The spectrum of equivariant Kasparov theory for cyclic groups of prime order
aa r X i v : . [ m a t h . K T ] S e p THE SPECTRUM OF EQUIVARIANT KASPAROV THEORY FORCYCLIC GROUPS OF PRIME ORDER
IVO DELL’AMBROGIO AND RALF MEYER
Abstract.
We compute the Balmer spectrum of the equivariant bootstrapcategory of separable G -C*-algebras when G is a group of prime order. Introduction and results
Let G be a second countable locally compact group. Kasparov’s G -equivariantKK-theory of complex separable G -C*-algebras [Kas88] defines a tensor triangu-lated category KK G . This fact was first recognized by Meyer and Nest [MN06],who used it reformulate the Baum–Connes assembly map elegantly and study itsfunctorial properties. The spectrum of an (essentially small) tensor triangulatedcategory T is a certain topological space Spc( T ) . This important invariant wasintroduced by Balmer [Bal05]. In [Del10], it was observed that the Baum–Connesconjecture would follow from the (also conjectural) surjectivity of the canonicalmap ⊔ H Spc(KK H ) → Spc(KK G ) , where H runs through the compact subgroupsof G .Unfortunately, a geometric result such as the above surjectivity still appears wellout of reach; this is despite recent major advances of tensor triangular geometry,the theory and computational techniques concerned with the spectrum of tensortriangulated categories (see the survey [Bal10b]). While numerous spectra havebeen computed in fields ranging from topology to modular representation theory,algebraic geometry and motivic theory (see [Bal19] for a large catalogue), in non-commutative geometry we are still striving to understand the most basic examples.The most general fact known to date is the existence, when G is a compact Liegroup, of a canonical continuous surjective map Spc KK G → Spec R( G ) onto theZariski spectrum of the complex character ring R( G ) ; and this holds for ratherformal reasons (see [Bal10a, Cor. 8.8]). In order to obtain sharper results, one mustseverely restrict the kind of groups and algebras under study.The present article contributes the first complete computation of a truly equi-variant example in this context, as well as techniques which may prove useful inother cases. Because of its good generation properties, we consider as in [Del14] thesubcategory Cell( G ) ⊂ KK G of G -cell algebras , that is, the localizing subcategoryof KK G generated by the function algebras C( G/H ) for H ≤ G ; and we aim atcomputing the spectrum of the tensor triangulated subcategory Cell( G ) c of com-pact objects (see Remark 2.3). In topology, this is analogous to considering the Date : September 14, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Equivariant Kasparov theory, triangulated categories, spectrum.First author partially supported by the Labex CEMPI (ANR-11-LABX-0007-01). spectrum of the equivariant stable homotopy category
SH( G ) c of finite G -spectra,as done in [BS17]. Moreover, we restrict attention to finite groups G . In this casethere is also a continuous surjection ρ G : Spc Cell( G ) c → Spec R( G ) , which is known to admit a canonical continuous splitting (see [Del10, Thm. 1.4]).This map was conjectured by the first author to be invertible.The main result of the present paper settles some cases of the conjecture:1.1. Theorem (See Section 6) . Let G = Z /p Z for a prime number p . Then thecanonical comparison map is a homeomorphism ρ G : Spc Cell( G ) c ∼ → Spec R( G ) . The representation ring R( G ) of Z /p is isomorphic to the group ring: R( G ) ∼ = Z [ G ] ∼ = Z [ x ] / ( x p − . A thick tensor-ideal subcategory of Cell( G ) c is a full subcategory C that is closedunder taking isomorphic – that is, equivariantly KK-equivalent – objects, mappingcones, suspensions, retracts and tensor products with arbitrary objects. A subsetof a topological space is called specialization closed if it is a union S = S i Z i ofclosed subsets Z i .Since Cell( G ) c is rigid and idempotent complete by Remark 2.3, Theorem 1.1may be combined with general abstract results (namely, [Bal10b, Thm. 14, Rmks. 12and 23]) to classify the thick tensor-ideal subcategories of Cell( G ) c :1.2. Corollary. If G ∼ = Z /p Z , then there is a canonical bijection between (1) thick tensor-ideal subcategories C of Cell( G ) c and (2) specialization closed subsets S of the Zariski spectrum Spec( Z [ x ] / ( x p − . The classification assigns to a thick tensor-ideal C the union of the supports supp( A ) of all algebras A in C , and to a specialization closed subset S the sub-category of all A with supp( A ) ⊆ S . Morally, the support of an object is thesubset of Spec( Z [ x ] / ( x p − on which it ‘lives’ (see Section 5 for the abstractdefinition and [Del10, §6] for a more concrete description). As a consequence of thecorollary, a compact G -cell algebra A can be built from another one B using thetensor-triangular operations if and only if supp( A ) ⊆ supp( B ) .If G = 1 is the trivial group, then Cell(1) ⊂ KK is just the usual bootstrapcategory and Cell(1) c consists of the C*-algebras in the bootstrap category withfinitely generated K-theory groups. In this case, our results are already known (see[Del10, Thm. 1.2] and also [Del11]).Theorem 1.1 is significantly harder to prove than the case of the trivial group andrequires new ingredients. Our proof strategy is roughly inspired by that of [BS17],which computes the spectrum of the above-mentioned equivariant stable homotopycategory SH( G ) c . Specifically, we divide the problem in two halves (see the begin-ning of Section 6) and settle the first half thanks to the separable monadicity ofrestriction functors proved in [BDS15]. For the second half, however, our proofsdiverge because of fundamental structural differences between equivariant stablehomotopy and KK-theory (see Remarks 6.8 and 6.9). The point is to compute thespectrum of a certain localization Q ( G ) of Cell( G ) (see Section 4). In order todo this, we show that the tensor triangulated category Q ( G ) is ‘weakly regular’,so that we may apply the results of [DS16]. (Incidentally, this method cannot beapplied to the equivariant stable homotopy category by Remark 6.9.) The weak HE SPECTRUM OF EQUIVARIANT KK-THEORY 3 regularity of Q ( G ) is shown using our last crucial ingredient, Köhler’s universalcoefficient theorem [Köh10].This strategy may well provide a proof of the bijectivity of ρ G for more generalfinite groups. Köhler’s result, however, only holds for groups of prime order andhence would need to be replaced by a more general argument.2. The equivariant bootstrap category
We collect here some structural results on equivariant KK-theory and cell alge-bras. Our sources are [MN06] and [Del14].Let G be a finite group (for simplicity, as many of the statements hold much moregenerally). First of all, recall from [MN06] that the Kasparov category of separable G -C*-algebras, KK G , is a tensor triangulated category , that is, it is triangulated inthe sense of Verdier and carries a symmetric monoidal structure. The exact tensorfunctor is the minimal tensor product of C*-algebras, equipped with the diagonalgroup action. The tensor unit object is the algebra C of complex numbers withthe trivial G -action. The category KK G is essentially small and has all countablecoproducts, provided by C -direct sums. The suspension functor Σ = C ( R ) ⊗ − of the triangulated structure of KK G is 2-periodic, that is, there is a natural iso-morphism Σ ∼ = id KK G , thanks to the Bott isomorphism β : C ∼ → C ( R ) . Let R( G ) be the representation ring of G . The graded endomorphism ring of the unit is End KK G ( ) ∗ ∼ = R( G )[ β ± ] . Definition ([Del14]) . Let
Cell( G ) := Loc( { C( G/H ) | H ≤ G } ) be the localizing subcategory of KK G generated by the G -C*-algebras C( G/H ) ofcomplex functions for all subgroups H ≤ G , with G -action induced by that of G onits cosets G/H . That is,
Cell( G ) is the smallest triangulated subcategory of KK G containing all C( G/H ) and closed under forming arbitrary (countable) coproducts.The algebras in Cell( G ) are called G -cell algebras .2.2. Remark ([DEM14, §3.1]) . The G -equivariant bootstrap category B G is definedsimilarly as the localizing subcategory of KK G generated by certain objects. It isshown in [DEM14, Sec. 3.1] that B G consists precisely of those separable G -C*-algebras that are equivariantly KK-equivalent to a G -action on a C*-algebra ofType I. If G is a finite cyclic group, then Cell( G ) = B G . For more general finitegroups, there is only an inclusion Cell( G ) ⊆ B G .2.3. Remark.
By [Del14, Prop. 2.9],
Cell( G ) is a rigidly-compactly generated tensortriangulated category, in a countable sense of the term (as here there are only count-able coproducts to work with). This entails, in particular, that its compact objects(those G -cell algebras A such that KK G ( A, − ) commutes with arbitrary countabledirect sums) coincide with its rigid objects (those A admitting an inverse A ∨ withrespect to the tensor product). It follows that Cell( G ) c , the full subcategory ofrigid-compact objects, is itself a tensor triangulated category. In addition, Cell( G ) c is essentially small, rigid (it admits the self-duality A A ∨ ), idempotent complete (every idempotent endomorphism splits), and it is generated as a thick subcategoryby the same set of objects: Cell( G ) c = Thick( { C( G/H ) | H ≤ G } ) . IVO DELL’AMBROGIO AND RALF MEYER
Recall that a subcategory is thick if it is triangulated and closed under retracts.2.4.
Remark.
For finite groups, the usual functors between the equivariant Kasparovcategories, such as restriction
Res GH , induction Ind GH , and inflation Inf G , preservecell algebras and also compact cell algebras. See [Del14, §2].3. Köhler’s universal coefficient theorem
From now on, we restrict attention to G = Z /p Z for a prime number p . For thesegroups we may use the universal coefficient theorem (“UCT”) due to Köhler [Köh10]in order to compute G -equivariant KK-theory groups by algebraic means, at leastin principle. We recall how this works and refer to [Köh10] as well as [Mey19, §4–5]for all proofs.Let ΣC( G ) → D → → C( G ) be the mapping cone sequence (distinguishedtriangle in Cell( G ) ) for the unit map → C( G ) , that is, the embedding of C asconstant functions on G . Let B := C ⊕ C( G ) ⊕ D be the sum of the three verticesof the triangle. Define the Z / -graded ring K := (cid:0) End
Cell( G ) ( B ) ∗ (cid:1) op . Given any G -C*-algebra A ∈ KK G , its Köhler invariant F ∗ ( A ) is defined as(3.1) F ∗ ( A ) := KK G ∗ ( B, A ) = KK G ∗ ( , A ) | {z } F ∗ ( A ) ⊕ KK G ∗ (C( G ) , A ) | {z } F ∗ ( A ) ⊕ KK G ∗ ( D, A ) , | {z } F ∗ ( A ) considered with its evident structure of Z / -graded countable left K -module. Theassignment A F ∗ ( A ) extends to a functor F ∗ = KK G ∗ ( B, − ) from KK G to Z / -graded countable left K -modules.Köhler’s UCT says that there is a natural short exact sequence(3.2) / / Ext K ( F ∗ +1 A, F ∗ A ′ ) / / KK G ∗ ( A, A ′ ) F ∗ / / Hom K ( F ∗ A, F ∗ A ′ ) / / of Z / -graded abelian groups for all A ∈ Cell( G ) and A ′ ∈ KK G . Here Hom K ( − , − ) denotes the graded Hom, which in degree zero is the group of degree-preserving K -linear maps and in degree one the group of degree-reversing maps; and similarly forthe extension group Ext K . The map denoted F ∗ is part of the functor F ∗ .3.3. Remark.
A standard consequence of the UCT is that every isomorphism be-tween the Köhler invariants F ∗ ( A ) and F ∗ ( B ) of two G -cell algebras A and B liftsto an isomorphism A ∼ = B in Cell( G ) . Köhler also characterizes the essential im-age of the functor F ∗ among the K -modules. Namely, it consists precisely of thosecountable Z / -graded K -modules which are exact in the sense that two sequencesbuilt out of M and the internal structure of K are exact. For M of the form F ∗ ( A ) ,the two sequences are those arising by applying KK G ( − , A ) to the triangle link-ing , C( G ) , D and its dual triangle under Baaj–Skandalis duality. It follows that A F ∗ ( A ) induces a bijection between isomorphism classes of G -cell algebras andisomorphism classes of exact countable Z / -graded left K -modules.3.4. Remark.
We have displayed in (3.1) the canonical decomposition of F ∗ ( A ) intothree parts. Note that any abstract K -module similarly decomposes as M = M ⊕ M ⊕ M , HE SPECTRUM OF EQUIVARIANT KK-THEORY 5 where the three direct summands are the images of the idempotent elements of K corresponding to the units of the endomorphism rings End( ) , End(C( G )) and End( D ) , respectively. The three parts are precisely the terms appearing in each ofthe two sequences defining the exactness of M as in Remark 3.3.3.5. Remark.
Both endomorphism rings
End( ) = R( G ) and End(C( G )) are canon-ically isomorphic to Z [ x ] / ( x p − . So M and M are Z [ x ] / ( x p − -modules for any K -module M . In view of later sections, let us detail what happens if we invert p . Thepolynomial x p − has two irreducible factors, x − and Φ p ( x ) := 1 + x + . . . + x p − .After inverting p , they give rise to the product decomposition Z [ x, p − ] / ( x p − ∼ = Z [ x, p − ] / ( x − | {z } ∼ = Z [ p − ] × Z [ x, p − ] / (Φ p ) | {z } ∼ = Z [ ϑ,p − ] (see Remark 6.4 below for more on this). Here we write ϑ for the ‘abstract’ primitive p -th root of unity, that is, the image of x in the right-hand side quotient. Nowsuppose that a K -module M is uniquely p -divisible , that is, it is acted on invertiblyby p . In other words, suppose it is a K [ p − ] -module. Then each of the two parts M and M decomposes into a sum of a Z [ p − ] -module and a Z [ ϑ, p − ] -module, viathe above actions and decomposition.4. A simplification of the bootstrap category
Let G = Z /p Z . We will use the UCT for G -cell algebras recalled in Section 3,together with closely related results from [Mey19], in order to study the followinglocalization of the G -equivariant bootstrap category.4.1. Notation.
Let
Loc(C( G )) ⊂ Cell( G ) be the localizing subcategory generatedby the G -C*-algebra C( G ) . Let Q G : Cell( G ) −→ Cell( G ) / Loc(C( G )) =: Q ( G ) be the Verdier quotient of the bootstrap category Cell( G ) by the localizing subcat-egory generated by the G -C*-algebra C( G ) .The quotient Q ( G ) is related to Cell( G ) by a nice localization sequence of tensortriangulated categories: Loc(C( G )) incl ⊥ / / Cell( G ) Q G ⊥ / / S j j Q ( G ) R i i The following omnibus lemma makes this more precise.4.2.
Lemma.
The quotient functor Q G admits a coproduct-preserving fully faithfulright adjoint R . The inclusion of its full kernel Ker( Q G ) = Loc(C( G )) also admitsa coproduct-preserving right adjoint S . (1) The full essential image of R , which we denote by N := Im( R ) = Im( R ◦ Q G ) ,is equal to the right orthogonal of Loc(C( G )) : N = Loc(C( G )) ⊥ = C( G ) ⊥ := { A ∈ Cell( G ) | KK G ∗ (C( G ) , A ) = 0 } . (2) The kernel
Ker( Q G ) = Loc(C( G )) equals the left orthogonal of N : Loc(C( G )) = ⊥ N = { A ∈ Cell( G ) | KK G ∗ ( A, B ) = 0 for all B ∈ N } . IVO DELL’AMBROGIO AND RALF MEYER (3)
There is an essentially unique distinguished triangle in
Cell( G ) of the form (4.3) Σ N / / P / / / / N, where P := incl ◦ S ( ) ∈ Loc(C( G )) and N := R ◦ Q G ( ) ∈ Loc(C( G )) ⊥ . (4) There are isomorphisms of endofunctors of
Cell( G )incl ◦ S ∼ = P ⊗ − and R ◦ Q G ∼ = N ⊗ − . Proof.
This all follows from standard results on rigidly-compactly generated cate-gories, modulo the fact that we are in the countably generated setting. All claimscan be deduced from [Del10, §2] for the case α = ℵ .More precisely, we begin by noticing that Thick(C( G )) is a tensor ideal in Cell( G ) c and Loc(C( G )) is a tensor ideal in Cell( G ) . Both follow from the iso-morphisms C( G ) ⊗ C( G/H ) ∼ = L G/H C( G ) for all H ≤ G and [Del14, Lem. 2.5].Then we apply [Del10, Thm. 2.28] to the rigidly-compactly generated category T := Cell( G ) (see Remark 2.3) and its tensor ideal of compact objects J :=Thick(C( G )) to conclude that L := Loc(C( G )) and L ⊥ = N form a pair of lo-calizing tensor ideals of Cell( G ) which are complementary as in [Del10, Def. 2.7].All the remaining claims then follow from [Del10, Prop. 2.26]. (cid:3) By construction, Q ( G ) enjoys the same structural properties as Cell( G ) but withan extra simplification: it is generated by its tensor unit.4.4. Corollary.
The category Q ( G ) is a tensor triangulated category that is rigidly-compactly generated ( in the countable sense ) . It is generated by its tensor unit, thatis, Q ( G ) = Loc( ) , and also Q ( G ) c = Thick( ) . The quotient functor Q G is anexact tensor functor and preserves coproducts. Moreover, Q G : Cell( G ) → Q ( G ) preserves compact objects; the image Q G (Cell( G ) c ) ⊆ Q ( G ) c identifies with theVerdier quotient Cell( G ) c / Thick(C( G )) and embeds fully faithfully in Q ( G ) c . Itsimage is dense , that is, any object of Q ( G ) c is a retract of one in Q G (Cell( G ) c ) .Proof. Again, these are standard consequences. Clearly, being the quotient of
Cell( G ) by a localizing tensor ideal, the category Q ( G ) inherits from Cell( G ) atensor triangulated structure and countable coproducts, and these are preserved bythe quotient functor Q G .The functor Q ( G ) has a coproduct-preserving right adjoint. A short computationusing this shows that it preserves compact objects. One verifies similarly that Q ( G ) is generated by the image under Q G of the compact generators of Cell( G ) . Since Q G (C( G )) ∼ = 0 by construction, the compact-rigid object Q ( G ) = Q G ( Cell( G ) ) suf-fices. The remaining claims, which are harder, are all part of Neeman’s localizationtheorem, in its countable form (see [Del10, Thm. 2.10]). (cid:3) Let us explicitly record another, immediate consequence of Lemma 4.2:4.5.
Corollary.
The right adjoint R of Q G restricts to a canonical equivalence Q ( G ) ∼ → Im( R ) = N of triangulated categories, which further restricts to an iso-morphism End Q ( G ) ( ) ∗ ∼ → End
Cell( G ) ( N ) ∗ of graded endomorphism rings. (cid:3) In the remainder of this section, we move beyond abstract generalities. Thenext two propositions classify the objects of Q ( G ) algebraically and compute thegraded endomorphism ring of its unit. We will actually work within N (thanksto Corollary 4.5) and use Köhler’s UCT as well as a refinement due to the second HE SPECTRUM OF EQUIVARIANT KK-THEORY 7 author (see [Mey19]). Köhler’s invariant F ∗ : KK G → K - Mod Z / ∞ is recalled inSection 3. Its target is the category of Z / -graded countable left K -modules. Recallthat any K -module decomposes canonically as M = M ⊕ M ⊕ M .4.6. Lemma.
Suppose that M ∈ K - Mod Z / ∞ is exact as in Remark and satisfies M = 0 . Then M is uniquely p -divisible and is entirely determined by M with itsnatural structure of graded Z [ ϑ, p − ] -module. Similarly, every K -linear map of suchmodules is uniquely determined by its restriction to their M -parts. This gives anequivalence { M exact and M = 0 } ∼ −→ Z [ ϑ, p − ] - Mod Z / ∞ , M M , between the full subcategory of exact K -modules M with M = 0 and the categoryof Z / -graded countable Z [ ϑ, p − ] -modules.Proof. Let M be an exact K -module whose M -part vanishes. In particular, andtrivially, the abelian group M is p -divisible. Then [Mey19, Thm. 7.2] applies to M and says that the whole group M is uniquely p -divisible and that its K -modulestructure is of the form described in [Mey19, Ex. 7.1]. In particular, M = X ⊕ Y, M = X ⊕ Z, M = Y ⊕ Σ Z, where X is some Z / -graded Z [ p − ] -module and Y, Z are some Z / -graded Z [ ϑ, p − ] -modules. The decompositions of M and M arise from the action of K [ p − ] as inRemark 3.5. Since M = 0 in our case, it follows that X = Z = 0 . So(4.7) M = Y, M = 0 , M = Y. The construction in [Mey19, Ex. 7.1] shows that the whole K [ p − ] -action on sucha K -module M is determined by the Z [ ϑ, p − ] -action on Y = M . Similarly, every Z [ x ] / ( x p − -linear map Y → Y ′ admits a unique K -linear extension M → M ′ tothe corresponding K -modules. This proves the lemma. (cid:3) Proposition.
The functor F ∗ restricts between N and the full subcategory of K - Mod Z / ∞ of those exact M such that M = 0 as in Lemma . In particular, the G -equivariant K-theory functor A F ∗ ( A ) := KK G ∗ ( , A ) induces a bijection between the isomorphism classes of objects in N and those in Z [ ϑ, p − ] - Mod Z / ∞ .Proof. We know from Köhler’s classification (Remark 3.3) that F ∗ induces a bijec-tion between the isomorphism classes of G -cell algebras and those of exact countablegraded modules. If A ∈ N = Loc(C( G )) ⊥ , then(4.9) F ∗ ( A ) := KK G ∗ (C( G ) , A ) = 0 . So F ∗ ( A ) is a K -module as in part (1). Hence the functor F ∗ restricts as claimed.Moreover, any (graded countable) Z [ ϑ, p − ] -module gives rise to a unique exact(countable Z / -graded) K -module of the form (4.7). Therefore, Köhler’s classifica-tion combines with Lemma 4.6 to yield the claimed classification for N . (cid:3) We conclude the section with this central computation:
IVO DELL’AMBROGIO AND RALF MEYER
Proposition.
As above, let G = Z /p Z for a prime number p . The gradedendomorphism ring of the tensor unit ∈ Q ( G ) is given by End Q ( G ) ( ) ∗ ∼ = Z [ ϑ, p − , β ± ] , where ϑ is a primitive p -th root of unity ( in the sense of Remark and set in degreezero ) and where β is the invertible Bott element ( in degree two ) . More precisely, therestriction End
Cell( G ) ( ) ∗ → End Q ( G ) ( ) ∗ of the localization functor Q G identifieswith the canonical grading-preserving ring map which inverts p and kills the idealgenerated by Φ p ( x ) = 1 + x + . . . + x p − : Z [ x ] / ( x p − β ± ] −→ Z [ x, p − ] / (Φ p ) [ β ± ] = Z [ ϑ, p − , β ± ] . Proof.
Recall the algebra P ∈ Loc(C( G )) of Lemma 4.2 (3). For any A belongingto N = Loc(C( G )) ⊥ we must have(4.11) KK G ∗ ( P, A ) = 0 and F ∗ ( A ) := KK G ∗ (C( G ) , A ) = 0 . The first vanishing group implies that the map → N of the distinghished trian-gle (4.3) induces a natural isomorphism(4.12) KK G ∗ ( N, A ) ∼ → KK G ∗ ( , A ) = F ∗ ( A ) for all such A ∈ N . Specializing this to the case A = N , we see that KK G ∗ ( N, N ) ∼ = F ∗ ( N ) . We claim that F ∗ ( N ) , and therefore KK G ∗ ( N, N ) , is a free Z [ ϑ, p − ] -module ofrank one and concentrated in degree zero. By Corollary 4.5 and after unwinding Z / -gradings by Bott periodicity, this would prove the proposition.By Proposition 4.8, there is an object R ∈ N such that F ∗ ( R ) (and thus F ∗ ( R ) )is a free Z [ ϑ, p − ] -module of rank one concentrated in degree zero. To prove theclaim, it now suffices to prove that N and R are isomorphic in N .Consider the UCT exact sequence (3.2) for an arbitrary object A ∈ N :(4.13) / / Ext K ( F ∗ +1 R, F ∗ A ) / / KK G ∗ ( R, A ) F ∗ / / Hom K ( F ∗ R, F ∗ A ) / / . By construction, F ∗ ( R ) is a free Z [ ϑ, p − ] -module of rank one in degree zero. Henceby Lemma 4.6, the Hom-term in (4.13) reduces to Hom Z [ ϑ,p − ] (cid:0) F ∗ ( R ) , F ∗ ( A ) (cid:1) ∼ = F ∗ ( A ) . Now, suppose that the Ext-term in (4.13) vanishes. We would obtain an isomor-phism F ∗ : KK G ∗ ( R, A ) ∼ → F ∗ ( A ) natural in A ∈ N . By combining it with the isomorphism (4.12) and applying theYoneda Lemma for the category N , this would show that R ∼ = N as wished. Thusit only remains to show that the Ext-term of (4.13) vanishes.Consider an arbitrary extension(4.14) / / F ∗ ( A ) / / M / / F ∗ +1 ( R ) / / of graded K -modules. By applying to (4.14) the idempotent element of K corre-sponding to the identity map of C( G ) we obtain an exact sequence / / F ∗ ( A ) / / M / / F ∗ +1 ( R ) / / HE SPECTRUM OF EQUIVARIANT KK-THEORY 9 where both outer terms vanish by hypothesis. Then M = 0 . In particular, M isuniquely p -divisible. Since M is exact as an extension of two exact K -modules, wemay apply [Mey19, Thm. 7.2] to it. Thus M also has the special form of Lemma 4.6and is uniquely determined by its M -part, viewed as a Z [ ϑ, p − ] -module. Nowconsider the extension of graded Z [ ϑ, p − ] -modules / / F ∗ ( A ) / / M / / F ∗ +1 ( R ) / / obtained by hitting (4.14) with the idempotent of K corresponding to the identityof . This extension must split because F ∗ +1 ( R ) is a free module. Moreover,by Lemma 4.6 once again, any Z [ ϑ, p − ] -linear section of M → F ∗ +1 ( R ) extendsto a K -linear section of M → F ∗ +1 ( R ) . Thus the original extension (4.14) of K -modules splits as well. As the latter extension was arbitrary, this implies that Ext K ( F ∗ +1 R, F ∗ A ) = 0 as required. (cid:3) The Balmer spectrum
We very briefly recall some basic notions of tensor triangular geometry, referringto [Bal10b] and the original references therein for more details.Let K be an essentially small tensor triangulated category, with tensor ⊗ andunit object . Its spectrum Spc K is the set of its prime ⊗ -ideals P , that is, thoseproper, full and thick subcategories P ( K which are prime tensor ideals for thetensor product: A ⊗ B ∈ P ⇔ A ∈ P or B ∈ P , for any objects A, B ∈ K . Thespectrum is endowed with the ‘Zariski’ topology, which has the family of subsets(5.1) supp( A ) := {P ∈ Spc
K |
A / ∈ P} ( A ∈ K ) as a basis of closed subsets.The ring End K ( ) is commutative. So we can consider the usual Zariski spectrumof its prime ideals, Spec End K ( ) . The assignment P 7→ ρ K ( P ) := { f ∈ End K ( ) | cone( f )
6∈ P} defines a continuous map between the two spectra:(5.2) ρ K : Spc K −→
Spec End K ( ) (see [Bal10a, Thm. 5.3]). In general, this comparison map is neither injective norsurjective. Surjectivity is more common and often much easier to prove.5.3. Remark.
The spectrum is functorial: Any exact tensor functor F : K → L defines a continuous map
Spc F : Spc L →
Spc K sending a prime P of L to (Spc F )( P ) := F − P . Moreover, Spc( F ◦ F ) = Spc F ◦ Spc F and Spc Id = Id .5.4.
Remark.
The comparison map (5.2) is natural (see [Bal10a, Thm. 5.3 (c)]); thatis, if F : K → L is an exact tensor functor, then the square
Spc L Spc F / / ρ K (cid:15) (cid:15) Spc K ρ L (cid:15) (cid:15) Spec End L ( ) Spec F / / Spec End K ( ) commutes; here the bottom arrow is the continuous map p F − p induced betweenZariski spectra by the ring homomorphism F : End K ( ) → End L ( ) . Computation of the spectrum
Finally, we are ready to prove Theorem 1.1.Let G = Z /p Z as before. Let Cell( G ) ⊂ KK G be the tensor triangulated cat-egory of G -cell algebras (Definition 2.1). Let Cell( G ) c ⊂ Cell( G ) be the tensortriangulated subcategory of its compact objects (Remark 2.3). Write ρ G := ρ Cell( G ) c : Spc Cell( G ) c → Spec R( G ) for the canonical map (5.2) comparing the triangular spectrum of Cell( G ) c to theZariski spectrum of the representation ring R( G ) = End Cell( G ) c ( ) = Z [ x ] / ( x p − .Our goal is to show that ρ G is a homeomorphism. To this end, we will adaptthe ‘divide and conquer’ strategy of [BS17] to the two exact tensor functors(6.1) Cell(1) Cell( G ) Res G o o Q G / / Q ( G ) given by restriction to the trivial group, Res G , and the Verdier quotient functor Q G studied in Section 4.6.2. Lemma.
The graded commutative ring
End Q ( G ) ( ) ∗ ∼ = Z [ ϑ, p − , β ± ] com-puted in Proposition is Noetherian and regular.Proof. The degree zero subring Z [ ϑ, p − ] is Noetherian because it is finitely gen-erated and commutative. It is regular because it is a localization of the classicalDedekind domain Z [ ϑ ] ∼ = Z [ ζ p ] ⊂ C , where ζ p is a complex primitive p -th root ofunit. It follows that our graded ring is also Noetherian and regular. (cid:3) Remark. If R ∗ is a Z -graded commutative ring, we can consider the Zariskispectrum Spec h R ∗ of its homogeneous prime ideals. The inclusion of the zerodegree subring R ⊂ R ∗ induces a continuous restriction map Spec h R ∗ → Spec R .If the graded ring R ∗ is 2-periodic and concentrated in even degrees – that is, R ∗ = R [ β ± ] with | β | = 2 – the latter map is easily seen to be a homeomorphism Spec h R ∗ ∼ → Spec R . Of course, this applies to our endomorphism rings.6.4. Remark.
Consider the Zariski spectrum of R( G ) = Z [ x ] / ( x p − . It has twoirreducible components, namely, the images of the two embeddings Spec Z ψ / / Spec Z [ x ] / ( x p −
1) Spec Z [ x ] / (Φ p ) ϕ o o induced by the two ring quotients Z Z [ x ] / ( x p − o o / / Z [ x ] / (Φ p ) given by killing x − and Φ p = 1 + x + . . . + x p − , respectively; these are the twoirreducible factors of x p − in Z [ x ] . Thus the maps ψ and ϕ are jointly surjective.The intersection of their images can be shown to consist exactly of one point lyingabove p , namely, the maximal ideal ψ (( p )) = ( p, x −
1) = ( p, Φ p ) = ϕ (( p )) . Byinverting p in Z [ x ] / (Φ p ) we get rid of precisely the preimage under ϕ of this commonpoint, thus eliminating the redundancy. In particular, we obtain a decomposition Spec R( G ) ∼ = Spec Z ⊔ Spec Z [ x, p − ] / (Φ p ) as sets. As before, we write Z [ ϑ, p − ] := Z [ x, p − ] / (Φ p ) . (This is all well-known;see, for instance, [BD08] and the references therein for explanations and context.) HE SPECTRUM OF EQUIVARIANT KK-THEORY 11
Now we know everything we need about the geometry of Q G . As for the restric-tion functor Res G : Cell( G ) c → Cell(1) c , we need the following:6.5. Lemma.
The equality supp C( G ) = Im(Spc Res G ) , between the support of theobject C( G ) and the image of the map Spc(Res G ) , holds in Spc Cell( G ) c .Proof. By [BDS15, Thm. 1.2] (see also [BD20, §2.4]), the restriction functor
Res G is a finite separable extension . More precisely, this is proved in [BDS15] for therestriction functor KK G → KK(1) between the whole
Kasparov categories. But theresult also holds for (compact) cell algebras. Let us briefly recall this. The two-sidedadjunction between
Ind G and Res G restricts to a two-sided adjunction between Cell( G ) and Cell(1) (Remark 2.4), and provides us with a separable commutativemonoid A G in Cell( G ) whose underlying object is Ind G ( ) = C( G ) . Exactly thesame proofs as in [BDS15] yield a canonical equivalence of tensor triangulatedcategories Cell(1) ≃ A G - Mod
Cell( G ) between the bootstrap category Cell(1) and the Eilenberg–Moore category of mod-ules in
Cell( G ) over the monoid A G . This equivalence identifies Res G with the‘free module’ functor F := A G ⊗ − : Cell( G ) → A G - Mod
Cell( G ) . It may also berestricted to compact objects: Cell(1) c ≃ ( A G - Mod
Cell( G ) ) c = A G - Mod
Cell( G ) c .As with any separable monoid, [Bal16, Thm. 1.5] shows the equality Im(Spc F ) =supp A G . Hence Im(Spc Res G ) = supp C( G ) by the above identifications. (cid:3) Proof of Theorem . We already know from [Del10, Thm. 1.4] that the map ρ G admits a continuous section – even for any finite group G . To show that it is ahomeomorphism, it will therefore suffice to prove its injectivity.We claim that the two functors (6.1) induce the following commutative diagram:(6.6) Spc Cell(1) cρ (cid:15) (cid:15) Spc Res G / / Spc Cell( G ) cρ G (cid:15) (cid:15) Spc Q ( G ) c Spc Q G o o ρ Q ( G ) c =: ρ Q (cid:15) (cid:15) Spec Z / / ψ / / Spec Z [ x ] / ( x p −
1) Spec Z [ ϑ, p − ] ϕ o o Indeed, the top row is obtained by restricting the two functors to rigid-compactobjects and applying the functoriality of
Spc( − ) (Remark 5.3). The three ver-tical maps are all instances of the canonical comparison (5.2). The two squarescommute because the latter is natural (see Remark 5.4). The bottom row is asin Remark 6.4: Indeed, the right arrow is given by inverting p and killing Φ p (byProposition 4.10) and the left arrow by mapping x (because it corresponds tothe rank homomorphism R( G ) → R(1) = Z ).By Remark 6.4, the bottom row of (6.6) is a disjoint-union decomposition of theset Spec Z [ x ] / ( x p − .We claim that the top row of (6.6) provides a similar decomposition, that is, itconsists of two injective and jointly surjective maps with disjoint images, so that Spc Cell( G ) c = Im(Spc Res G ) ⊔ Im(Spc Q G ) . (6.7)Indeed, we obviously have Spc Cell( G ) c = {P | C( G ) / ∈ P} ⊔ {P | C( G ) ∈ P} . Moreover, by Corollary 4.4 the restriction of Q G on compact objects factors as aVerdier quotient with kernel Thick(C( G )) followed by a full dense embedding: Cell( G ) c −→ Cell( G ) c / Thick(C( G )) ֒ −→ Q ( G ) c . By basic tensor-triangular results [Bal05, Prop. 3.11 and Prop. 3.13], the inducedmap
Spc Q G is therefore injective with image Im(Spc Q G ) = {P | Thick(C( G )) ⊆ P} = {P | C( G ) ∈ P} . Let us consider the other half of the decomposition. Recall the inflation functor
Inf G : Cell(1) c → Cell( G ) c (Remark 2.4), which endows each A ∈ Cell(1) c with thetrivial G -action. It is an exact tensor functor such that Res G ◦ Inf G = Id . Since Spc is a contravariant functor (Remark 5.3), we deduce that
Spc(Inf G ) ◦ Spc(Res G ) = Spc(Res G ◦ Inf G ) = Spc(Id Cell(1) c ) = Id . Thus
Spc(Res G ) is injective. By Lemma 6.5 and the definition of support (5.1), itsimage is Im(Spc Res G ) = supp C( G ) = {P | C( G ) / ∈ P} . This concludes the proof of the decomposition (6.7).The above decompositions of the triangular and Zariski spectra and the commu-tative diagram (6.6) imply that the middle vertical map ρ G is bijective if and onlyif both ρ and ρ Q are. We know from [Del10, Thm. 1.2] that ρ is bijective. Asfor ρ Q ( G ) , we will appeal to [DS16].The comparison map has a graded version ρ ∗Q : Spc Q ( G ) c −→ Spec h End( ) ∗ , whose target is the homogeneous spectrum of the graded endomorphism ring of Q ( G ) (see Remark 6.3); this follows from [Bal10a, Thm. 5.3] for the choice u = Σ( ) of grading object. By Corollary 4.4, Q ( G ) c is generated by its tensor unit asa thick subcategory. By Lemma 6.2, the graded ring End Q ( G ) c ( ) ∗ is Noether-ian and regular. Therefore, [DS16, Thm. 1.1] shows that the graded comparisonmap ρ ∗Q is bijective. The map ρ Q is bijective as well because the isomorphism Spec h End( ) ∗ ∼ = Spec End( ) of Remark 6.3 identifies ρ ∗ G with ρ Q (see [Bal10a,Cor. 5.6.(b)]). This completes the proof. (cid:3) Remark.
As already mentioned, the present proof of Theorem 1.1 is looselyinspired by the analogous determination (as a set) of the spectrum of
SH( G ) c , thestable homotopy category of compact G -spectra; more precisely, by the proof of[BS17, Thm. 4.9]. The latter argument works for any finite group G by induc-tion on its order, and this induction could be adapted to yield a homeomorphism ρ : Spc Cell( G ) c ∼ → Spec R( G ) for general G , provided we knew that a certain ringis regular for all G (in order to invoke [DS16]). The ring in question is the gradedendomorphism ring of the tensor unit in the tensor triangulated category Cell( G ) / Loc( { C( G/H ) | H (cid:12) G } ) which we currently do not know how to compute. In other words, we would needto find a general replacement for our use of Köhler’s UCT in Section 4.6.9. Remark.
Besides the proofs’ analogies, the result in [BS17] is actually quitedifferent from ours. Most strikingly, the comparison map ρ SH( G ) c is very far frombeing injective. In particular, [DS16] cannot be applied to the case of G -spectra; HE SPECTRUM OF EQUIVARIANT KK-THEORY 13 in this case, the role of [DS16] in the proof’s structure is played instead by the factthat the composite of inflation followed by localization SH Inf G / / SH( G ) / / SH( G ) / Loc( { Σ ∞ + G/H | H (cid:12) G } ) is an equivalence of tensor triangulated categories (see [BS17, §2 (H)]). The analo-gous result for KK-theory is false. By Proposition 4.10, it fails for G ∼ = Z /p Z . References [Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories.
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