullback square of G -spectra, L T ( n +1) K ( D coBor ) (cid:15) (cid:15) / / L T ( n +1) TC(Perf( τ ≥ R ) coBor ) (cid:15) (cid:15) L T ( n +1) K (Perf( π R ) coBor ) / / L T ( n +1) TC(Perf( π R ) coBor ) . Now we obtain from [LRRV19, Th. 1.2] that the G -spectra TC(Perf( τ ≥ R ) coBor ) , TC(Perf( τ ≥ R ) coBor )are modulo p induced from the family of cyclic subgroups of G ; in particular, their geometric fixedpoints at non-cyclic subgroups vanish modulo p . Finally, the term L T ( n +1) K (Perf( π R ) coBor ) van-ishes for n ≥ n = 0, it follows from Theorem 6.3 and Swan’s inductiontheorem from [Swa60] (reproved below as Theorem 7.5) that L T ( n +1) K (Perf( π R ) coBor ) is inducedfrom the family of cyclic subgroups.Thus, we find that L T ( n +1) K ( D coBor ) = L T ( n +1) K ( D Bor ) = L T ( n +1) K G ( R ) is complete for thefamily of cyclic subgroups. In particular, the T ( n + 1)-local geometric fixed points vanish for non-cyclic subgroups. Suppose then that G is cyclic and has order divisible by p ; we must show that L T ( n +1) Φ G K ( D Bor ) = 0. In fact, since there is an inclusion H E G with G/H ≃ C p , we have thetransfer map ( K ( D Bor ) H ) hC p → K ( D Bor ) G . This map is T ( n + 1)-locally an equivalence thanks to Theorem B (cf. also Proposition 4.1).Since it factors through the map ( E P ⊗ K ( D Bor )) G → K ( D Bor ) G for P the family of propersubgroups, it follows that this last map has T ( n + 1)-local image containing the unit, whence L T ( n +1) Φ G K ( D Bor ) = 0 as desired. (cid:3)
Next, we prove Theorem G, which was inspired by the generalized character theory of Hopkins,Kuhn, and Ravenel [HKR00] as well as the results of [MNN19]. We will apply this below to recoversome cases of the chromatic bounds on K -theory spectra. Proposition 6.5.
Fix a prime p and a non-negative integer n . Let R be an E ∞ -ring spectrum and G = C × np . Suppose that the sum of the rationalized transfer maps (6.4) M H ( G R ( BH ) ⊗ Q → R ( BG ) ⊗ Q is a surjection (or equivalently has image containing the unit). Then L T ( n + i ) R ≃ L K ( n + i ) R ≃ for all i ≥ (at the prime p ).Proof. A T ( n )-local ring spectrum is contractible if and only if its K ( n )-localization is contractible,cf. [LMMT20, Lem. 2.3]. Therefore, it suffices to prove that L K ( n + i ) R = 0. To verify the desiredvanishing, we can replace R by the E ∞ - R -algebra L K ( n + i ) ( E n + i ⊗ R ); by naturality of the trans-fer map, the hypotheses of the result are preserved by this replacement. Thus, we may assumethroughout that R = L K ( n + i ) R receives an E ∞ -map from E n + i . By the main result of [MNN15]then, it suffices to show π R ⊗ Q = 0.When n = 0, the left hand side of (6.4) is 0, so π R ⊗ Q = 0. So it suffices to consider the case n >
0. Since each transfer map factors through a maximal proper subgroup, the surjectivity of(6.4) is equivalent to the surjectivity of(6.5) M C × ( n − p ∼ = H ( G R ( BH ) ⊗ Q → R ( BG ) ⊗ Q . The left hand side of this equation contains p n − p − -copies of R ( BC × ( n − p ). Since R is a K ( n + i )-local E n + i -module, it follows that for any k , one has that C ∗ ( BC × kp , R ) = C ∗ ( BC × kp , E n + i ) ⊗ E n + i R is a free R -module of rank p ( n + i ) k (cf. Lemma 4.2). In particular, the right-hand-side of (6.5) isfree over π ( R ) ⊗ Q of rank p ( n + i ) n , while each summand on the left-hand-side has rank p ( n + i )( n − .Using the surjectivity of (6.5), we find that if π ( R ) ⊗ Q = 0, then we would conclude the inequalityof ranks, p ( n + i )( n − p n − p − ≥ p ( n + i ) n . However, we see easily that this inequality cannot hold if i ≥ n >
0. This contradictionproves the result. (cid:3)
Theorem 6.6.
Let p be a prime, n ≥ and R an E ∞ -ring spectrum. Suppose that R -based Swaninduction holds for the family of proper subgroups of C × np . Then L T ( i ) K ( R ) = 0 for i ≥ n at theprime p .Proof. We write G = C × np and consider the G -spectrum K G ( R ), i.e., the equivariant algebraic K -theory of R , with G acting trivially on R . Since R is an E ∞ -ring spectrum, K G ( R ) has the structureof an E ∞ -algebra in Sp G . By assumption, K G ( R ) Q is nilpotent for the family of proper subgroups,cf. Proposition 3.4. There is a natural map of E ∞ -algebras in Sp G of the form K G ( R ) Q → (cid:0) K G ( R ) Bor (cid:1) Q , so the target is also nilpotent for the family of proper subgroups. But K G ( R ) Bor is simply the Borel-equivariant G -spectrum associated to the trivial G -action on K ( R ), so thecondition that (cid:0) K G ( R ) Bor (cid:1) Q should be nilpotent for the family of proper subgroups is exactly thatthe map (6.4) (with K ( R ) replacing R ) should be a surjection. Thus, the result follows fromProposition 6.5. (cid:3) ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 29 Swan induction theorems; proof of Theorem E
In this section, we establish several examples of Swan induction theorems for structured ringspectra, and prove Theorem E. In particular, we show that one always has Swan induction forthe family of abelian subgroups for
M U (Theorem 7.4), for the cyclic groups for Z (Theorem 7.5,recovering results of [Swa60]) or for S [1 / | G | ] (Theorem 7.6), for the rank ≤ KU (Theorem 7.13), and for the rank ≤ n + 1 abelian subgroups for E n at p = 2 (Theorem 7.12).7.1. Geometric arguments.
Throughout, let R be an E ∞ -ring spectrum. We first observe thefollowing basic features of the Swan induction property. Remark 7.1. (1) If R is an E ∞ -ring such that one has R -based Swan induction with respectto a family of subgroups F of some group G , and R ′ is an E ∞ -ring admitting a map from R (even an E -map suffices), then R ′ -based Swan induction for F and G holds as well.(2) In order to prove that R -based Swan induction holds with respect to a family F of subgroupsof G , it suffices to show that for every subgroup H ⊆ G which is not in F , then one hasSwan induction with respect to the family of proper subgroups of H . This is an elementaryobservation about Green functors, cf. [MNN17, Prop. 6.40].(3) Suppose G ։ G ′ is a surjection, and R -based Swan induction holds for the family of propersubgroups of G ′ . Then R -based Swan induction holds for the family of proper subgroupsof G .In this subsection, we give geometric proofs of Swan induction in several cases. Our basic toolis the following. Proposition 7.2.
Suppose M is a G -space such that: (1) M admits a finite G -CW structure. (2) M has isotropy in the family F of subgroups of G . (3) There is an equivalence (7.1) R ⊗ M + ≃ n M k =1 Σ i k R ∈ Fun(
BG,
Perf( R )) for some integers i , . . . , i n . Here we equip the Σ i k R with the trivial G -action.Then R -based Swan induction holds for the family F .Proof. We consider the object X = R ⊗ M + ∈ Fun(
BG,
Perf( R )) and calculate its K -class [ X ] intwo different ways.(1) By assumption, M has a finite G -CW decomposition with equivariant cells of the form G/H × D n . The G -cells necessarily satisfy H ∈ F by hypothesis on the isotropy of M . Itfollows that there exist integers n H , H ∈ F such that(7.2) [ X ] = X H ∈ F n H [ R ⊗ G/H + ] = X H ∈ F Ind GH ( n H ) ∈ Rep(
G, R ) . (2) The assumption (3) gives an equivalence in Fun( BG,
Perf( R )) between X and a direct sumof n > X ] = n ∈ Rep(
G, R ) . Equating (7.2) and (7.3), we obtain the result. (cid:3)
The first condition in Proposition 7.2 will be satisfied if, for example, M is a compact smoothmanifold with G -action, by the equivariant triangulation theorem [Ill78]. We can check the condition(3) of Proposition 7.2 via the following result. Proposition 7.3.
Suppose that M is a G -space with the homotopy type of a finite G -CW complex.Then M satisfies condition (3) of Proposition 7.2 if and only if: (1) The R ∗ -cohomology R ∗ ( M hG ) is a free module on generators in even degrees over R ∗ ( BG ) . (2) The natural map R ∗ ( ∗ ) ⊗ R ∗ ( BG ) R ∗ ( M hG ) → R ∗ ( M ) is an isomorphism.Proof. In fact, using (1), we can produce a G -equivariant map from a sum of shifts of the unit into R ⊗ D M + ∈ Fun(
BG,
Perf( R )), by choosing a basis of R ∗ ( M hG ) = π −∗ Hom
Fun(
BG,
Perf( R ) ( R, R ⊗ D M + ); the induced map is an equivalence in Fun( BG,
Perf( R )) by the second condition. (cid:3) Theorem 7.4.
Suppose R is complex-orientable (as an E -ring, meaning there is an E -ring map M U → R ). Then R -based Swan induction holds for the family of abelian subgroups (for any finitegroup G ).Proof. We fix an embedding G ⊆ U ( n ) and consider the action on the flag variety M = F = U ( n ) /T for T ⊆ U ( n ) a maximal torus. As a smooth G -manifold, M admits a finite G -CW structure. Thestabilizers of the G -action are abelian (as they are contained in conjugates of T ). By [MNN17,Prop. 7.49], we obtain an equivalence of the form (7.1). Alternatively, we can use Proposition 7.3and the projective or flag bundle formula to see this. Therefore, we can apply Proposition 7.2 toconclude. (cid:3) When R is a discrete commutative ring, a classical theorem of Swan [Swa60] states that one hasSwan induction for the family of cyclic subgroups. We give a geometric proof of Swan’s theorem inthe spirit of some of our other results. Theorem 7.5 (Swan [Swa60]) . H Z -based Swan induction holds for the family of cyclic groups (forany finite group G ).Proof. By Theorem 7.4 and downward induction based on Remark 7.1, we see that it sufficesto consider G = C × p for some prime p . We consider the p -dimensional projective Heisenbergrepresentation of G on C p , given by the matrices(7.4) A = ζ p ζ p . . . ζ p − p , B = . Here A is a diagonal matrix whose eigenvalues are the powers of a primitive p th root ζ p of unity,and B is the permutation matrix for a cyclic permutation. Since the matrices A and B commuteup to scalars, they define a projective representation of G , yielding an embedding G ⊆ P GL p ( C ).The group P GL p ( C ) acts naturally on CP p − , and the action of the subgroup G ⊆ P GL p ( C )has no fixed points. It follows that the class [ H Z ⊗ CP p − ] in Rep( G, Z ) is a sum of classes inducedfrom proper subgroups. To calculate the class [ H Z ⊗ CP p − ] in another manner, we can alsoconsider the finite Postnikov filtration { τ ≤ i ( H Z ⊗ CP p − ) } whose successive subquotients are even ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 31 suspensions Σ i Z . The G -action on each of the (shifted discrete) associated graded terms is trivialbecause the G -action extend to an action of the connected group P GL p ( C ). Therefore, we findthat [ H Z ⊗ CP p − ] = p ∈ Rep( G, Z ). It follows that we have integers n H for each H ( G such that (cid:3) (7.5) p = X H ( G n H Ind GH (1) ∈ Rep( G, Z ) . Theorem 7.6.
Let G be a finite group. Then S [1 / | G | ] -based Swan induction holds for the familyof cyclic subgroups of G .Proof. Let R = S [1 / | G | ]. We have Fun( BG,
Perf( R )) ≃ Perf( R [ G ]); equivalently, an R [ G ]-moduleis perfect if and only if its underlying R -module is perfect, and similarly for every subgroup of G .This follows from the fact that taking homotopy G -fixed points commutes with arbitrary colimitsin the ∞ -category of R -modules. Recall that if A is a connective E -ring, then the natural map K ( A ) → K ( π A ) is an isomorphism. We thus find a chain of isomorphismsRep( G, R ) = K (Fun( BG,
Perf( R ))) ≃ K ( R [ G ]) ≃ K ( π R [ G ]) ≃ K (Fun( BG,
Perf( Z [1 / | G | ]))) . Applying Swan’s theorem (Theorem 7.5 above), we conclude the result. (cid:3)
Next, we prove a Swan induction result for KU . We give a geometric argument here that onlyworks at small primes; we will prove the result in full generality later in Theorem 7.13. Theorem 7.7.
For any finite group G , KU -based Swan induction holds for the family of abeliansubgroups of G whose p -part for p ∈ { , , } has rank ≤ .Proof. In view of Theorem 7.4, we may assume that G is abelian. We then reduce to provingthat one has Swan induction for G = C × p and p ∈ { , , } for the family of proper subgroups(cf. Remark 7.1). Since p ≤
5, we have a non-toral embedding
G ֒ → Γ for Γ a suitable simplyconnected compact Lie group: by the results of [Bor61], it suffices to choose Γ such that H ∗ (Γ; Z )has p -torsion in its cohomology, e.g., we can take Γ = E . Let T ⊆ Γ be a maximal torus andconsider the Γ-action on the flag variety Γ /T , as well as its restricted G -action. We will show thatthe G -space Γ /T satisfies the hypotheses of Proposition 7.2, for F the family of proper subgroups.First of all, G acts without fixed points since it is a non-toral subgroup of Γ. By [MNN17, Cor. 8.17],we have an equivalence Γ /T + ⊗ KU ≃ | W | M KU ∈ Fun( B Γ , Perf( KU )) , where W is the Weyl group of Γ. Restricting to G , this proves hypothesis (3) of Proposition 7.2and thus our result. (cid:3) Next, we include some results which are specific to the prime 2, based on the use of representationspheres; they have the advantage of applying at arbitrary chromatic heights. We first need twolemmas that will enable us to recognize the triviality of group actions.
Lemma 7.8.
Let E be an even E -ring spectrum such that π ∗ E is torsion-free and let p be a prime.Let M ∈ Fun( BC p , Perf( E )) be such that: (1) The underlying E -module of M is equivalent to a direct sum of copies of E . (2) The C p -action on π ∗ M is trivial. Then M is equivalent to a direct sum of copies of the unit in Fun( BC p , Perf( E )) .Proof. Let { x i } ⊆ π M be a basis of the free π ∗ E -module π ∗ M . For each i , we want to produce a C p -equivariant map of E -modules(7.6) E → M which carries 1 x i in homotopy. Taking the direct sum of these maps, we will have the desiredequivalence. Equivalently, to produce (7.6), we need to show that the image of π ∗ ( M hC p ) → π ∗ M contains each x i . However, the E -term of the homotopy fixed point spectral sequence for π ∗ ( M hC p )is concentrated in even total degree by our assumptions, and thus collapses. This shows that thereare no obstructions to producing the maps (7.6) and thus to providing the equivalence of thelemma. (cid:3) Lemma 7.9.
Fix a group G . Let E be an E -ring spectrum such that π ∗ ( C ∗ ( BG ; E )) is even andtorsion-free. Let p be a prime, and let M ∈ Fun( B ( C p × G ) , Perf( E )) be such that: (1) The underlying object of
Fun(
BG,
Perf( E )) is equivalent to a direct sum of copies of theunit. (2) The C p -action on π ∗ ( M hG ) is trivial.Then M is equivalent to a direct sum of copies of the unit in Fun( B ( C p × G ) , Perf( E )) : in particular,the C p × G -action is trivial.Proof. We use that Fun( B ( C p × G ) , Perf( E )) = Fun( BC p , Fun(
BG,
Perf( E ))). The thick subcate-gory of Fun( BG,
Perf( E )) generated by the unit is equivalent to Perf( C ∗ ( BG ; E )), via the functor( − ) hG . Thus, the result follows from Lemma 7.8 applied to the object M hG ∈ Fun( BC p , Perf( C ∗ ( BG ; E ))). (cid:3) The next result establishes a very weak result towards the general expectation that the complexityof the representation theory over E ∞ -rings should stabilize once the rank of the group is a bitlarger than the chromatic complexity of the coefficients. A more subtle such result would be ourConjecture 7.22. Proposition 7.10.
Let p be a prime, n ≥ and G be an elementary abelian p -group of rank n + 2 . Let R be a K ( n ) -local, even E ∞ -ring under E n such that π ∗ R is torsion-free. Let M ∈ Fun(
BG,
Perf( R )) . Suppose that for each H ( G , the object Res GH M ∈ Fun(
BH,
Perf( R )) isequivalent to a direct sum of copies of the unit. Then M is equivalent to a direct sum of copies ofthe unit in Fun(
BG,
Perf( R )) .Proof. Let G ′ ( G be a maximal proper subgroup, and fix a complement C p ≃ H ⊆ G , so that G = G ′ × H . By our assumptions and Lemma 7.9, it suffices to show that the H -action on π ∗ ( M hG ′ )is trivial (cf. Lemma 4.2, which shows that C ∗ ( BG ′ ; R ) is even and torsion-free).To see this, we claim that the map of C ∗ ( BG ′ , R )-modules(7.7) M hG ′ → Y G ′′ ≃ C × np ( G ′ M hG ′′ is injective on homotopy. Since M is G ′ -equivariantly isomorphic to a sum of copies of the unit, itsuffices to verify the injectivity of π ∗ ((7.7)) with M replaced by R ; this case follows because(7.8) C ∗ ( BG ′ , R ) → Y G ′′ ≃ C × np ( G ′ C ∗ ( BG ′′ , R ) ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 33 is injective on π ∗ ( − ) ⊗ Q by Hopkins-Kuhn-Ravenel character theory [HKR00] as G ′ ≃ C × ( n +1) p and R is L n -local, cf. [MNN19, Prop. 5.36], and both sides are torsion-free.For any G ′′ ( G ′ , we have an induced H -action on the C ∗ ( BG ′′ , R )-module M hG ′′ , and themap in (7.7) is H -equivariant. Since M restricts to a direct sum of copies of the unit for everyproper subgroup of G , the H -action on M hG ′′ is trivial; indeed, this follows because the action ofthe proper subgroup generated by G ′′ and H on M is trivial. It follows from the injectivity onhomotopy of (7.7) and this observation that the H -action on M hG ′ is trivial on homotopy groups.This completes the proof. (cid:3) We now start considering representation spheres. For a based space B , let B h n i be the ( n − B , so the first potentially non-trivial homotopy group is in degree n . Let M O h n i be the Thom spectrum associated to the J -homomorphism BO h n i → BO → BGL S . We definethe function φ for all integers n ≥ φ ( n ) = 8 a + 2 b when n = 4 a + b + 1 with 0 ≤ b ≤ Lemma 7.11.
Let n ≥ , G = C × n and consider the characters { η i } ≤ i ≤ n of G obtained bypulling back the sign character along the n projection maps. Define α = Q ni =1 (1 − η i ) ∈ RO ( G ) .Then for any M O h φ ( n ) i -oriented E -ring spectrum R , there is an equivalence S α ⊗ R ≃ R in Fun(
BG,
Perf( R )) .Proof. We claim that the map(7.9) BC × n Q ni =1 (1 − η i ) −−−−−−−−→ BO lifts to BO h φ ( n ) i . This implies that for any M O h φ ( n ) i -oriented R , there is an equivalence S α ⊗ R ≃ R in Fun( BG,
Perf( R )) as desired.By Bott periodicity, φ ( n ) is the degree of the n th nonzero homotopy group of BO , starting with φ (1) = 1. We argue inductively on n ≥ η is the sign representation of C , then the virtual representation sphere S − η is classified by a map BC ≃ BO (1) → BO which lifts to BO h φ (1) i = BO h i because BC is connected. This settles the base case n = 1.Suppose now we have a lifting of (7.9) to BO h φ ( n ) i for some n ≥
1. The next classifying mapis obtained as follows: BC × n +12 ≃ BC × BC × n − η n +1 ) ⊗ Q ni =1 (1 − η i )) −−−−−−−−−−−−−−−→ BO (1) ∧ BO h φ ( n ) i → BO ∧ BO ⊗ −→ BO.
Since BO (1) ∧ BO h φ ( n ) i is φ ( n )-connected, the composite of the last two maps lifts to BO h φ ( n +1) i as desired. (cid:3) Theorem 7.12.
Let G be an abelian 2-group. Let R be an E ∞ -ring spectrum. Suppose that forsome n ≥ , we have either: (1) R = E n , a Lubin-Tate theory of height n at the prime . (2) R is M O h φ ( n + 2) i -orientable (as an E -ring and with φ as defined before Lemma 7.11).Then R -based Swan induction holds for the family of subgroups of G of rank at most n + 1 .Proof. By pulling back along the map to a maximal elementary abelian quotient of G , it sufficesto treat the case where G = C × n +22 and prove that R -based Swan induction holds for the family ofproper subgroups, cf. Remark 7.1. Let ǫ , . . . , ǫ n +2 be independent sign characters G → {± } , i.e., { ǫ i } is a basis for the F -vector space Hom( G, {± } ). Let η i ∈ RO ( G ) (1 ≤ i ≤ n + 2) be the classof the associated one-dimensional representation. We consider the class α = n +2 Y i =1 (1 − η i ) ∈ RO ( G ) , and the associated representation G -sphere S α ∈ Sp G . Note that for any proper subgroup H ( G , α restricts to a class in RO ( H ) which is divisible by 2 and therefore comes from the complexrepresentation ring. This follows because α belongs to the ( n + 2)-th power of the augmentationideal, and for any m , the augmentation ideal in RO ( C × m ) / F [ C × m ] has its ( m + 1)th powerequal to zero.Given a nontrivial character µ of G , considered as a 1-dimensional real representation, thecofiber sequence S ( µ ) + → S → S µ shows that the class [ S µ ] ∈ Rep( G, S ) has the property that[ S µ ] − S ( µ ) + ] is induced from a proper subgroup, namely the kernel of µ . Consequently, if V isa sum of nontrivial characters in RO ( G ), then [ S V ] − ∈ Rep( G, S ) is a sum of classes induced fromproper subgroups in Rep( G, S ). Expanding out the product defining α into a sum of characters,we find only a single trivial representation (since the ǫ i are linearly independent). It follows thatin Rep( G, S ), one has(7.10) [ S α ] = − [ S α − ] = − C, for C a sum of classes induced from proper subgroups.We also claim that S α ⊗ R ≃ R in Fun( BG,
Perf( R )) . Under the first hypothesis, this follows from Proposition 7.10 since the restriction of α to a propersubgroup is a complex representation sphere, and thus trivializable. Under the second hypothesis,this follows from Lemma 7.11. Consequently, [ S α ⊗ R ] = 1 ∈ Rep(
G, R ). By (7.10), it follows that1 = − C , so 2 ∈ Rep(
G, R ) is a sum of classes induced from proper subgroups, as desired. (cid:3)
Note that this argument cannot work at odd primes, since all representations of C p are complexfor p > Proof of Theorem E, (3).
This follows from Theorem 7.6 to handle the prime-to-2 case combinedwith Theorem 7.12, which handles the prime 2. (cid:3)
Swan induction for KU . In this subsection, we prove the Swan induction theorem for KU . Note that we have already given (geometric) proofs earlier for the p -part with p ≤
5, seeTheorem 7.7.
Theorem 7.13.
Let G be any finite group. Then KU -based Swan induction holds for the familyof abelian subgroups of rank ≤ . To prove Theorem 7.13, it suffices (cf. Remark 7.1) to treat the case of G = C × p for an arbitraryprime p . Our proof will rely essentially on twisted K -theory. We will first need various preliminaries. Construction 7.14 (Twists of K -theory) . There is a natural map of spaces K ( Z , → BGL ( KU ) , where BGL ( KU ) is the classifying space of trivial invertible KU -modules, cf. [ABG10, Sec. 7]for an account, which induces the identity on π . Consequently, for any finite group G , we have a ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 35 natural map(7.11) H ( G ; Z ) → Pic(Fun(
BG,
Perf( KU ))) , where the right-hand-side is the Picard group of the symmetric monoidal ∞ -category Fun( BG,
Perf( KU )).Given a class τ ∈ H ( G ; Z ), we let KU τ be the associated object of Fun( BG,
Perf( KU )).We will be especially interested in the case G = C × p . Choosing a nonzero class τ ∈ H ( G ; Z ) = F p , we obtain an invertible object KU τ ∈ Fun( B ( C × p ) , Perf( KU )). The induced map B ( C × p ) → BGL ( KU ) is nontrivial, as it lifts uniquely to the 3-connective cover τ ≥ BGL ( KU ), and K ( Z , KU τ is not equivalent to the unit inFun( BG,
Perf( KU )).In the next lemma, to distinguish the factors, we write C ap ⊆ C × p for the first factor and C bp ⊆ C × p for the second. Note that KU ( BC bp ) is isomorphic to the completion of the representationring R ( C bp ) at the augmentation ideal by the Atiyah–Segal completion theorem [AS69, Ati61]. If ζ is a nontrivial character of C bp , then R ( C bp ) is free on the powers of [ ζ ]. Lemma 7.15.
Let τ ∈ H ( C × p ; Z ) be a nontrivial element. (1) The underlying object KU τ | C bp in Fun( BC bp , Perf( KU )) is isomorphic to the unit. (2) The residual C ap -action on ( KU τ ) hC bp ≃ C ∗ ( BC bp , KU ) has the property that the action bya generator in C ap acts by multiplication by [ ζ ] i , for some < i < p .Proof. The first assertion follows because τ restricts to zero in H ( C p ; Z ) = 0. The second assertionfollows because the action of a generator is necessarily given by multiplication by an element of KU ( BC p ) whose p th power is the identity. Moreover, this generator is necessarily nontrivial orthe entire C × p -action on KU τ would be trivial by Lemma 7.9. (cid:3) Our key tool is the following result. We consider the C × p -action on CP p − arising from theprojective representation on C p as in the proof of Theorem 7.5. We identify the KU -linearizationof this action. Proposition 7.16.
We have a decomposition in
Fun( B ( C × p ) , Perf( KU ))(7.12) KU ⊗ DCP p − ≃ M τ ∈ H ( C × p ; Z ) KU τ . Proof.
Again, we label the first and second factors of C × p by C ap , C bp . We first calculate the C bp -equivariant KU -theory of CP p − . Fix a nontrivial character ζ of C bp . The underlying C bp -space of CP p − is the projectivization of the representation 1 ⊕ ζ ⊕ · · · ⊕ ζ ⊗ ( p − of C bp .By the projective bundle theorem, it follows that there is an isomorphism of R ( C bp )-algebras, KU ∗ C bp ( CP p − ) ≃ R ( C bp )[ x ] / p − Y i =0 ( x − [ ζ i ]) , cf. [Seg68, Prop. 3.9]. Here x is the class of the tautological line bundle on CP p − , made C bp -equivariant. We have a residual C ap -action on this R ( C bp )-algebra. Using the definition of x as the class of a tautological bundle, one checks that a generator of C ap carries x to x [ ζ i ] for an appropriate i = 0. From this, it follows that ( KU ⊗ DCP p − ) | C bp is a direct sum of p copies of the unit in Fun( BC bp , Perf( KU )).Therefore, we have ( KU ⊗ DCP p − ) hC bp ≃ L p − i =1 C ∗ ( BC bp , KU ). By the comparison between equi-variant and Borel-equivariant K -theory, and the above calculation, we see that the residual C ap actson the i th factor by multiplication by [ ζ i ] ∈ R ( C bp ) → KU ( BC bp ) (up to renumbering factors).Now we prove the desired equivalence. It suffices to compare the C bp -homotopy fixed points ofboth sides of (7.12), C ap -equivariantly as free modules over the even, torsion-free E ∞ -ring spectrum C ∗ ( BC bp ; KU ). We will do this by a homotopy fixed-point spectral sequence argument. On π ,we have seen from the previous paragraph and Lemma 7.15 that π (cid:16) ( KU ⊗ DCP p − ) hC bp (cid:17) and π ( L τ KU hC bp τ ) are isomorphic as π ( C ∗ ( BC bp , KU ))-modules equipped with a C ap -action. Usingthe homotopy fixed point spectral sequence (and observing that there is no room for obstructions ),we can produce C ap -equivariant maps KU hC bp τ → ( KU ⊗ DCP p − ) hC bp for each τ , and the direct sumof these is an equivalence. (cid:3) Proof of Theorem 7.13.
It suffices to show that KU -based Swan induction holds for C × p and forthe family of proper subgroups. We first observe that Rep( − , KU ) ⊗ Q is a Green functor and thusRep( G, KU ) ⊗ Q receives a map from the rationalized Burnside ring A ( G ) ⊗ Q . For any finite group G , we have complementary idempotents e G , e e G in A ( G ) ⊗ Q (which is isomorphic to a product ofcopies of Q over conjugacy classes of subgroups H ⊆ G ) such that:(1) e G is a Q -linear combination of the classes of the G -sets G/H, H ( G .(2) For each H ( G , the restriction of e G to A ( H ) ⊗ Q is equal to 1. Equivalently, for each H ( G , the homomorphism A ( G ) ⊗ Q → Q which sends a G -set T to | T H | carries e G to 1.(3) e G + e e G = 1.Let M be a rational Green functor for the group G , so that we have a ring map A ( G ) ⊗ Q → M ( G ).Then M is induced from the family of proper subgroups (equivalently, 1 ∈ M ( G ) is a sum of classesinduced from proper subgroups) if and only if this map carries e G to 1 (or, equivalently, sends e e G to zero); indeed, this follows because multiplication by e G acts as the identity on classes inducedfrom a proper subgroup.Consider Rep( C × p , R ) for any E ∞ -ring R . In this case, we have another expression for the imageof e C × p under A ( C × p ) ⊗ Q → Rep( C × p , R ) ⊗ Q (for which we will simply write e C × p ). In fact, weclaim that(7.13) [ R ⊗ CP p − ] /p = e C × p ∈ Rep( C × p , R ) ⊗ Q , for the C × p -action on CP p − arising from the p -dimensional projective representation as above. Inother words, pe C × p is the class of R ⊗ CP p − ∈ Fun( BC × p , Perf( R )) in rationalized K . To seethis, we first observe that any finite G -CW complex has a well-defined Euler characteristic takingvalues in A ( G ) which can be calculated by taking the Euler characteristic of the cellular chains.Now CP p − is a finite fixed-point-free C × p -complex such that the fixed points under any proper Explicitly, we consider the C bp -equivariant line bundle on CP p − given by the set of pairs ( x, v ) for x ∈ CP p − and v ∈ x ; the C bp -equivariant structure is by action on the pair. The claim follows by noting that C ap , C bp act on C p ,but their actions fail to commute by a p th root of unity. Note here that all the summands π ( KU hC bp τ ) for τ = 0 have trivial higher C ap -cohomology. ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 37 subgroup have Euler characteristic p . This implies the associated class in A ( C × p ) is pe C p × C p asdesired, by the above characterization of the idempotent e G , whence the claim.We specialize now to the case where R = KU . Let τ be a generator of H ( C × p ; Z ) = F p and let x = [ KU τ ]. Then, combining (7.13) and the decomposition of Proposition 7.16, we conclude1 + x + · · · + x p − p = e C × p ∈ Rep( C × p , KU ) ⊗ Q . Note that x p = 1, so the left hand side is clearly idempotent. This also determines the complemen-tary idempotent, using the decomposition of the group ring Z [1 /p ][ C p ] ≃ Z [1 /p ] × Z [1 /p, ζ p ]. Wetherefore have(7.14) 1 p p − Y j =1 (1 − x j ) = e e C p × C p , since this is the complementary idempotent in the group ring.Our goal is to show that e e C × p = 0 in Rep( C × p , KU ) ⊗ Q , which is equivalent to the Swaninduction claim. Given an elementary abelian p -group G of rank rk( G ) ≥
2, one has e e G = Q φ : G ։ G ′ φ ∗ e e G ′ , where the product ranges over all surjections G ։ G ′ with rk( G ′ ) = rk( G ) − φ is an idempotent in A ( G ) ⊗ Q with trivial restrictionto proper subgroups and with image image under the G -fixed point map A ( G ) → Q equal to 1.Therefore, we can express e e C × p as the product(7.15) Y φ : C × p ։ C × p φ ∗ e e C × p , using the pullback in the representation ring. We will now analyze this using group rings.For any finite group G , we have a natural map H ( BG ; Z ) → Pic(Fun(
BG,
Perf( KU ))) , whichdefines a map of commutative rings(7.16) ϕ G : Q [ H ( BG ; Z )] → Q ⊗ Z Rep(
G, KU ) , which is compatible with pullback in G . By (7.14), there exists a class in the group ring Q [ H ( BC × p ; Z )]whose image under ϕ C p is precisely the idempotent e e C × p . Using the expression (7.15), we see thatthere is a class u ∈ Q [ H ( BC × p ; Z )] whose image under ϕ C p is e e C × p . Moreover, u restricts to zeroin Q [ H ( BH ; Z )] for all proper subgroups H ( C p . By the next two lemmas, this is enough toforce u = 0, which proves the theorem. (cid:3) Lemma 7.17.
Let X ≃ C × p be a rank elementary abelian p -group, so H ( X ; Z ) is also arank elementary abelian p -group. As Z ⊆ X ranges over the rank subgroups of X , the maps H ( X ; Z ) → H ( Z ; Z ) ≃ F p range over the nonzero maps H ( X ; Z ) → F p , up to scalars.Proof. The construction which sends Z ⊆ X to the kernel of the surjection H ( X ; Z ) → H ( Z ; Z )establishes a map(7.17) Ψ : { Z ⊆ X } → (cid:8) hyperplanes in H ( X ; Z ) (cid:9) . We need to show that (7.17) is a bijection. Note that both sides are finite sets of the same cardinality,and that the map is Aut( X ) = GL ( F p )-equivariant (using the induced action on H ( X ; Z )).Choose a decomposition X = V ⊕ W where V has rank 2 and W has rank 1. By the universalcoefficient theorem, we have a natural short exact sequence(7.18) 0 → H ( V ; Z ) → H ( X ; Z ) → Tor ( H ( V ; Z ) , H ( W ; Z )) → , where Tor ( H ( V ; Z ) , H ( W ; Z )) has rank two. On the left-hand-side of (7.17), we consider thecollection C of subspaces Z ⊆ X such that the composite Z ⊆ X ։ V is not surjective; equivalently, Z = L ⊕ W for some L ⊆ V a 1-dimensional subspace. Note that | C | = p +1. On the right-hand-side,consider the collection D of hyperplanes in H ( X ; Z ) which contain H ( V ; Z ); the exact sequence(7.18) also easily shows | D | = p + 1.We claim that Ψ − ( D ) = C . In fact, given a two-dimensional subspace Z ⊆ X such that H ( V ; Z ) → H ( X ; Z ) → H ( Z ; Z ) is zero, it follows easily that the map Z → X ։ V fails to besurjective, and conversely. The group Aut( V ) ⊆ Aut( X ) (via the diagonal embedding, fixing W )preserves and acts transitively on C . Moreover, Aut( V ) ⊆ Aut( X ) preserves and acts transitively on D , because we have an Aut( V )-equivariant identification H ( V ; Z ) ≃ H ( V ; Q / Z ) = Hom( V, F p ),and D is identified with the set of lines in H ( V ; Z ). Therefore, for c ∈ C , we necessarily have thatΨ − (Ψ( c )) consists of a single point since the fibers of Ψ at points of D must all have the samecardinality. Since Aut( X ) acts transitively on the set of 2-dimensional subspaces of X , it followseasily that (7.17) is an isomorphism as desired. (cid:3) Lemma 7.18.
Let A be a finite abelian group. Let x ∈ Q [ A ] be an element such that for every map A → C , for C a cyclic group, the image of x under Q [ A ] → Q [ C ] is zero. Then x = 0 .Proof. We can extend scalars to C . Then we have a natural isomorphism C [ A ] ≃ Q A ∨ C , for A ∨ the group of characters of A . Our assumption is that for any map C ′ → A ∨ with C ′ cyclic, therestriction Q A ∨ C → Q C ′ C annihilates x ; this clearly forces x = 0. (cid:3) Applications to chromatic complexity.
In this subsection, we record the applications ofthe above Artin induction theorems to chromatic bounds for the K -theory of certain ring spectra.We recover another new proof of Mitchell’s theorem and are able to treat some special cases ofTheorem A. Corollary 7.19 (Mitchell [Mit90]) . For i ≥ , we have L T ( i ) K ( Z ) ≃ (for any prime p ).Proof. Combine Theorem 6.6 and Swan induction for H Z (Theorem 7.5). (cid:3) Corollary 7.20.
Let E n be a height n ≥ Lubin-Tate theory at the prime and G ⊆ G n a finitesubgroup of the extended Morava stabilizer group. Then L T ( n + m ) K ( E hGn ) = 0 for all m ≥ .Proof. When G is the trivial subgroup, this follows directly from Theorem 7.12 and Theorem 6.6.The general case then follows from the Galois descent theorem [CMNN20, Thm. 1.10], which gives L T ( n + m ) K ( E hGn ) ≃ ( L T ( n + m ) K ( E n )) hG ≃ . (cid:3) We next recover the following result. At p ≥
5, the result is a consequence of the calculations ofAusoni–Rognes [AR02] and Ausoni [Aus10], which determine the mod ( p, v ) homotopy groups of K ( l ) (resp. K ( ku )) at such primes (and in particular yield the stronger Lichtenbaum–Quillen styleclaim that K ( ku ) / ( p, v ) agrees with its T (2)-localization in high degrees). For p ≤
3, this resulthas been recently proved by Angelini-Knoll–Salch [AKS20] using trace methods. The result is alsoa special case of Theorem A.
Corollary 7.21.
For i ≥ , we have L T ( i ) K ( KU ) = L T ( i ) K ( KO ) ≃ (at any prime p ).Proof. By Galois descent [CMNN20], it suffices to handle the case of KU . In this case, Theorem 6.6together with Theorem 7.13 imply the result. (cid:3) Motivated by the above results, we conjecture the following Swan induction result for E n ; wehave proved it at p = 2 in Theorem 7.12, or for n = 1 in Theorem 7.13. ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 39 Conjecture 7.22.
Let p be a prime, n ≥ E n a Lubin-Tate theory of height n at the prime p and G finite group. Then E n -based Swan induction holds for the family of those abelian subgroupsof G for which:(1) The prime-to- p part is cyclic.(2) The p -part has rank ≤ n + 1. Remark 7.23 (A purely algebraic question) . Finally, Theorem 6.6 can be used to prove that L K (1) K ( F p ) = 0, which is a consequence of the stronger result K ( F p ; Z p ) = H Z p proved byQuillen; indeed, one sees that H F p -based Swan induction holds for the trivial family in C p us-ing the filtration of the regular representation F p [ C p ] by trivial representations. One also knowsthat L K (1) K ( Z /p n ) = 0 for any n ≥
1, cf. [LMMT20, BCM20, Mat20] for three proofs. Can thisresult also be proved using Theorem 6.6, i.e., does H Z /p n -based Swan induction hold for the trivialfamily in C p ? Appendix A. Mackey functors and orthogonal G -spectra This appendix provides a fairly self-contained proof of the fact that, for a finite group G , thesymmetric monoidal ∞ -categories afforded by orthogonal G -spectra and by spectral Mackey func-tors are equivalent. This result is due originally to Guillou and May [GM11] (ignoring the monoidalstructure), and was revisited by Barwick and Barwick-Glasman-Shah [Bar17, BGS20] in the contextof more general parametrized homotopy theory, see specifically [Nar16, Thm. A.4]. Compared totheir work, our approached is streamlined by ignoring all models (as used by [GM11]), and by notaddressing any universal properties of Mackey functors (as in [Bar17, BGS20]).The motivation for giving our proof of their result is the immediate need of the present paper:We use categorical methods to construct Mackey functors, and then apply descent results provenfor the homotopy theory of orthogonal G -spectra to them. Our work also yields a new proof of theequivariant Barratt-Priddy-Quillen theorem (which however uses the non-equivariant one).Throughout, let G denote a finite group. We refer the reader to [MNN17, Sec. 5] for a quickaccount of the symmetric monoidal ∞ -category Sp G extracted from the model category of orthog-onal G -spectra. We denote by O ( G ) the orbit category of G , by S the ∞ -category of spaces,by S G := Fun( O ( G ) op , S ) the presentable, cartesian closed ∞ -category of G -spaces (see [BH17,Lem. 2.1]), by S G, • ≃ S G, ∗ / the presentable, closed symmetric monoidal ∞ -category of based G -spaces, and by Σ ∞ G : S G, • −→ Sp G the suspension spectrum functor. We consider S G with itscartesian monoidal structure. In the appendix, we will write the units of G -spectra and spectralMackey functors by 1 rather than S .To set the notation for Mackey functors, we denote by Fin G the category of finite G -sets, bySpan(Fin G ) the (2 , G (cf. [BH17, App. C]) and setMack G := Mack G (Sp) = Fun × (Span(Fin G ) op , Sp) , the category of finite product-preserving presheaves on Span(Fin G ) with values in the ∞ -categorySp of spectra. Note that Span(Fin G ) = Burn eff G as recalled in Definition 2.1, but we stick to theformer notation in the Appendix, in order to be compatible with our main references. We recallthat Mack G ≃ P Σ (Span(Fin G )) ⊗ Sp, cf. Remark 2.3. Below, we will recall the suspension functorin the Mackey context, to be denoted Σ ∞M : S G, • −→ Mack G . The cartesian product on Fin G induces a symmetric monoidal structure on Span(Fin G ), we endowMack G with the symmetric monoidal structure given by Day-convolution and denote it by ⊗ . Thisis the unique symmetric monoidal structure which is bicocontinuous and compatible with Σ ∞M .Our main result is the following. Theorem A.1.
There is a unique symmetric monoidal left-adjoint L : Sp G → Mack G such that L ◦ Σ ∞ G ≃ Σ ∞M , and L is an equivalence. The rest of this section will provide a proof of this result.The construction of L rests on the following result, and we thank Markus Hausmann for providinga key reference in its proof. Compare also [GM20, App. C] for a treatment. Theorem A.2.
The suspension Σ ∞ G : S G, • −→ Sp G is the initial example of a presentably sym-metric monoidal functor which inverts the functor S V ⊗ − for all finite-dimensional, orthogonalrepresentations V of G .Proof. We have the symmetric monoidal suspension functor Σ ∞ G : S G, • −→ Sp G . By constructionof Sp G , the representation spheres map to invertible objects in Sp G . Now the initial presentablysymmetric monoidal ∞ -category C equipped with a cocontinuous, symmetric monoidal functorfrom S G, • inverting the representation spheres is discussed (in a more general context) in [Rob15,Sec. 2.1], cf. also [BH17, Lem. 4.1]; note that the representation spheres are symmetric objectsby [GM20, Lem. C.5]. Equivalently, we can also perform this construction at the level of smallfinitely cocomplete symmetric monoidal ∞ -categories by restricting to the compact objects. By[Rob15, Prop. 2.19, Cor. 2.22], we find that this formal inversion (in P r L st ) is given by the colimitof smashing with S V on S G, • , as V ranges over G -representations. By construction, we obtaina canonical, cocontinuous symmetric monoidal functor C → Sp G . It follows from the above thatthe mapping spaces between the finite G -sets T, T ′ are computed in the same way (namely, aslim −→ V Hom S G, • ( S V ∧ T + , S V ∧ T ′ + )), so C → Sp G is fully faithful on compact generators, whence theresult. (cid:3) To construct Σ ∞M , recall from [BH17, § ι : Fin G, + → Span(Fin G ) and the symmetric monoidal equivalence P Σ (Fin G, + ) ≃ S G, • ([BH17, Lem. 2.1]). This induces Σ ∞M , to be defined as the compositionΣ ∞M := (cid:18) S G, • ≃ P Σ (Fin G, + ) P Σ ( ι ) −−−→ P Σ (Span(Fin G )) = Fun × (Span(Fin G ) op , S ) → Mack G (cid:19) , where the final map is the stabilization. By construction, Σ ∞M is a map in CAlg(Pr L ).Theorem A.2 tells us that to construct the functor L in Theorem A.1, we need to see that Σ ∞M inverts all representation spheres. We will do this by constructing from scratch on Mack G whatwill a posteriori turn out to be geometric fixed point functors, and by establishing some of theirbasic properties. Denote by P the family of proper subgroups of G and recall the cofiber sequencein S G, • , defining g E P : lim −→ G/H ∈O ( G ) P G/H + ≃ E P + −→ ∗ + = S −→ g E P , (cf. [MNN19, Appendix A.1]). The least formal part of our argument is the following. In other words, a map in CAlg(Pr L ). ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 41 Lemma A.3.
We have an equivalence (cid:16) Σ ∞M ( g E P ) (cid:17) ( G/G ) ≃ S in Sp (since the source is an E ∞ -ring, the equivalence is uniquely specified). To see this, we will need the following result on manipulating colimits. For an ∞ -category C , wedenote by C ⊲ the result of freely adjoining a final object to C , and by C ≃ the maximal underlyingsubgroupoid of C . The construction C 7→ C ≃ is right adjoint to the inclusion S ≃ G rp ∞ ⊆ C at ∞ of ∞ -groupoids into all ∞ -categories. Proposition A.4.
Let C be an ∞ -category and F : C ⊲ → S the functor defined by F ( c ) = (cid:0) ( C ⊲ ) /c (cid:1) ≃ . Then the canonical map of spaces lim −→ C F → lim −→ C ⊲ F is equivalent to the inclusion C ≃ ⊆ ( C ⊲ ) ≃ .Proof. The closely related functor F ′ : C ⊲ → C at ∞ defined by F ′ ( c ) := ( C ⊲ ) /c classifies the cocarte-sian codomain fibration cd : Fun(∆ , C ⊲ ) −→ C ⊲ given by evaluation on 1 [Lur09, Cor. 2.4.7.12].It follows that F = ( − ) ≃ ◦ F ′ classifies the left fibration cd ′ : Fun(∆ , C ⊲ ) left −→ C ⊲ obtain bypassing from Fun(∆ , C ⊲ ) to the sub-simplicial set Fun(∆ , C ⊲ ) left ⊆ Fun(∆ , C ⊲ ) consisting of allsimplices all of whose edges are cd -cocartesian. Informally then, the objects of Fun(∆ , C ⊲ ) left arethe morphisms in C ⊲ , and the morphisms are the commuting squares in which the map betweensources is an equivalence.We now observe that evaluation at zero , ez : Fun(∆ , C ) left → C ≃ , is a Cartesian fibration whichhas all fibers contractible (because each of them has an initial object). In particular, ez is a weakequivalence, and an inverse equivalence is provided by sending objects to identity morphisms. Wehave thus seen that lim −→ C F ≃ C ≃ .Furthermore, the canonical map lim −→ C F → lim −→ C ⊲ F is equivalent to the obvious map Fun(∆ , C ) left → Fun(∆ , C ⊲ ) left , the target of which is equivalent to the fiber over the cone point, namely ( C ⊲ ) ≃ .One checks that this identifies the canonical map with the inclusion C ≃ ⊆ ( C ⊲ ) ≃ , as claimed. (cid:3) Proof of Lemma A.3.
Applying Proposition A.4 with C = O ( G ) P the category of orbits with properisotropy (hence C ⊲ = O ( G )), we obtain a cofiber sequence in spaceslim −→ G/H ∈O ( G ) P ( O ( G ) / ( G/H )) ≃ ≃ O ( G ) ≃P ֒ → O ( G ) ≃ −→ ∗ ⊔ +where the final map sends all orbits with proper isotropy to +, and sends G/G to ∗ . We canconsider this as a cofiber sequence in pointed spaces S • of the formlim −→ G/H ∈O ( G ) P ( O ( G ) / ( G/H )) ≃ + ≃ O ( G ) ≃P , + −→ O ( G ) ≃ + −→ S = ∗ + where the final map sends all orbits with proper isotropy to the base-point +, and sends G/G to ∗ . Applying the free commutative monoid functor P : S • → CMon( S ) yields a cofiber sequence inCMon( S ):(A.1) lim −→ G/H ∈O ( G ) P (Fin G / ( G/H )) ≃ −→ Fin ≃ G −→ Fin ≃ in which the final map is identified with taking G -fixed points. To see this, observe that O ( G ) / ( G/H ) ≃O ( H ) and that P ( O ( H ) ≃ + ) ≃ Fin ≃ H , as can be checked most easily using the general formula P ( Z ) = W n ≥ (cid:0) Z × n × Σ n E Σ n + (cid:1) .We denote by ( − ) + the group completion on CMon( S ), and observe thatΩ ∞ (Σ ∞M ( G/H + )( G/G )) = Hom
Span(Fin G ) ( G/G, G/H ) + ≃ (Fin G / ( G/H )) ≃ , + . We thus see that the delooping of the group completion of the cofiber sequence (A.1) is a cofibersequence in Sp of the form(Σ ∞M ( E P )) ( G/G ) −→ Σ ∞M ( S )( G/G ) −→ Σ ∞M ( g E P )( G/G ) ≃ S , using the Barratt–Priddy–Quillen theorem that Ω ∞ ( S ) ≃ Fin ≃ , + . (cid:3) Next, we will need to discuss restriction for Mackey functors.
Construction A.5 (Restriction for Mackey functors) . Let H ⊆ G be a subgroup.(1) We have a symmetric monoidal and coproduct preserving functorRes GH : Span(Fin G ) −→ Span(Fin H ) . This sends a G -set U to the underlying H -set of U , and behaves accordingly on correspon-dences (cf. [BH17, App. C.3]).(2) We also have a functor G × H ( − ) : Span(Fin H ) −→ Span(Fin G ) , which takes a H -set T to the G -set G × H T , and behaves analogously on correspondences.Note that the construction T G × H T on finite H -sets preserves fiber products. Proposition A.6.
Both the functors (Res GH ( − ) , G × H ( − )) : Span(Fin G ) ⇄ Span(Fin H ) and thefunctors ( G × H ( − ) , Res GH ( − )) : Span(Fin H ) ⇄ Span(Fin G ) are biadjoint.Proof. If S is a finite H -set and T is a finite G -set, then we have an equivalence of categories(Fin G ) / ( G × H S ) × T ≃ −→ (Fin H ) /S × Res GH ( T ) , given by pulling back along the H -map S × T → ( G × H S ) × T . Using that Hom Span(Fin G ) ( X, Y ) ≃ (Fin G ) ≃ /X × Y , the result follows. See also [BH17, App. C.3] for a more general treatment of Span( − )as an ( ∞ , (cid:3) We now define restriction for Mackey functors, essentially by left Kan extension. Namely, wedefine the symmetric monoidal, cocontinuous functor Res
GH, M : Mack G → Mack H to be Res GH, M := P Σ (Res GH ) ⊗ id Sp G .As an example, note that for F ∈ Mack G and subgroups H ′ ⊆ H ⊆ G we haveRes GH, M ( F )( H/H ′ ) ≃ F ( G/H ′ ) . To see this, since both sides are colimit preserving functors of F , it suffices to check the case when F is the suspension of an orbit, and then the claim is immediate from the second adjunction inProposition A.6. Proposition A.7.
Assume F ∈ Mack G is such that for all proper subgroups H ⊆ G we have Res
GH, M ( F ) ≃ ∗ . Then the canonical map F ( G/G ) −→ (cid:16) Σ ∞M ( g E P ) ⊗ F ) (cid:17) ( G/G ) is an equivalence.Proof. First observe that for all subgroups H ⊆ G , the suspension Σ ∞M ( G/H + ) ∈ Mack G is self-dual.It then follows that for all proper subgroups H ⊆ G , the spectrum( F ⊗ Σ ∞M ( G/H + ))( G/G ) ≃ F ( G/H ) ≃ Res
GH, M ( F )( H/H ) ≃ ∗ is contractible, and hence that ( F ⊗ Σ ∞M ( E P + )) ( G/G ) ≃ ∗ . The result follows. (cid:3) ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 43 We next introduce geometric fixed points in the Mackey context. The fixed point functor ( − ) G :Fin G −→ Fin commutes with pullbacks and hence induces a functor on span categories. Thisfunctor preserves finite coproducts and the cartesian product, hence the functorΦ G M := P Σ (Span(( − ) G )) ⊗ id Sp : Mack G −→ P Σ (Span(Fin)) ⊗ Sp ≃ Spcommutes with all colimits and is symmetric monoidal. By construction, it takes the expected valueson orbits, namely Φ G M (Σ ∞M ( G/H + )) is contractible for a proper subgroup H ⊆ G , and equivalentto S for H = G . In fact, it is clear that, more generally, for each X ∈ S G, • we haveΦ G M (Σ ∞M ( X )) ≃ Σ ∞ ( X ( G/G )) . For a subgroup H ⊆ G , we denote Φ G,H M := Φ H M ◦ Res
GH, M .We will need to know that our geometric fixed points are given by the familiar contruction: Proposition A.8.
We have an equivalence of functors Φ G M ( − ) ≃ (cid:16) Σ ∞M ( g E P ) ⊗ ( − ) (cid:17) ( G/G ) .Proof. First, we have a natural transformation for F ∈ Mack G given by F ( G/G ) → Φ G M ( F ), since F ( G/G ) is corepresented by the unit and Φ G M ( − ) is symmetric monoidal by construction. Since thetarget is unaffected by replacing F by Σ ∞M ( g E P ⊗ F ), we obtain a map (cid:16) Σ ∞M ( g E P ⊗ ( − )) (cid:17) ( G/G ) → Φ G M ( − ). This map is an equivalence on all orbits, because for a subgroup H ⊆ G we can computethat (cid:16) Σ ∞M ( g E P ) ⊗ Σ ∞M ( G/H + ) (cid:17) ( G/G ) ≃ (cid:16) Σ ∞M (cid:16)g E P ⊗ ( G/H + ) (cid:17)(cid:17) ( G/G )is contractible if H is proper, and is S if H = G by Lemma A.3. Therefore, the result follows sinceboth functors preserve colimits. (cid:3) This allows to easily establish the basic properties of geometric fixed points in the Mackeycontext:
Proposition A.9.
The family { Φ G,H M } H ⊆ G of symmetric monoidal left adjoints is jointly conser-vative.Proof. It only remains to see the joint conservativity, so assume Φ
G,H M ( F ) ≃ ∗ for all H ⊆ G , andwe need to see that F ≃ ∗ .This is clear for trivial G , and we argue by induction on the group order in general. We can thusassume that Res GH, M ( F ) ≃ ∗ for all proper subgroups H ⊆ G . In particular then, for all propersubgroups H ⊆ G we know that F ( G/H ) = Res
GH, M ( F )( H/H ) = ∗ is contractible, and need to see that F ( G/G ) is as well. But combining Proposition A.7 andProposition A.8, we see that F ( G/G ) ≃ Φ G M ( F ) = Φ G,G M ( F ), and this is contractible by assumption. (cid:3) This finally lets us check that suspension for Mackey functors inverts all representation spheres.
Proposition A.10.
For every representation V of G , Σ ∞M ( S V ) ∈ Mack G is invertible. The equivalence is constructed in the proof, we will only need an abstract equivalence.
Proof.
We first note that Σ ∞M ( S V ) ∈ Mack G is at least dualizable. Since Mack G is stable, thedualizable objects are stable under finite colimits, and S V is a finite colimits of orbits. It thussuffies to remark that the orbits are dualizable (in fact, self-dual) already in Span(Fin G ). Once weknow Σ ∞M ( S V ) ∈ Mack G is dualizable, it will be invertible if and only if it becomes so after applyingany family of jointly conservative symmetric monoidal functors. By Proposition A.9 it will thussuffice to see that for every subgroup H ⊆ G , the spectrum Φ G,H M (Σ ∞M ( S V )) is invertible, but thisfollows from a direct computation:Φ G,H M (Σ ∞M ( S V )) = Φ H M (Res GH, M (Σ ∞M ( S V ))) ≃ Σ ∞ (( S V ) H ) ≃ S dim( V H ) . (cid:3) We can now complete the proof of our main result.
Proof of Theorem A.1.
Theorem A.2 and Proposition A.10 show that there is a unique symmetricmonoidal left adjoint L : Sp G −→ Mack G such that L ◦ Σ ∞ G ≃ Σ ∞M . It remains to see that L isan equivalence. Denote by R the right adjoint of L . Since both Sp G and Mack G are generatedunder colimits by dualizable objects (namely the suspensions of orbits), it follows from [BDS16,Thm. 1.3] that R admits itself a right adjoint, hence preserves colimits, and that the adjunction( L, R ) satisfies a projection formula. Furthermore, R is conservative because the image of its leftadjoint L contains a set of generators. We can thus apply [MNN17, Prop. 5.29] to conclude thatthe adjunction ( L, R ) induces an adjoint equivalenceMod Sp G ( R (1 Mack G )) ≃ Mack G , and it remains to see that the counit of the adjunction(A.2) 1 Sp G −→ R ( L (1 Sp G )) ≃ R (1 Mack G )is an equivalence.Now we use induction on the group order. Given a proper subgroup H ( G , we have a commu-tative diagram in CAlg(Pr L ), Sp G (cid:15) (cid:15) / / Mack G (cid:15) (cid:15) Sp H / / Mack H , by the universal property of Sp G . The inductive hypothesis gives that the bottom horizontal arrowis an equivalence. This implies that if X ∈ Sp G , then Hom Sp G ( G/H + , X ) = Hom Mack G ( G/H + , X )since both sides are calculated as maps out of the unit in Sp H (resp. Mack H ). In particular, thisimplies that (A.2) restricts to an equivalence after restriction to proper subgroups; therefore, itsuffices to see that Φ G (i.e., geometric fixed points for orthogonal spectra) turns this map into anequivalence. Since Φ G (1 Sp G ) = S and we are looking at a map of commutative algebras, it sufficesin fact to see that there is an equivalence of spectra Φ G ( R (1 Mack G )) ≃ S . This follows from thefollowing computation:Φ G ( R (1 Mack G )) ≃ (cid:16) Σ ∞ G ( g E P ) ⊗ R (1 Mack G )) (cid:17) G ≃ (cid:16) R h L (Σ ∞ G ( g E P )) ⊗ Mack G ) i(cid:17) G ≃≃ L (Σ ∞ G ( g E P ))( G/G ) ≃ (Σ ∞M ( g E P ))( G/G ) ≃ S . ESCENT AND VANISHING IN CHROMATIC ALGEBRAIC K -THEORY VIA GROUP ACTIONS 45 This computation used in turn: The definition of Φ G , the projection formula for ( L, R ), the factthat ( R ( − )) G ≃ ( − )( G/G ) (by adjointness of L and R ), the fact that L ◦ Σ ∞ G ≃ Σ ∞M , and finallyLemma A.3. (cid:3) As promised earlier, our account yields the following proof of the equivariant Barratt–Priddy–Quillen theorem (originally due to [GM11]), which by-passes any loop-space theory (but uses thenon-equivariant version).
Corollary A.11.
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