A more general method to classify up to equivariant KK-equivalence II: Computing obstruction classes
AA MORE GENERAL METHOD TO CLASSIFY UP TOEQUIVARIANT KK-EQUIVALENCE II: COMPUTINGOBSTRUCTION CLASSES
RALF MEYER
Abstract.
We describe Universal Coefficient Theorems for the equivariantKasparov theory for C ∗ -algebras with an action of the group of integers or overa unique path space, using KK-valued invariants. We compare the resultingclassification up to equivariant KK-equivalence with the recent classificationtheorem involving a K-theoretic invariant together with an obstruction class ina certain Ext -group and with the classification by filtrated K-theory. This isbased on a general theorem that computes these obstruction classes. Introduction
Objects in a triangulated category, such as equivariant KK-categories, may beclassified up to isomorphism using a primary homological invariant and a secondary“obstruction class” provided they have a projective resolution of length 2 in a suitablesense. This method was applied in [4] to objects in the circle-equivariant KK-category KK T , the equivariant KK-category KK X for C ∗ -algebras over a finite, unique pathspace X , and graph C ∗ -algebras with finite ideal lattice. Here we compute theobstruction classes that occur in this classification. This makes the classification forobjects of KK T and KK X more explicit.The result suggests, in fact, a different invariant for objects in these categoriesthat is fine enough to admit a Universal Coefficient Theorem. The price to pay isthat the invariant uses bivariant K-theory instead of ordinary K-theory. We explainthis for the equivariant bootstrap class in the category KK Z ; the latter is equivalentto KK T by Baaj–Skandalis duality.An object of KK Z is a C ∗ -algebra A with a Z -action, which is generated by asingle automorphism α . The most obvious homological invariant on KK Z maps thisto the K-theory K ∗ ( A ) with the module structure over the group ring Z [ x, x − ] of Z that is induced by α . Let A be the category of countable, Z / Z [ x, x − ]. This is a stable Abelian category. The K-theory described above definesa stable homological functor F Z K : KK Z → A . The category A has cohomologicaldimension 2, that is, any object has a projective resolution of length 2. Let A δ bethe additive category of pairs ( A, δ ) with A ∈∈ A and δ ∈ Ext A (Σ A, A ); morphismsfrom (
A, δ ) to ( A , δ ) in A δ are morphisms f from A to A in A with δ f = f δ . Itis shown in [4] that isomorphism classes of objects in the bootstrap class in KK Z arein bijection with isomorphism classes of objects in A δ . In particular, any object A of A lifts to an object in the bootstrap class in KK Z . The lifting is, however, notunique: different liftings of A are in bijection with Ext A (Σ A, A ); here two liftings B , B are identified if there is an isomorphism B ∼ = B that induces the identitymap on A = F Z K ( B ) = F Z K ( B ).In this article, we compute the obstruction class of an object of KK Z explicitly. Thisquestion remained open in [4]. Let ( A, α ) be a C ∗ -algebra with an automorphism α Mathematics Subject Classification.
Primary 19K35; Secondary 18E30, 46L35.Supported by the DFG grant “Classification of non-simple purely infinite C*-algebras.” a r X i v : . [ m a t h . K T ] A p r RALF MEYER as above. The exact sequence in the Universal Coefficient Theorem splits, so thatKK ( A, A ) ∼ = Hom (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) ⊕ Ext (cid:0) K ∗ +1 ( A ) , K ∗ ( A ) (cid:1) . While the splitting above is not natural, it is canonical enough to associate welldefined elements α ∈ Hom (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) and α ∈ Ext (cid:0) K ∗ +1 ( A ) , K ∗ ( A ) (cid:1) to α .Here α is the action on K ∗ ( A ) induced by α , which is part of the Z [ x, x − ]-modulestructure on F Z K ( A ). We show how to compute the obstruction class from α .Namely, it is µ ∗ ( α ) for a canonical map µ ∗ : Ext (cid:0) K ∗ +1 ( A ) , K ∗ ( A ) (cid:1) → Ext Z [ x,x − ] (cid:0) K ∗ +1 ( A ) , K ∗ ( A ) (cid:1) . Thus an object (
A, α ) of KK Z is determined uniquely up to isomorphism by A as anobject of KK together with the class [ α ] of α in KK ( A, A ). We treat the pair ( A, [ α ])as an object of a certain exact category KK [ Z ] with a suspension automorphism. Theforgetful functor F Z : KK Z → KK [ Z ] is a stable homological functor. The fact thatit gives a complete invariant for objects of KK Z follows from a Universal CoefficientTheorem computing KK Z ∗ ( A, B ). This Universal Coefficient Theorem is alreadyproven in [11], and it is shown there to be equivalent to a Pimsner–Voiculescu likeexact sequence for KK Z ∗ ( A, B ).The Universal Coefficient Theorem for the invariant F Z : KK Z → KK [ Z ] is basedon projective resolutions of length 1 for the stable homological ideal ker F Z . We usethese resolutions to compute the obstruction class related to the homological invariant F Z K ( A ) := K ∗ ( A ). Namely, let 0 → B → B → A → F Z in KK Z . This remains exact with respect to the largerstable homological ideal ker F Z K . The objects B and B are no longer projectivefor ker F Z K , but they have projective resolutions of length 1. Thus KK Z ( B , B ) iscomputed by a Universal Coefficient Theorem, which splits non-naturally. So thearrow B → B gives a class in Ext Z [ x,x − ] (cid:0) K ∗ +1 ( B ) , K ∗ ( B ) (cid:1) . It must determinethe obstruction class of A because A is KK Z -equivalent to the cone of the map B → B . The result is the formula for the obstruction class of A asserted above.This computation of obstruction classes is carried out in Section 3 in the samegenerality as in [4], using a triangulated category with a stable homological idealwith enough projectives.The Universal Coefficient Theorem for KK Z is developed in Section 2.1. This isfollowed by an analogous treatment for the category of C ∗ -algebras over a uniquepath space X in Section 2.2. In Section 4, we compare the classification theoremsthat our two invariants give for objects in the bootstrap classes in KK Z and KK X .In Section 5, we study KK X in the special case where X is totally ordered. Thenfiltrated K-theory is a third complete invariant for the same objects (see [12]), andwe compare this invariant with the new ones. This is a rather long homologicalcomputation. In Section 5.2, the result of Section 5 is formulated more nicely if X has only two points, so that C ∗ -algebras over X are extensions of C ∗ -algebras.2. Universal Coefficient Theorems with KK-valued functors
This section develops Universal Coefficient Theorems for the equivariant Kasparovcategories KK Z and KK X for a unique path space X , which are based on forgetfulfunctors to KK . We shall use the general framework developed in [11] for doingrelative homological algebra in triangulated categories. The starting point for thisis a stable homological ideal I in a triangulated category T . Being stable andhomological means that there is a stable homological functor F : T → A to somestable Abelian category A with I ( A, B ) = ker F ( A, B ) := { g ∈ T ( A, B ) : F ( g ) = 0 } . OMPUTING OBSTRUCTION CLASSES 3
Here stability of A and F means that A is equipped with a suspension automorphismand that F ◦ Σ T = Σ A ◦ F .A stable homological ideal I allows to carry over various notions from homologicalalgebra to T . In particular, there are I -exact chain complexes, I -projective objectsand I -projective resolutions in T , which then allow to define I -derived functors. Weshall only be interested in cases where there are enough I -projective objects. Thethick subcategory h P I i generated by the I -projective objects is an analogue of the“bootstrap class” in Kasparov theory. If A ∈∈ h P I i has an I -projective resolutionof length 1, then the graded group T ∗ ( A, B ) for B ∈∈ T may be computed by aUniversal Coefficient Theorem (UCT). The Hom and Ext groups in this UCT arethose in a certain Abelian category, namely, the target category of the universal I -exact stable homological functor F u . The functor F u strongly classifies objectsof h P I i with an I -projective resolution of length 1. That is, any isomorphism F u ( A ) ∼ = F u ( B ) between such objects is induced by an isomorphism A ∼ = B in T .In many cases of interest, the universal functor F u is quite explicit. The followingtheorem allows to recognise it: Theorem 2.1 ([11, Theorem 57]) . Let T be a triangulated category, let I ⊆ T be a stable homological ideal, and let F : T → A be an I -exact stable homologicalfunctor into a stable Abelian category A . Let PA be the class of projective objectsin A . Suppose that idempotent morphisms in T split. The functor F is the uni-versal I -exact stable homological functor to a stable Abelian category and there areenough I -projective objects in T if and only if (1) A has enough projective objects; (2) the adjoint functor F ‘ of F is defined on PA ; (3) F ◦ F ‘ ( A ) ∼ = A for all A ∈∈ PA . The ideals to be treated in this section are defined as the kernel on morphismsof a triangulated functor F : T → S to another triangulated category S . Ideals ofthis form are already treated in [11], using a construction of Freyd to embed S into an Abelian category. Here we unify these two cases by allowing the functor F to take values in an exact category (see [5, 7]). An exact category is an additivecategory with a chosen class of “admissible” extensions. A typical example is a full,additive subcategory of an Abelian category that is closed under extensions. AnAbelian category A is exact where all extensions are admissible. And a triangulatedcategory S is exact where only split extensions are admissible. So exact categoriescontain the two cases treated in [11]. In many examples of classification results usinga UCT, the range of the universal I -exact functor F u old is a certain exact subcategoryof a module category, and only objects in this exact subcategory have projectiveresolutions of length 1. This holds for the UCTs in [2, 3, 8, 12] (see Remark 5.6 forone of these cases). The phenomenon is understood better in [6].Quillen’s Embedding Theorem allows to embed any exact category into an Abelianone in a fully faithful, fully exact way. So all results in the general theory carryover to the case where F takes values in an exact category. In particular, we maymodify the definition of the universal I -exact stable homological functor by allowingexact categories as target. Let F u old : T → A be the universal I -exact functor intoan Abelian category as in Theorem 2.1. Let E ⊆ A be the exact subcategoryof A generated by the range of F u old . Let F u : T → E be F u old viewed as a functorto E . This is the universal I -exact functor in the new sense. Its universal propertyfollows from the one of F u old and Quillen’s Embedding Theorem. If I is defined bya forgetful functor F : T → S to another triangulated category, then the universalhomological functor to an Abelian category typically uses Freyd’s embedding of S into an Abelian category. In contrast, we shall see that in many examples the RALF MEYER universal ker F -exact functor to an exact category gives an exact category that isvery closely related to S . Remark . Assume that all objects in T have an I -projective resolution of finitelength. Then the image of F u old is contained in the subcategory of objects in A witha finite-length projective resolution. This subcategory is exact. And it is alreadygenerated as an exact category by the projective objects in A : this is proved byinduction on the length of a resolution. Hence the image of F u new must be equal tothe subcategory of objects with a finite-length projective resolution. So F u new = F u old if and only if all objects of A have a finite-length projective resolution.2.1. Actions of the group of integers.
We now treat a concrete example, namely,the case T := KK Z . This is a triangulated category. Its objects are pairs ( A, α ) witha separable C ∗ -algebra A and α ∈ Aut( A ); the latter generates an action of Z on A by automorphisms. The arrows in KK Z are the Kasparov groups KK Z ( A, B ), andthe composition is the Kasparov product; we have dropped the automorphisms fromour notation as usual to avoid clutter. The triangulated category structure on KK Z is described in [9, Appendix]. The relative homological algebra in KK Z is alreadystudied in [11]. The main result is the following variant of the Pimsner–Voiculescusequence for KK Z ∗ ( A, B ). Let A and B be C ∗ -algebras with automorphisms α and β ,respectively. Then there is an exact sequence(2.3) KK ( A, B ) KK Z ( A, B ) KK ( A, B )KK ( A, B ) KK Z ( A, B ) KK ( A, B ) forget α ∗ β − ∗ − α ∗ β − ∗ − (see [11, Section 5.1]); here ( α ∗ β − ∗ − x ) = β − ◦ x ◦ α − x for all x ∈ KK ∗ ( A, B ).We may rewrite this as a pair of short exact sequencescoker (cid:0) α ∗ β − ∗ − i ( A, B ) → KK i ( A, B ) (cid:1) (cid:26) KK Z i ( A, B ) (cid:16) ker (cid:0) α ∗ β − ∗ − i ( A, B ) → KK i ( A, B ) (cid:1) for i = 0 ,
1. We shall explain that the long exact sequence (2.3) is an instance of aUniversal Coefficient Theorem as in [11, Theorem 66]. In particular, the cokerneland kernel in it are the Ext and Hom groups in a certain exact category. This isalready proven in [11], but the consequences for classification are not explored there.We shall also treat other examples by the same method later.The forgetful functor R Z : KK Z → KK , ( A, α ) A, is triangulated. So its kernel on morphisms I Z ( A, B ) := { f ∈ KK Z ( A, B ) : R Z ( f ) = 0 in KK ( A, B ) } is a stable homological ideal in KK Z . We now describe the universal I Z -exactfunctor. The main point here is to describe its target category. Let KK [ Z ] bethe additive category of functors Z → KK with natural transformations as arrows.Equivalently, an object of KK [ Z ] is an object A ∈∈ KK with a group homomorphism a : Z → KK ( A, A ) × , n a n , where KK ( A, A ) × denotes the multiplicative groupof invertible elements in the ring KK ( A, A ). And an arrow (
A, a ) → ( B, b ) in KK [ Z ] is an arrow f ∈ KK ( A, B ) that satisfies b n ◦ f = f ◦ a n for all n ∈ Z . Thehomomorphism a is determined by its value at 1 ∈ Z , which may be any element a ∈ KK ( A, A ) × . So we also denote objects of KK [ Z ] as ( A, a ). An element OMPUTING OBSTRUCTION CLASSES 5 f ∈ KK ( A, B ) is an arrow (
A, a ) → ( B, b ) in KK [ Z ] if and only if b ◦ f = f ◦ a or,equivalently, b − ◦ f ◦ a − f = 0. Thusker (cid:0) α ∗ β − ∗ − ( A, B ) → KK ( A, B ) (cid:1) ∼ = Hom KK [ Z ] (cid:0) ( A, [ α ]) , ( B, [ β ]) (cid:1) . Here we have implicitly used the functor F Z : KK Z → KK [ Z ] , ( A, α ) ( A, [ α ]) , where [ α ] ∈ KK ( A, A ) × is the KK -class of the automorphism α . It has the samekernel on morphisms I Z as the forgetful functor R Z .We call a kernel–cokernel pair K (cid:26) E (cid:16) Q in KK [ Z ] admissible if it splits in KK ,that is, E ∼ = K ⊕ Q as objects of KK . This turns KK [ Z ] into an exact category(see [5, Exercise 5.3]). We use this exact structure to define projective objects andprojective resolutions in KK [ Z ].Given A ∈∈ KK and a free Abelian group G = Z [ I ] on a countable set I , wedefine A ⊗ G ∈∈ KK by(2.4) A ⊗ G := M i ∈ I A. In particular, A ⊗ Z := A . This construction is an additive functor in G . Namely,let G and H be free Abelian groups and let f : G → H be a group homomorphism.The Universal Coefficient Theorem implies KK ( C ⊗ G, C ⊗ H ) ∼ = Hom( G, H ). Sowe get a functorial f C ∈ KK ( C ⊗ G, C ⊗ H ). Identifying A ⊗ G = A ⊗ ( C ⊗ G )(with the minimal tensor product of C ∗ -algebras), we get a functorial f A := id A ⊗ f C ∈ KK ( A ⊗ G, A ⊗ H ) . Let A ∈∈ KK . Then A ⊗ Z [ x, x − ] with the invertible element x A = id A ⊗ x ∈ KK (cid:0) A ⊗ Z [ x, x − ] , A ⊗ Z [ x, x − ] (cid:1) induced by multiplication with the invertible element x ∈ Z [ x, x − ] is an object of KK [ Z ]. It behaves like a free module over A becauseHom KK [ Z ] (cid:0) ( A ⊗ Z [ x, x − ] , x A ) , ( B, b ) (cid:1) ∼ = KK ( A, B ) . In particular, ( A ⊗ Z [ x, x − ] , x A ) is projective in the exact category KK [ Z ]. Theinvertible element x A in the KK-endomorphism ring of ( A ⊗ Z [ x, x − ] , x A ) lifts tothe shift automorphism τ ∈ Aut(C ( Z , A )) , ( τ f )( n ) := f ( n − . That is, F Z (C ( Z , A ) , τ ) ∼ = ( A ⊗ Z [ x, x − ] , x A ). And(2.5) KK Z (C ( Z , A ) , B ) ∼ = KK ( A, B ) ∼ = Hom KK [ Z ] (cid:0) ( A ⊗ Z [ x, x − ] , x A ) , B (cid:1) . The first isomorphism here is [9, Equation (20)]. It applies the forgetful functorKK Z (cid:0) (C ( Z , A ) , τ ) , ( B, β ) (cid:1) → KK (C ( Z , A ) , B )and then composes with the inclusion ∗ -homomorphism A → C ( Z , A ) that maps A identically onto the summand at 0 ∈ Z . Equation (2.5) says that the partial adjointof the functor F Z is defined on ( A ⊗ Z [ x, x − ] , x A ) and maps it to (C ( Z , A ) , τ ). Lemma 2.6.
Any object of KK [ Z ] has a free resolution of length .Proof. The trivial representation of the group Z on Z corresponds to Z made amodule over Z [ x, x − ] by letting x ∈ Z [ x, x − ] act by 1. This module has thefollowing free resolution of length 1:(2.7) 0 → Z [ x, x − ] mult( x − −−−−−−−→ Z [ x, x − ] ev −−→ Z → , RALF MEYER where ev : Z [ x, x − ] → Z is the homomorpism of evaluation at 1, so ev ( x n ) = 1 forall n ∈ Z . The extension in (2.7) splits as an extension of Abelian groups because Z is free as an Abelian group.The resolution (2.7) induces a chain complex(2.8) 0 → ( A ⊗ Z [ x, x − ] , [ α ] ⊗ x ) → ( A ⊗ Z [ x, x − ] , [ α ] ⊗ x ) → ( A, [ α ]) → KK [ Z ]. That is, the maps are arrows in KK [ Z ]. Theextension (2.7) splits by a group homomorphism, and the tensor product constructionis additive. Hence (2.8) is exact, that is, it splits in KK . The arrow A ⊗ Z [ x, x − ] = M n ∈ Z A L n ∈ Z [ α n ] −−−−−−−→ M n ∈ Z A = A ⊗ Z [ x, x − ]in KK is an isomorphism( A ⊗ Z [ x, x − ] , id A ⊗ x ) ∼ = ( A ⊗ Z [ x, x − ] , [ α ] ⊗ x ) . Thus (2.8) is isomorphic to a free resolution(2.9) 0 → ( A ⊗ Z [ x, x − ] , ⊗ x ) → ( A ⊗ Z [ x, x − ] , ⊗ x ) → ( A, [ α ]) → , in KK [ Z ], where the boundary map on ( A ⊗ Z [ x, x − ] , ⊗ x ) has changed to [ α ] − ⊗ mult( x ) − id A ⊗ Z [ x,x − ] . (cid:3) Lemma 2.6 implies that the exact category KK [ Z ] has enough projective objectsand that an object of KK [ Z ] is projective if and only if it is a direct summand of afree object. Since the partial adjoint of the functor F Z is defined on all free objects,it is defined also on all projective objects of KK [ Z ], and it is inverse to F Z on thissubcategory. Idempotents in the category KK Z split because it is triangulated andhas countable direct sums. Proposition 2.10.
The functor F Z : KK Z → KK [ Z ] is the universal I Z -exact stablehomological functor from KK Z to an exact category.Proof. To prove this, we first embed KK [ Z ] into an Abelian category by Quillen’sEmbedding Theorem. One way to do this is to embed KK into an Abelian category A by Freyd’s Embedding Theorem and then form A [ Z ]. The Abelian category A [ Z ]has enough projective objects, and all its projective objects already belong to KK [ Z ].So Theorem 2.1 applies and shows that the functor to A [ Z ] is the universal I -exactfunctor to an Abelian category. The subcategory KK [ Z ] ⊆ A [ Z ] is exact and containsthe range of the functor. In fact, we shall show soon that the functor KK Z → KK [ Z ]is surjective on objects (see Theorem 2.12). Taking this for granted, Remark 2.2shows that the functor KK Z → KK [ Z ] is the universal I -exact functor to an exactcategory. (cid:3) Equip C ( Z , A ) with the Z -action generated by the shift automorphism τ . Weuse (2.7) to lift the resolution (2.9) in KK [ Z ] to the following I Z -projective resolutionof length 1 in KK Z :(2.11) 0 → C ( Z ) ⊗ A ϕ −→ C ( Z ) ⊗ A p −→ ( A, [ α ]) → ϕ = [ τ ] ⊗ [ α ] − −
1, and p is the counit of the adjunction (2.5); that is, thefirst isomorphism in (2.5) maps p to the identity element in KK ( A, A ). In otherwords, when we forget the Z -actions, then p restricts to the identity map at the 0thsummand in C ( Z ) ⊗ A = L n ∈ Z A . When we use the resolution (2.11) to computederived functors, we getExt KK [ Z ] , I Z (cid:0) ( A, [ α ]) , ( B, [ β ]) (cid:1) ∼ = coker (cid:0) ( α ∗ ) − β ∗ − ( A, B ) → KK ( A, B ) (cid:1) . The map ( α ∗ ) − β ∗ −
1, has the same cokernel as α ∗ β − ∗ − OMPUTING OBSTRUCTION CLASSES 7
A rather deep result says that the I Z -projective objects generate KK Z . Equiva-lently, if R Z ( A ) ∼ = 0 in KK , then already A ∼ = 0 in KK Z . This is related to the proofof the Baum–Connes conjecture for the group Z . It follows immediately from thePimsner–Voiculescu sequence (2.3). Theorem 2.12.
Let ( A, α ) and ( B, β ) be objects of KK Z . Any isomorphism F Z ( A ) ∼ = F Z ( B ) in KK [ Z ] lifts to an isomorphism A ∼ = B in KK Z . And anyobject of KK [ Z ] is isomorphic to F Z ( A ) for some A ∈∈ KK Z , which is unique up toisomorphism.If t ∈ KK ( A, B ) × is a KK -equivalence with t ◦ [ α ] = [ β ] ◦ t , then there is anisomorphism in KK Z ( A, B ) that is mapped to t by the forgetful functor KK Z → KK .Proof. The Universal Coefficient Theorem for KK Z ( A, B ) and the ideal I Z allows tolift any map t : ( A, [ α ]) → ( B, [ β ]) in KK [ Z ] to an element ˆ t ∈ KK Z (cid:0) ( A, α ) , ( B, β ) (cid:1) .The naturality of the Universal Coefficient Theorem implies that the compositionvanishes for two elements of the Ext -part of KK Z ( A, B ). Hence ˆ t is invertible if t isinvertible. Any object of KK [ Z ] has a projective resolution of length 1. This allowsto lift it to an object of KK Z (see [4, Proposition 2.3]). This proves both assertionsin the first paragraph. The second paragraph only describes isomorphisms in KK [ Z ]more concretely. (cid:3) We make the isomorphism criterion in Theorem 2.12 more explicit under theassumption that the C ∗ -algebras A and B belong to the bootstrap class. Let A ± beC ∗ -algebras in the bootstrap class withK ( A + ) = K ( A ) , K ( A + ) = 0 , K ( A − ) = 0 , K ( A − ) = K ( A ) . Then K ∗ ( A + ⊕ A − ) ∼ = K ∗ ( A ). So A ∼ = A + ⊕ A − by the Universal CoefficientTheorem for KK. We use such a KK-equivalence to map α to an element ofKK ( A + ⊕ A − , A + ⊕ A − ). We rewrite this as a 2 × (cid:18) α ++ α + − α − + α −− (cid:19) , α ++ ∈ KK ( A + , A + ) ∼ = Hom (cid:0) K ( A ) , K ( A ) (cid:1) ,α + − ∈ KK ( A − , A + ) ∼ = Ext (cid:0) K ( A ) , K ( A ) (cid:1) ,α − + ∈ KK ( A + , A − ) ∼ = Ext (cid:0) K ( A ) , K ( A ) (cid:1) ,α −− ∈ KK ( A − , A − ) ∼ = Hom (cid:0) K ( A ) , K ( A ) (cid:1) . Here we have used the Universal Coefficient Theorem for the C ∗ -algebras A ± .A similar decomposition B ∼ = B + ⊕ B − allows us to map β to an element ofKK ( B + ⊕ B − , B + ⊕ B − ), which we then rewrite as a 2 × (cid:18) β ++ β + − β − + β −− (cid:19) , β ++ ∈ KK ( B + , B + ) ∼ = Hom (cid:0) K ( B ) , K ( B ) (cid:1) ,β + − ∈ KK ( B − , B + ) ∼ = Ext (cid:0) K ( B ) , K ( B ) (cid:1) ,β − + ∈ KK ( B + , B − ) ∼ = Ext (cid:0) K ( B ) , K ( B ) (cid:1) ,β −− ∈ KK ( B − , B − ) ∼ = Hom (cid:0) K ( B ) , K ( B ) (cid:1) . And we may also transfer an element t ∈ KK ( A, B ) to such a 2 × (cid:18) t ++ t + − t − + t −− (cid:19) , t ++ ∈ KK ( A + , B + ) ∼ = Hom (cid:0) K ( A ) , K ( B ) (cid:1) ,t + − ∈ KK ( A − , B + ) ∼ = Ext (cid:0) K ( A ) , K ( B ) (cid:1) ,t − + ∈ KK ( A + , B − ) ∼ = Ext (cid:0) K ( A ) , K ( B ) (cid:1) ,t −− ∈ KK ( A − , B − ) ∼ = Hom (cid:0) K ( A ) , K ( B ) (cid:1) . The naturality of the exact sequence in the Universal Coefficient Theorem impliesthat the Kasparov product of an element of Ext (cid:0) K ∗ ( A ) , K ∗ ( B ) (cid:1) with an elementof KK ( B, C ) depends only on the image of the latter in Hom (cid:0) K ∗ ( B ) , K ∗ ( C ) (cid:1) . Thus RALF MEYER the product of two Ext-terms always vanishes. Hence the condition t [ α ] = [ β ] t inTheorem 2.12 is equivalent to four equations t ++ α ++ = β ++ t ++ , t + − α −− + t ++ α + − = β + − t −− + β ++ t + − ,t −− α −− = β −− t −− , t − + α ++ + t −− α − + = β − + t ++ + β −− t − + . The equations on the left assert that the two diagonal entries of t [ α ] and [ β ] t areequal; those on the right assert equality of the two off-diagonal entries. Theorem 2.12says that ( A, α ) and (
B, β ) are KK Z -equivalent if and only if there is an invertibleelement t for which these equations hold. The equations on the left mean thatthe isomorphism t ∗ = ( t ++ , t −− ) ∈ Hom (cid:0) K ∗ ( A ) , K ∗ ( B ) (cid:1) induced by t intertwinesthe automorphisms α ∗ ∈ Aut(K ∗ ( A )) and β ∗ ∈ Aut(K ∗ ( B )). Equivalently, t ∗ is anisomorphism of Z / Z [ x, x − ]-modules. The equations on the right meanthat there is t := ( t + − , t − + ) ∈ Ext (cid:0) K ∗ ( A ) , K ∗ ( B ) (cid:1) with t α ∗ − β ∗ t = β t ∗ − t ∗ α , where we abbreviate α ∗ = ( α ++ , α −− ) and α = ( α + − , α − + ), and similarly for β .The choice of t has no effect on the invertibility of t . So the criterion in Theorem 2.12is whether the image of β t ∗ − t ∗ α vanishes in the cokernel of(2.13) Ext (cid:0) K ∗ ( A ) , K ∗ ( B ) (cid:1) → Ext (cid:0) K ∗ ( A ) , K ∗ ( B ) (cid:1) , t t α ∗ − β ∗ t . We shall later identify this cokernel with the group Ext Z [ x,x − ] (cid:0) K ∗ ( A ) , K ∗ ( B ) (cid:1) and show that the image of β t ∗ − t ∗ α in this cokernel is the relative obstructionclass for ( A, α ) and (
B, β ) and a Z [ x, x − ]-module isomorphisms t ∗ : K ∗ ( A ) ∼ = K ∗ ( B ).This gives the rule to translate between the classifying invariants in Theorem 2.12and in [4]. Remark . If A = B is a unital Kirchberg algebra (separable, nuclear, unital,purely infinite and simple), then a much finer classification theorem for automor-phisms is proved by Nakamura [13].2.2. C ∗ -Algebras over unique path spaces. Now we prove a Universal Coeffi-cient Theorem for KK X for a unique path space X . Let X be a countable set andlet → be a relation on X , which says for which points x, y ∈ X there is an edge x → y . Equip X with the partial order generated by ← , that is, x (cid:22) y if and onlyif there is a chain of edges x = x ← x ← · · · ← x ‘ = y with some ‘ ≥ x , . . . , x ‘ − ∈ X . Equip X with the Alexandrov topology generated by this partialorder. We assume ( X, → ) to be a unique path space , that is, there is at most onechain of edges x = x ← x ← · · · ← x ‘ = y between any two points x, y ∈ X .For x ∈ X , the subset U x := { y ∈ X : x (cid:22) y } is the minimal open subsetcontaining x . A C ∗ -algebra over X is equivalent to a C ∗ -algebra A with fixedideals A ( U x ) / A for all x ∈ X , such that A ( U x ) ⊆ A ( U y ) for all x, y ∈ X with x → y or, equivalently, with U x ⊆ U y . The equivariant Kasparov category KK X forC ∗ -algebras over X has separable C ∗ -algebras over X as objects and the KK-groupsKK X ( A, B ) as arrows (see also [10], where this category is denoted KK ( X )). Theforgetful functor R X : KK X → Y x ∈ X KK , A ( A ( U x )) x ∈ X , is a triangulated functor between triangulated categories. Its kernel on morphisms I X ( A, B ) := { f ∈ KK X ( A, B ) : f ( U x ) = 0 in KK ( A ( U x ) , B ( U x )) for all x ∈ X } is a stable homological ideal. We now describe the universal I X -exact stablehomological functor as in Section 2.1. OMPUTING OBSTRUCTION CLASSES 9
Let KK [ X ] be the category of functors ( X, (cid:23) ) → KK with natural transformationsas arrows. Since X is a unique path space, the category associated to the partiallyordered set ( X, (cid:23) ) is the path category of the directed graph ( X, → ). By theuniversal property of the path category, an object of KK [ X ] is given by A x ∈∈ KK for x ∈ X and α y,x ∈ KK ( A x , A y ) for x, y ∈ X with x → y , without any relationson the α y,x . This uniquely determines KK-classes α y,x ∈ KK ( A x , A y ) for x, y ∈ X with x (cid:23) y such that α x,x = id A x and α z,y ◦ α y,x = α z,x for all x, y, z ∈ X with x (cid:23) y (cid:23) z . An arrow ( A x , α y,x ) → ( B x , β y,x ) is a family of arrows f x ∈ KK ( A x , B x )for x ∈ X with f y α y,x = β y,x f x in KK ( A x , B y ) for all x, y ∈ X with x → y ; then f y α y,x = β y,x f x holds for all x, y ∈ X with x (cid:23) y . Define F X : KK X → KK [ X ]by mapping a C ∗ -algebra A over X to the object of KK [ X ] where A x := A ( U x )and where α y,x ∈ KK ( A x , A y ) for x, y ∈ X with x → y is the KK-class of theinclusion map A ( U x ) , → A ( U y ). Then α y,x ∈ KK ( A x , A y ) for x, y ∈ X with x (cid:23) y is the KK-class of the inclusion map as well. A kernel–cokernel pair K → E → Q in KK [ X ] is called admissible if it splits pointwise, that is, K x → E x → Q x is asplit extension in KK for all x ∈ X ; so E x ∼ = K x ⊕ Q x in KK for all x ∈ X , butthe sections Q x → E x are not compatible with the structure maps E x → E y and Q x → Q y for x → y . This turns KK [ X ] into an exact category (see [5, Exercise 5.3]).Let z ∈ X and A ∈∈ KK . As in [10], let i z ( A ) ∈∈ KK X be the C ∗ -algebra A with i z ( A )( U x ) := ( A if z ∈ U x , that is, x (cid:22) z, X ( i z ( A ) , B ) ∼ = KK (cid:0) A, B ( U z ) (cid:1) for all B ∈∈ KK X by [10, Proposition 3.13]. This isomorphism applies the restrictionmap KK X ( i z ( A ) , B ) → KK ( i z ( A )( U z ) , B ( U z )) and then identifies i z ( A )( U z ) = A .The object j z ( A ) := F X ( i z ( A )) ∈∈ KK [ X ]has j z ( A ) x = A for x (cid:22) z and j z ( A ) x = 0 otherwise, and the map j z ( A ) x → j z ( A ) y for x (cid:23) y is the identity map in KK ( A, A ) if z (cid:23) x and the zero map inKK (0 , j z ( A ) y ) otherwise. We compute(2.16) Hom KK [ X ] (cid:0) j z ( A ) , ( B x , β y,x ) (cid:1) ∼ = KK ( A, B z ) . Equations (2.15) and (2.16) imply(2.17) KK X ( i z ( A ) , B ) ∼ = KK ( A, B ( U z )) ∼ = Hom KK [ X ] (cid:0) j z ( A ) , F X ( B ) (cid:1) for all B ∈∈ KK X . Theorem 2.18.
Let ( X, → ) be a countable unique path space. The objects of theexact category KK [ X ] of the form L z ∈ X j z ( A z ) for A z ∈∈ KK for z ∈ X areprojective. For any object A = ( A x , α x,y ) of KK [ X ] , there is an admissible extension (2.19) M x → y j y ( A x ) ι (cid:26) M x ∈ X j x ( A x ) π (cid:16) A, which is a projective resolution of length . The exact category KK [ X ] has enoughprojective objects, and the partial adjoint of F X is defined on all projective objectsof KK [ X ] and is a section for F X there. The functor F X : KK X → KK [ X ] is theuniversal I X -exact stable homological functor to an exact category. Proof.
The objects j z ( A z ) are projective by (2.16). This is inherited by the directsum L z ∈ X j z ( A z ). The identity map on A x has an adjunct a x : j x ( A x ) → A by (2.16). These maps induce the map π = ( a x ) x ∈ X : L x ∈ X j x ( A x ) → A . For eachedge x → y in the directed graph ( X, → ), the map (id A x , − α y,x ) : A x → A x ⊕ A y isadjunct to a map a x → y : j y ( A x ) → j x ( A x ) ⊕ j y ( A y ) ⊆ M x ∈ X j x ( A x )by (2.16). It satisfies π ◦ a x → y = 0. The maps a x → y combine to a map ι : M x → y j y ( A x ) → M x j x ( A x )with π ◦ ι = 0.Now we prove that the maps π and ι mapped to Q x ∈ X KK by the forgetful functorform a split exact sequence. We consider the entries at a fixed z ∈ X . The entryof A at z ∈ X is simply A z . The entry of L x ∈ X j x ( A x ) at z ∈ X is the direct sumof A x over all x ∈ X with x (cid:23) z . The entry of L x → y j y ( A x ) at z ∈ X is the directsum of A x for all edges x → y in X with y (cid:23) z . The entry of π at z is( α z,x ) x (cid:23) z : M x (cid:23) z A x → A z . This is split surjective with the canonical section that maps A z identically onto thesummand A z for x = z . The entry of ι at z maps the summand A x for x → y (cid:23) z to A x ⊕ A y ⊆ L t (cid:23) z A t using (id A x , − α y,x ). We are going to define a map s : M t (cid:23) z A t → M x → y (cid:23) z A x which together with s forms a contracting homotopy for the short chain complexformed by ι | z and π | z . By assumption, if t (cid:23) z then there is a unique chain t = x → x → · · · → x ‘ = z . We let ( s ) z | A t for t (cid:23) z map the summand A t in L t (cid:23) z A t to the direct sum of A x i for the edges x i → x i +1 for i = 0 , . . . , ‘ − α x i ,t to map A t to A x i . If x → y (cid:23) z , then y = x in the abovechain. Therefore, s ◦ ι | A x is a map to L ‘j =0 A x j , where the entry at A x j is α x j ,x − α x j ,y ◦ α y,x = 0 for j = 1 , . . . , ‘ , and the identity map for j = 0. So s ◦ ι isthe identity map. Finally, we claim that s ◦ π + ι ◦ s is the identity map on L t (cid:23) z A t .This is checked on each summand A t separately. Let t = x → x → · · · → x ‘ = z bethe unique chain as above. Then ι ◦ s is a telescoping sum of ± α x j ,t for j = 0 , . . . , ‘ ,where α t,t occurs only with sign + and α z,t only with sign − . And s ◦ π is themap α z,t . Thus s ◦ π + ι ◦ s is the identity map on A t . This finishes the proofthat (2.19) is I X -exact. Its entries are I X -projective. So it is an I X -projectiveresolution.The projective resolution (2.19) implies that KK [ X ] has enough projective objectsand that an object is projective if and only if it is a direct summand of an object ofthe form L z ∈ X j z ( B z ) for some separable C ∗ -algebras B z for z ∈ X . Equation (2.17)says that the adjoint functor to F X : KK X → KK [ X ] is defined on j z ( A ) and mapsit to i z ( A ). Since the partial adjoint commutes with direct sums, it is also definedon L z ∈ X j z ( B z ) for any B z ∈∈ KK and maps it to L z ∈ X i z ( B z ). Idempotents inthe category KK X split because it is triangulated and has countable direct sums.Therefore, the partial adjoint of F X is defined on all projective objects of KK [ X ]and is a section for F X there. An argument as in the proof of Proposition 2.10shows that F X is the universal I X -exact functor to an exact category. The targetcategory is not smaller because the functor KK X → KK [ X ] is surjective on objectsby Corollary 2.20. (cid:3) OMPUTING OBSTRUCTION CLASSES 11
Unlike in Section 2.1, the I X -projective objects in KK X do not generate KK X . Thelocalising subcategory h P I i generated by them is equal to the localising subcategorygenerated by objects of the form i x ( A ) for x ∈ X , A ∈∈ KK . If X is finite, then itis described in several equivalent ways in [10, Definition 4.7]. It contains all nuclearC ∗ -algebras over X . This is the subcategory on which Theorem 2.18 implies aUniversal Coefficient Theorem, using Hom and Ext groups in the category KK [ X ].This implies the following classification theorem: Corollary 2.20.
Let ( X, → ) be a countable unique path space and let A and B be C ∗ -algebras over X that belong to the localising subcategory h P I i generated byobjects of the form i x ( B ) for x ∈ X , B ∈∈ KK . Any isomorphism between F X ( A ) and F X ( B ) in KK [ X ] lifts to an isomorphism in KK X . And any object of KK [ X ] liftsto an object of h P I i . An isomorphism F X ( A ) ∼ −→ F X ( B ) is a family of invertibleelements t x ∈ KK ( A ( U x ) , B ( U x )) for x ∈ X for which the diagrams A ( U x ) A ( U y ) B ( U x ) B ( U y ) t x t y commute in KK for all x, y ∈ X with x → y . The criterion above may be made more explicit if, in addition, A ( U x ) and B ( U x ) belong to the bootstrap class for all x ∈ X . As in Section 2.1, we identify A ( U x ) ∼ = A ( U x ) + ⊕ A ( U x ) − and rewrite the classes of the inclusion maps α y,x ∈ KK ( A ( U x ) , A ( U y )) and β y,x ∈ KK ( B ( U x ) , B ( U y )) for x → y as 2 × t x ∈ KK ( A ( U x ) , B ( U x )). As in the case of KK Z , the equality β y,x t x = t y α y,x for x → y in Corollary 2.20 may be rewritten as four equalitiesof matrix coefficients. The equality of the diagonal terms says that the diagrams (cid:0) K ∗ ( A ( U x )) , K ∗ ( α y,x ) (cid:1) and (cid:0) K ∗ ( B ( U x )) , K ∗ ( β y,x ) (cid:1) of countable Z / x, y ∈ X with x → y , the following diagramcommutes: K ∗ ( A ( U x )) K ∗ ( A ( U y ))K ∗ ( B ( U x )) K ∗ ( B ( U y )) K ∗ ( t x ) K ∗ ( α y,x ) K ∗ ( t y )K ∗ ( β y,x ) The equality of the off-diagonal terms will be studied in Section 4.2.3.
Computation of the obstruction class
We recall the setup of [4]. Let T be a triangulated category with countable directsums. Let I be a stable homological ideal in T with enough projective objects. Let F : T → A be the universal I -exact stable homological functor (in this article, weallow A to be exact). We assume that A is paired as in [4, Definition 2.14], thatis, A = A + × A − with Σ A + = A − and Σ A − = A + . For instance, A could be thecategory of Z -graded or Z / A + and A − may be taken to be the gradedmodules concentrated in even or odd degrees, respectively. We want to compute theobstruction class of an object A ∈∈ T . This is only meaningful if A is constructedfrom simpler ingredients.We assume that there is an exact, I -exact triangle(3.1) B ϕ −→ B p −→ A i −→ Σ B , where B and B are objects of h P I i with projective resolutions of length 1. Then A ∈∈ h P I i as well. The I -exactness assumption says that i ∈ I . Equivalently, F ( i ) = 0, F ( ϕ ) is monic, and F ( p ) is epic. The objects B and B are uniquelydetermined up to isomorphism by F ( B ) and F ( B ) because they have projectiveresolutions of length 1 (compare [4, Proposition 2.3]). We may split B i ∼ = B + i ⊕ B − i with F ( B ± i ) ∈ A ± for i = 0 ,
1. Then ϕ becomes a 2 × ϕ = (cid:18) ϕ ++ ϕ + − ϕ − + ϕ −− (cid:19) with ϕ − + : B +1 → B − , and so on. The two diagonal entries give an element of T ( B +1 , B +0 ) ⊕ T ( B − , B − ) ∼ = Hom A (cid:0) F ( B ) , F ( B ) (cid:1) , and the two off-diagonal entries give an element of T ( B +1 , B − ) ⊕ T ( B − , B +0 ) ∼ = Ext A (cid:0) Σ F ( B ) , F ( B ) (cid:1) ;here we have used the Universal Coefficient Theorem to compute T ( B ± , B ± ). Theresults for these four groups only have a single Hom or a single Ext group becauseof the parity assumptions. Thus the splitting of ϕ into ϕ := ϕ ++ + ϕ −− and ϕ := ϕ + − + ϕ − + splits the exact sequenceExt A (cid:0) Σ F ( B ) , F ( B ) (cid:1) (cid:26) T ∗ ( B , B ) (cid:16) Hom A (cid:0) F ( B ) , F ( B ) (cid:1) . We shall compute the obstruction class of A in terms of ϕ .By assumption, there is a short exact sequence F ( B ) F ( B ) F ( A ) . F ( ϕ ) F ( p ) This induces a long exact sequence0 ← Ext A (cid:0) Σ F ( A ) , F ( A ) (cid:1) ∂ ←− Ext A (cid:0) Σ F ( B ) , F ( A ) (cid:1) F ( ϕ ) ∗ ←−−−− Ext A (cid:0) Σ F ( B ) , F ( A ) (cid:1) F ( p ) ∗ ←−−−− Ext A (cid:0) Σ F ( A ) , F ( A ) (cid:1) ← · · · because Ext k A (cid:0) F ( B i ) , F ( A ) (cid:1) = 0 for i = 0 , k ≥ Theorem 3.2.
The obstruction class of A is ∂ ( p ∗ ϕ ) = ∂ ( F ( p ) ◦ ϕ ) ∈ Ext A (cid:0) Σ F ( A ) , F ( A ) (cid:1) , where F ( p ) ∈ Hom A (cid:0) Σ F ( B ) , Σ F ( A ) (cid:1) and ϕ ∈ Ext A (cid:0) Σ F ( B ) , F ( B ) (cid:1) .Proof. We shall recall the construction of obstruction classes in [4] along the way.It starts with an I -projective resolution of A of length 2. So first we have toconstruct this. We use the projective resolutions of F ( B i ) of length 1, which existby assumption. They lift canonically to I -projective resolutions(3.3) 0 → P i d i −−→ P i d i −−→ B i in T for i = 0 , d i is I -monicand d i is I -epic. The arrow ϕ ∈ T ( B , B ) lifts to a chain map(3.4) P P B P P B d Φ d Φ ϕd d between the I -projective resolutions (3.3) (see [11, Proposition 44]). We write (cid:26) for I -monic and (cid:16) for I -epic maps. We claim that(3.5) 0 → P
11 ( − d , Φ ) −−−−−−→ P ⊕ P
01 (Φ ,d ) −−−−−→ P p ◦ d −−−→ A → OMPUTING OBSTRUCTION CLASSES 13 is an I -projective resolution of A of length 2. The entries are I -projective byconstruction. Next we prove that (3.5) is a resolution, that is, it becomes an exactchain complex when we apply F to it.When we apply F to the diagram (3.4), the two rows become short exact sequencesin A , and the vertical maps become a chain map between them. The mapping coneof this chain map is again an exact chain complex in A . It has the form0 → F ( P ) → F ( P ) ⊕ F ( P ) → F ( P ) ⊕ F ( B ) → F ( B ) → . The map F ( ϕ ) : F ( B ) → F ( B ) is monic with cokernel F ( A ). So the directsummand F ( B ) and its image in F ( B ) together form a contractible subcomplex.The quotient by it is again an exact chain complex in A . This is what we get byapplying F to (3.5). So this is a resolution as asserted.An axiom for triangulated categories provides an exact triangle P P ⊕ P D Σ P − d , Φ ) containing ( − d , Φ ). Similarly, the map p ◦ d : P → A in (3.5) is part of anexact triangle(3.6) D P A Σ D . γ p ◦ d The long exact sequences for F applied to these two exact triangles show that F ( D )is the cokernel of the monomorphism F ( − d , Φ ), and that F ( D ) is the kernel ofthe epimorphism F ( p ◦ d ). The exactness of (3.5) implies F ( D ) ∼ = F ( D ). Since B and B belong to h P I i , so do D and D . And F ( D ) has a projective resolutionof length 1 by construction. Hence the Universal Coefficient Theorem applies to D and D . Thus the isomorphism F ( D ) ∼ = F ( D ) lifts to an isomorphism D ∼ = D . Weshall identify D = D .The Universal Coefficient Theorem for D gives a short exact sequence(3.7) Ext A (cid:0) Σ F ( D ) , F ( P ) (cid:1) T ( D, P ) Hom A (cid:0) F ( D ) , F ( P ) (cid:1) . F We split D = D + ⊕ D − and P = P +00 ⊕ P − into objects of even and odd parity asin the construction of ϕ above the theorem. Then T ( D, P ) splits accordingly as a2 × T ( D + , P +00 ) ⊕ T ( D − , P − ) is isomorphicto Hom A (cid:0) F ( D ) , F ( P ) (cid:1) , whereas the sum of the off-diagonal terms T ( D − , P +00 ) ⊕ T ( D + , P − ) is isomorphic to Ext A (cid:0) Σ F ( D ) , F ( P ) (cid:1) . This is the (unnatural) splittingof the Universal Coefficient Theorem exact sequence (3.7) that follows because A is paired. In particular, we decompose γ = γ + γ into its parity-preserving andparity-reversing parts.Let A be another object of h P I i with an isomorphism F ( A ) ∼ = F ( A ). Theargument above shows that both A and A are cones of some γ, γ ∈ T ( D, P )as in (3.6), which lift the inclusion map F ( D ) (cid:26) F ( P ) that we get from theresolution (3.5). The relative obstruction class is defined as follows: compose γ − γ ∈ Ext A (cid:0) Σ F ( D ) , F ( P ) (cid:1) ⊆ T ( D, P )with the map F ( p ◦ d ) : F ( P ) → F ( A ) and apply the boundary map for theextension F ( D ) (cid:26) F ( P ) (cid:16) F ( A ). That is, plug γ − γ into the map(3.8) Ext A (cid:0) Σ F ( D ) , F ( P ) (cid:1) F ( p ◦ d ) ∗ −−−−−−→ Ext A (cid:0) Σ F ( D ) , F ( A ) (cid:1) ∂ DP A −−−−−→ Ext A (cid:0) Σ F ( A ) , F ( A ) (cid:1) . Let γ : D → P be the unique parity-preserving arrow that lifts the inclusion map F ( D ) (cid:26) F ( P ) and let A be its cone. The obstruction class of A is the relativeobstruction class for A and A . That is, we plug γ := γ − γ into (3.8).Finally, we relate the obstruction class defined above to ϕ . The solid square inthe following diagram commutes:(3.9) B B A Σ B D P A Σ D ϕ pγε p ◦ d d Σ ε By the third axiom of triangulated categories, there is an arrow ε making all threesquares commute. We shall only use the left square. Since D , B , B and P haveprojective resolutions of length 1 and A is paired, the Universal Coefficient Theoremallows us to split them into even and odd parts. Hence each of the arrows γ , ϕ , ε and d splits into a parity-preserving and a parity-reversing part. We have alreadyused the splittings ϕ = ϕ + ϕ and γ = γ + γ , and now we also split ε = ε + ε .The arrow d is parity-preserving because T ( P , B ) ∼ = Hom A (cid:0) F ( P ) , F ( B ) (cid:1) hasno parity-reversing part. The left commuting square in (3.9) implies(3.10) d ◦ γ = ϕ ◦ ε + ϕ ◦ ε = ϕ ◦ ε + ϕ ◦ ε. The second step uses that the composite of two odd terms always vanishes, that is,the Ext -term in the Universal Coefficient Theorem is nilpotent.Composing γ ∈ Ext A (cid:0) Σ F ( D ) , F ( P ) (cid:1) with F ( p ◦ d ) as in (3.8) has thesame effect as composing with p ◦ d because the exact sequence in the UniversalCoefficient Theorem is natural. Thus the obstruction class of A is the image of p ◦ d ◦ γ ∈ Ext A (cid:0) Σ F ( D ) , F ( A ) (cid:1) under the boundary map ∂ DP A : Ext A (cid:0) Σ F ( D ) , F ( A ) (cid:1) → Ext A (cid:0) Σ F ( A ) , F ( A ) (cid:1) Equation (3.10) and p ◦ ϕ = 0 imply p ◦ d ◦ γ = p ◦ ϕ ◦ ε + p ◦ ϕ ◦ ε = p ◦ ϕ ◦ ε. The composite p ◦ ϕ = p ∗ ( ϕ ) ∈ Ext A (cid:0) Σ F ( B ) , F ( A ) (cid:1) also appears in the statementof the theorem. Composing with ε in T has the same effect as composing with F ( ε ) in the graded category Ext ∗ A . When we apply F to the morphism of exacttriangles (3.9), we get the following morphism of extensions in A : F ( B ) F ( B ) F ( A ) F ( D ) F ( P ) F ( A ) F ( ϕ ) F ( p ) F ( γ ) F ( ε ) F ( p ) ◦ F ( d ) F ( d ) Since boundary maps in Ext-theory are natural, the boundary map ∂ : Ext A (cid:0) Σ F ( B ) , F ( A ) (cid:1) → Ext A (cid:0) Σ F ( A ) , F ( A ) (cid:1) for the top row is equal to the composite of F ( ε ) and the boundary map ∂ DP A .Thus the obstruction class of A is ∂ ( F ( p ) ◦ ϕ ) as asserted. (cid:3) Comparison of classification theorems
In this section, we apply Theorem 3.2 in several cases to relate the classificationtheorem involving the obstruction class to other classification theorems. We firstcompare the classification for Z -actions in Theorem 2.12 with the one obtained in [4].Then we compare the classification for C ∗ -algebras over a unique path space ( X, → )in Corollary 2.20 with the one in [4]. In both cases, the invariants can be translatedinto each other rather directly. OMPUTING OBSTRUCTION CLASSES 15
Actions of the integers.
The Universal Coefficient Theorem for Z -actionsin Section 2.1 is based on the stable homological ideal I Z defined by the forgetfulfunctor KK Z → KK . Now we use another ideal I Z K . Let A Z be the category ofcountable Z / Z [ x, x − ]-modules. Let F Z K : KK Z → A Z , ( A, α ) (cid:0) K ∗ ( A ) , K ∗ ( α ) (cid:1) , that is, we map ( A, α ) ∈∈ KK Z to the Z / ∗ ( A ) with the Z [ x, x − ]-module structure given by the automorphism K ∗ ( α ) of K ∗ ( A ). Let I Z K ( A, B ) := { ϕ ∈ KK Z ( A, B ) : F Z K ( ϕ ) = 0 } be the kernel of F Z K on morphisms. This example is treated in [4], but there KK Z is disguised as KK T for the circle group T . These two categories are equivalent byBaaj–Skandalis duality (see [1]). The functor F Z K corresponds to the functor on KK T that maps a C ∗ -algebra with a continuous T -action to K T ∗ ( A ) with the canonicalmodule structure over the representation ring Z [ x, x − ] of T , which is used in [4].If B ∈∈ KK Z , then (2.5) impliesKK Z (C ( Z ) , B ) ∼ = KK ( C , B ) ∼ = K ( B ) ∼ = Hom A Z (cid:0) Z [ x, x − ] , F Z K ( B ) (cid:1) . Thus the partial adjoint of F Z K is defined on the rank-1 free module Z [ x, x − ]and maps it to C ( Z ). Then it is defined on all free modules, also in odd parity.Since idempotents in KK Z split, the partial adjoint ( F Z K ) ‘ of F Z K is defined onall projective objects of A Z . Since F Z K (C ( Z )) is the rank-1 free module again,we get ( F Z K ) ◦ ( F Z K ) ‘ ( P ) ∼ = P for all projective objects of A Z . Any object in A Z has a projective resolution of length 2. Hence it belongs to the image of F Z K by[4, Lemma 2.4]. Remark 2.2 and Theorem 2.1 show that F Z K is the universal I Z K -exactstable homological functor both to an exact and to an Abelian category.The category A Z is paired in an obvious way, using the subcategories A Z ± ofcountable Z [ x, x − ]-modules concentrated in degree 0 and 1, respectively. So theobstruction class is defined for any object of KK Z . Assume that ( A, α ) ∈∈ KK Z is such that A belongs to the bootstrap class in KK . Equivalently, ( A, α ) belongsto the localising subcategory of KK Z generated by the I Z K -projective object C ( Z ).Then the main result of [4] shows that A is determined uniquely up to isomorphismby (cid:0) K ∗ ( A ) , K ∗ ( α ) (cid:1) ∈∈ A Z and the obstruction class.A chain complex in KK Z that is I Z -exact is also I Z K -exact because I Z ⊆ I Z K .So the I Z -projective resolution in (2.11) is I Z K -exact. Its entries C ( Z , A ) are nolonger I Z K -projective. We claim, however, that they have I Z K -projective resolutionsof length 1. Let P (cid:26) P (cid:16) K ∗ ( A ) be a resolution of the Z / ∗ ( A ) by countable free Abelian groups. Then Z [ x, x − ] ⊗ P (cid:26) Z [ x, x − ] ⊗ P (cid:16) Z [ x, x − ] ⊗ K ∗ ( A )is a projective resolution of F Z K (C ( Z , A )) ∼ = Z [ x, x − ] ⊗ K ∗ ( A ) . Here the tensor products with Z [ x, x − ] carry the module structure defined bymultiplication with x in the tensor factor Z [ x, x − ]. So we are in the situationof (3.1). Theorem 3.2 implies the following formula for the obstruction class of A : Theorem 4.1.
Let A be a C ∗ -algebra in the bootstrap class in KK and α ∈ Aut( A ) .Split [ α ] ∈ KK ( A, A ) into parity-preserving and parity-reversing parts α ∈ Hom (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) and α ∈ Ext (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) . Define γ : Ext Z (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) → Ext Z (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) , x α ◦ x − x ◦ α. Then
Ext Z [ x,x − ] (cid:0) Σ F Z K ( A ) , F Z K ( A ) (cid:1) ∼ = coker γ and the obstruction class of A is theimage of − α ( α ) − in this cokernel.Proof. In our case, the map ϕ in (3.1) is the map C ( Z ) ⊗ A → C ( Z ) ⊗ A in (2.11).Split A = A + ⊕ A − into its even and odd parts as before. ThenC ( Z ) ⊗ A ∼ = C ( Z ) ⊗ A + ⊕ C ( Z ) ⊗ A − is the parity decomposition of C ( Z ) ⊗ A . The translation τ and the identity areparity-preserving. Decompose [ α ] and [ α − ] into their even and odd parts. Theparity-reversing part of the map on C ( Z ) ⊗ A is ϕ = [ τ ] ⊗ [ α − ] . The map p ∈ KK Z (C ( Z ) ⊗ A, A ) satisfies p ( τ ⊗ id A ) = [ α ] p because it is Z -equivariant, andit restricts to the identity map on the 0th summand of C ( Z ) ⊗ A . The isomorphismKK Z (cid:0) (C ( Z , A ) , τ ) , ( B, β ) (cid:1) ∼ = KK ( A, B ) in (2.5) forgets the Z -action and thenevaluates at 0. Therefore, the image of ϕ in KK ( A, A ) is the restriction of p ◦ ϕ to the 0 th summand. And this is [ α ][ α − ] ∈ KK ( A, A ). Since αα − = id A andproducts of two parity-reversing KK-classes always vanish, [ α ][ α − ] + [ α ] [ α − ] =[id A ] = 0. Theorem 3.2 now says that the obstruction class for A is the image of[ α ][ α − ] = − [ α ] [ α − ] = − α ( α ) − ∈ Ext Z (cid:0) K ∗ ( A ) , K ∗ ( A )) (cid:1) under the boundary map to Ext Z [ x,x − ] (cid:0) Σ F Z K ( A ) , F Z K ( A ) (cid:1) . The description of thelatter Ext group in the theorem follows when we compute it with the projectiveresolution in (3.5) defined by the length-1 resolutions of C ( Z ) ⊗ A above. Thiscomputation has already been done in [4, Section 3.2]. When the Ext groups aredescribed in this way, the boundary map in Theorem 3.2 becomes a trivial map,mapping an element of Ext Z (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) to its image in coker γ . (cid:3) The formula for the obstruction class depends, of course, on the isomorphismExt Z [ x,x − ] (cid:0) Σ F Z K ( A ) , F Z K ( A ) (cid:1) ∼ = coker γ , and this depends on the I Z -projective reso-lution from which it is obtained. Our theorem uses the most obvious resolution. Onemay apply the automorphism of Ext Z [ x,x − ] (cid:0) Σ F Z K ( A ) , F Z K ( A ) (cid:1) that composes withthe automorphism − Σ α on Σ F Z K ( A ). This replaces the obstruction class − α ( α ) − by α . So the images of − α ( α ) − and α in Ext Z [ x,x − ] (cid:0) Σ F Z K ( A ) , F Z K ( A ) (cid:1) containthe same information.Theorem 4.1 and the computations after Theorem 2.12 allow to deduce theclassification by F Z K and the obstruction class from the classification by the in-variant F Z in Section 2.1. Let ( A, α ) and (
B, β ) belong to the bootstrap classin KK Z . Assume that there is an isomorphism t : F Z K ( A ) ∼ −→ F Z K ( B ) in A Z , that is,a grading-preserving Z [ x, x − ]-module isomorphism K ∗ ( A ) ∼ = K ∗ ( B ). We use thisisomorphism to identify K ∗ ( A ) = K ∗ ( B ). By Theorem 4.1, the relative obstructionclass vanishes if and only if α − β ∈ Ext (cid:0) K ∗ +1 ( A ) , K ∗ ( A ) (cid:1) vanishes in coker( γ ).Equivalently, there is t ∈ Ext (cid:0) K ∗ ( A ) , K ∗ ( B ) (cid:1) for which t = t + t ∈ KK ( A, B )satisfies [ β ] ◦ t = t ◦ [ α ]. Then t is an isomorphism F Z ( A ) ∼ = F Z ( B ) in KK [ Z ], andTheorem 2.12 shows that such an isomorphism lifts to an isomorphism in KK Z ( A, B ).Recall that Baaj–Skandalis duality is an equivalence of triangulated categories KK T ∼ = KK Z . Hence everything said above about Z -actions carries over to T -actions.The functor A K ∗ ( A ) becomes B K T ∗ ( B ) on KK T , equipped with the naturalmodule structure over the representation ring Z [ x, x − ] of T . The automorphism α is replaced by an automorphism of B (cid:111) T , namely, the generator β of the dualaction of Z . Since K ∗ ( B (cid:111) T ) ∼ = K T ∗ ( B ), the Universal Coefficient Theorem splitsKK ( B (cid:111) T , B (cid:111) T ) into the parity-preserving part Hom (cid:0) K T ∗ ( B ) , K T ∗ ( B ) (cid:1) and theparity-reversing part Ext (cid:0) K T ∗ ( B ) , K T ∗ ( B ) (cid:1) . The obstruction class of B (cid:111) T is the OMPUTING OBSTRUCTION CLASSES 17 class of − β ( β ) − ∈ Ext (cid:0) K T ∗ ( B ) , K T ∗ ( B ) (cid:1) in the cokernel of the map γ : Ext (cid:0) K T ∗ ( B ) , K T ∗ ( B ) (cid:1) → Ext (cid:0) K T ∗ ( B ) , K T ∗ ( B ) (cid:1) , x β x − xβ . As above, a grading-preserving Z [ x, x − ]-module isomorphism t : K T ∗ ( A ) → K T ∗ ( B )is compatible with the obstruction classes if and only if it lifts to an isomorphismbetween A (cid:111) T and B (cid:111) T in KK [ Z ], and such an isomorphism lifts further to anisomorphism between A (cid:111) T and B (cid:111) T in KK Z . By Baaj–Skandalis duality, thelatter is equivalent to an invertible element in KK T ( A, B ).4.2. C ∗ -Algebras over unique path spaces. Now we return to the setup ofSection 2.2. So ( X, → ) is a countable directed graph with the unique path property.Let (cid:22) be the partial order generated by ← and equip X with the Alexandrovtopology defined by (cid:22) . Let KK X be the category of C ∗ -algebras over X . Objectsin the appropriate bootstrap class in KK X are classified in [4] under the extraassumption that X be finite. Actually, the arguments in [4] work in the same way ifthe set X is countable. Here we treat this more general case right away.Let A X be the category of all functors from X to the category of countable Z / A X is a family of count-able Z / G x for x ∈ X with grading-preserving grouphomomorphisms γ y,x : G x → G y for all x, y ∈ X with x → y . Morphisms( G x , γ y,x ) → ( H x , η y,x ) in A X are families of grading-preserving group homomor-phisms t x : G x → H x that satisfy t y ◦ γ y,x = η y,x ◦ t x for all x, y ∈ X with x → y .We define the functor F X K : KK X → A X by mapping a C ∗ -algebra over X to the diagram of Z / ∗ ( A ( U x )) for x ∈ X with the maps induced by the inclusion maps A ( U x ) / A ( U y )for x → y . The target category A X is a paired, stable Abelian category, where A X ± ⊆ A X are the subcategories of Z / F X K is a stable homological functor. Let I X K be its kernelon morphisms. Equation (2.17) impliesKK X ( i z ( C ) , B ) ∼ = KK (cid:0) C , B ( U z ) (cid:1) ∼ = K (cid:0) B ( U z ) (cid:1) ∼ = Hom A X (cid:0) j z ( C ) , F X K ( B ) (cid:1) . Hence the partial adjoint ( F X K ) ‘ of F X K is defined on j z ( C ) for all z ∈ X . Asin Section 2.2, any object of A X is a quotient of a direct sum of objects of theform j z ( C ) for z ∈ X . Hence ( F X K ) ‘ is defined on all projective objects of I X K and F X K ◦ ( F X K ) ‘ ( P ) = P for all projective objects P of A X ; this is proved likethe corresponding statement about F Z K in Section 4.1. Therefore, the functor F X K : KK X → A X is the universal I X K -exact stable homological functor into an exactcategory or into an Abelian category by Theorem 2.1 and Remark 2.2. And F X K restricts to an equivalence of categories between the I X K -projective objects in KK X and the projective objects in A X . Let B X ⊆ KK X be the localising subcategorygenerated by the I X K -projective objects. This is the analogue of the bootstrap classin KK X . Lemma 4.2.
Let G = ( G x , γ y,x ) and H = ( H x , η y,x ) be objects of A X . (1) There is a projective resolution for G of length in A X . (2) There is A ∈ B X with F X K ( A ) ∼ = G . (3) Write
Ext for the
Ext of Abelian groups. The group Ext A X ( G, H ) isnaturally isomorphic to the cokernel of the map (4.3) Y x ∈ X Ext( G x , H x ) → Y x → y Ext( G x , H y ) , ( t x ) x ∈ X ( η y,x ◦ t x − t y ◦ γ y,x ) x → y , Proof.
Since I X ⊆ I X K , any I X -exact chain complex in KK X is also I X K -exact. Inparticular, the I X -projective resolution in (2.19) is I X K -exact. We claim that itsentries have I X K -projective resolutions of length 1. Hence there is a projectiveresolution of F X K ( A ) of length 2 as in (3.5). To prove the same for all objects G of A X , we carry over (2.19).Let J z ( B ) ∈∈ A Z for a countable Z / B be the diagramwith J z ( B ) x = B if z (cid:23) x and J z ( B ) x = 0 otherwise, with the identity map J z ( B ) x → J z ( B ) y for z (cid:23) x → y and the zero map otherwise. Then(4.4) Hom A X (cid:0) J z ( B ) , H (cid:1) ∼ = Hom (cid:0) B, H z (cid:1) , where the isomorphism simply restricts a morphism in A X to the object z ∈ X . In particular, if y (cid:23) z , then J y ( B ) z = B and so there is a canonical map J y (cid:23) z ( B ) : J z ( B ) → J y ( B ) in A X . Explicitly, this map is the identity map on J z ( B ) x if z (cid:23) x and the zero map on J z ( B ) x = 0 if z (cid:15) x . The proof that (2.19) is exactalso proves the exactness of(4.5) 0 → M x → y J y ( G x ) ψ −→ M x ∈ X J x ( G x ) q −→ G → ψ restricted to the summand J y ( G x ) for x → y is the map ( J x → y ( G x ) , J y ( γ y,x ))to J x ( G x ) ⊕ J y ( G y ), and q maps J x ( G x ) to G by the adjunct of the identity mapon G x under the adjunction in (4.4). Explicitly, the entry of L x ∈ X J x ( G x ) at z ∈ X is L x (cid:23) z G x , which is mapped to G z by ( γ z,x ) x (cid:23) z . The proof that (4.5) is exactshows that the chain complexes of Z / → M x → y J y ( G x ) z ψ z −−→ M x ∈ X J x ( G x ) z q z −→ G z → z ∈ X .For each x ∈ X , there is a resolution(4.6) P x, d x, P x, d x, G x of G x of length 1 by countable free Z / γ y,x : G x → G y for x → y lifts to a morphism of extensions(4.7) P x, P x, G x P y, P y, G yd x, γ y,x d x, γ y,x γ y,x d y, d y, The construction J z above is an exact functor, and it maps free Abelian groups toprojective objects of A X by the adjunction (4.4). Hence M x → y J y ( P x, ) (cid:26) M x → y J y ( P x, ) (cid:16) M x → y J y ( G x ) , M x ∈ X J x ( P x, ) (cid:26) M x ∈ X J x ( P x, ) (cid:16) M x ∈ X J s x ( G x )are projective resolutions of length 1 in A X . The resolution in (4.5) gives a projectiveresolution of G of length 2 as in (3.5). This proves the first assertion. OMPUTING OBSTRUCTION CLASSES 19
Now [4, Lemma 2.4] shows that there is A ∈∈ B X with G = F X K ( A ); the propertythat KK X ( A, B ) = 0 for all I X K -contractible B ∈∈ KK X is equivalent to A ∈∈ B X .The projective resolution of G built above has the form0 → M x → y J y ( P x, ) → M x → y J y ( P x, ) ⊕ M x ∈ X J x ( P x, ) → M x ∈ X J x ( P x, ) → G, where the maps L x → y J y ( P x,i ) → L x ∈ X J x ( P x,i ) for i = 0 , γ iy,x : P x,i → P y,i in (4.7). We use this resolution to compute the group Ext A X ( G, H ). An element ofExt A X ( G, H ) is represented by a map L x → y J y ( P x, ) → H . By the adjunction (4.4),this corresponds to a family of maps f x → y : P x, → H y . This family represents 0in Ext A X ( G, H ) if and only if the corresponding map L x → y J y ( P x, ) → H factorsthrough L x → y J y ( P x, ) ⊕ L x J x ( P x, ). Using the adjunction (4.4) again, a map onthis direct sum corresponds to families of maps g x → y : P x, → H y and h x : P x, → H x .The resulting map L x → y J y ( P x, ) → H corresponds to the family of maps − g x → y ◦ d x, + η y,x h x − h y γ y,x : P x, → H y . The resolutions (4.6) compute Ext( G x , D ) for any Z / D . Soeach f x → y : P x, → H y represents an element of Ext( G x , H y ), and it represents thezero element if and only if it is of the form g x → y ◦ d x, for some g x → y : P x, → H y .The elements h x above represent elements of Ext( G x , H x ). If they represent thezero element of Ext( G x , H x ), then the term η y,x h x − h y γ y,x above may be rewrittenin the form − g x → y ◦ d x, . Now we get the formula for Ext A X ( G, H ) in the thirdstatement in the lemma. (cid:3)
Theorem 4.8.
Let A and B belong to B X . An isomorphism t : F X K ( A ) ∼ −→ F X K ( B ) in A X lifts to an invertible element in KK X ( A, B ) if and only if tδ A = δ B t holdsin Ext A X (cid:0) Σ F X K ( A ) , F X K ( B ) (cid:1) , where δ A and δ B are the obstruction classes of A and B .Proof. The lemma verifies all the conditions to apply the classification methodof [4]. (cid:3)
The proof of the lemma also gives all the ingredients needed in Section 3. So wemay now compute obstruction classes:
Theorem 4.9.
Let B be an object of B X . Let B x := B ( U x ) for x ∈ X . Let β y,x ∈ KK ( B x , B y ) be the KK -class of the inclusion map B x , → B y . Split β y,x = β y,x + β y,x with a parity-preserving part β y,x ∈ Hom (cid:0) K ∗ ( B x ) , K ∗ ( B y ) (cid:1) and a parity-reversing part β y,x ∈ Ext (cid:0) K ∗ ( B x ) , K ∗ ( B y ) (cid:1) . The obstruction class of B is theclass in the cokernel of the map in (4.3) that is represented by ( β y,x ) x → y ∈ Y x → y Ext (cid:0) K ∗ ( B x ) , K ∗ ( B y ) (cid:1) . Proof.
The projective resolution (2.19) in KK [ X ] lifts to an exact triangle M x → y i y ( B x ) ϕ −→ M x ∈ X i x ( B x ) p −→ B → Σ M x → y i y ( B x )in KK X by [4, Proposition 2.3]. This is how the Universal Coefficient Theorem in[11, Theorem 66] is proved. Here the map ϕ restricted to the summand i y ( B x )is the difference of two maps: the map i y ( β y,x ) to i y ( B y ) and the canonical map i x,y ( B x ) : i y ( B x ) → i x ( B x ) that is the adjunct of the identity map B x → i x ( B x ) y = B x under the adjunction (2.15). And the map p restricted to i x ( B x ) is the adjunctof the identity map B x → B x under the adjunction (2.15). Both L x → y i y ( B x ) and L x ∈ X i x ( B x ) belong to B X and have I X K -projectiveresolutions of length 1 (see (4.6)). So we are in the situation of (3.1). We spliteach B x := B ( U x ) into its even and odd parity part B x = B + x ⊕ B − x . Then i y ( B + x )and i y ( B − x ) are of even or odd parity, respectively. Split β y,x = β y,x + β y,x into aparity-preserving and a parity-reversing part. So β y,x ∈ Hom (cid:0) K ∗ ( B x ) , K ∗ ( B y ) (cid:1) and β y,x ∈ Ext (cid:0) K ∗ ( B x ) , K ∗ ( B y ) (cid:1) by the Universal Coefficient Theorem forKK, see the discussion after Theorem 2.12. The induced maps i y ( β y,x ) and i y ( β y,x ) are parity-preserving and parity-reversing, respectively. And the map i x,y ( B x ) is parity-preserving. So the parity-reversing part ϕ of ϕ is the mapthat restricts to i y ( β y,x ) : i y ( B x ) → i y ( B y ) on the summand for x → y . Thecomposite p ◦ ϕ is the map L x → y i y ( B x ) → B whose restriction to the sum-mand i y ( B x ) is adjunct to β y,x : B x → B y . These maps define an element of Q x → y Ext (cid:0) K ∗ ( B x ) , K ∗ ( B y ) (cid:1) . The obstruction class of B is its image under theboundary map to Ext A X (cid:0) Σ F X K ( B ) , F X K ( B ) (cid:1) by Theorem 3.2. We have describedExt A X ( G, H ) in Lemma 4.2 in such a way that this boundary map becomes tauto-logical: it simply maps an element of Q x → y Ext( G x , H y ) to its class in the cokernelof the map in (4.3). In particular, the obstruction class of B is represented by( β y,x ) x → y ∈ Q x → y Ext (cid:0) K ∗ ( B x ) , K ∗ ( B y ) (cid:1) . (cid:3) Theorem 4.9 allows us to compare the classification theorems for C ∗ -algebrasin B X that use the invariant F X ( B ) or F X K ( B ) with the obstruction class. Let A, B ∈∈ B X . An isomorphism t : F X K ( A ) ∼ −→ F X K ( B )is equivalent to a family of isomorphisms t x : K ∗ ( A ( U x )) ∼ −→ K ∗ ( B ( U x ))that make the diagrams K ∗ ( A ( U x )) K ∗ ( B ( U x ))K ∗ ( A ( U y )) K ∗ ( B ( U y )) K ∗ ( α y,x ) t x ∼ = K ∗ ( β y,x ) t y ∼ = commute for all edges x → y . Here the maps K ∗ ( α y,x ) and K ∗ ( β y,x ) are induced bythe inclusion maps of our C ∗ -algebras over X . The obstruction class for the isomor-phism ( t x ) x ∈ X vanishes if and only if there are t x ∈ Ext (cid:0) K ∗ ( A ( U x )) , K ∗ ( B ( U x )) (cid:1) for x ∈ X such that(4.10) t y ◦ K ∗ ( α y,x ) − K ∗ ( β y,x ) ◦ t x = β y,x ◦ t x − t y ◦ α y,x for all edges x → y . Here α y,x ∈ Ext (cid:0) K ∗ ( A ( U x )) , K ∗ ( A ( U y )) (cid:1) , β y,x ∈ Ext (cid:0) K ∗ ( B ( U x )) , K ∗ ( B ( U y )) (cid:1) are the parity-reversing parts of the KK-classes of the inclusion maps. The elements t x and t x together form t x ∈ KK ( A ( U x ) , B ( U x )). And (4.10) is equivalent to t y ◦ α y,x = β y,x ◦ t x in KK ( A ( U x ) , B ( U y )).Corollary 2.20 says that any family of KK-equivalences t x ∈ KK ( A ( U x ) , B ( U x ))with t y ◦ α y,x = β y,x ◦ t x for all edges x → y lifts to an invertible element inKK X ( A, B ). The classification using the obstruction class says that a family of iso-morphisms t x : K ∗ ( A ( U x )) ∼ −→ K ∗ ( B ( U x )) lifts to an invertible element in KK X ( A, B )if and only if there exist t x satisfying (4.10). Corollary 2.20 makes a slightly strongerassertion because it says that any choice of elements t x satisfying (4.10) may berealised by an invertible element in KK X ( A, B ). OMPUTING OBSTRUCTION CLASSES 21
The result in Corollary 2.20 about the existence of liftings is also slightly strongerthan the corresponding result using the invariant F X K and the obstruction classbecause it does not require the C ∗ -algebras A ( U x ) to belong to the bootstrap class.5. Filtrated K-theory for totally ordered spaces
Now we consider the special unique path space X = { ← ← · · · ← n } for some n ∈ N ≥ . We are going to compare the classification for C ∗ -algebras over X that follows from the Universal Coefficient Theorem in [12] to the classifications inSection 4.2. The invariant used in [12] is filtrated K-theory. This is the diagram ofK-theory groups formed by K ∗ ( A ( S )) for all locally closed subsets S ⊆ X . Here asubset is locally closed if and only if it is of the form[ a, b ] := { a, a + 1 , . . . , b − , b } , ≤ a ≤ b ≤ n. The maps in the filtrated K-theory diagram are those that come from naturaltransformations. We are going to describe these below.Let I X fil be the kernel on morphisms of the filtrated K-theory functor, thatis, f ∈ KK X ( A, B ) belongs to I X fil ( A, B ) if and only if it induces the zero mapK ∗ ( A [ a, b ]) → K ∗ ( B [ a, b ]) for all 1 ≤ a ≤ b ≤ n . This is a stable homological ideal.It is shown in [11] that any object A ∈∈ B X has an I X fil -projective resolution(5.1) 0 → P ϕ −→ P f −→ A of length 1. Then it follows that A is isomorphic to the cone of ϕ . And there is aUniversal Coefficient Theorem for objects of B X based on Hom and Ext groupsbetween their filtrated K-theory diagrams. Thus any isomorphism between thefiltrated K-theory diagrams of A, B ∈∈ B X lifts to an equivalence in KK X ( A, B ).The minimal open subset U x containing x is [ x, n ] for each x ∈ X . SinceK ∗ ( A ( U x )) is part of the filtrated K-theory of A , the ideal I X fil is contained in theideal I X K that is used above for a general unique path space. So any I X fil -exactchain complex is also I X K -exact. If P ∈∈ B X is I X fil -projective, then the Abeliangroups K ∗ ( P ( S )) are free for all locally closed subsets S ⊆ X (this follows from[12, Theorem 3.12] and will become manifest below). The computation of Ext A X inLemma 4.2 shows that Ext A X ( F X K ( P ) , D ) = 0 for all D ∈∈ A X if P ( U x ) is free foreach x ∈ X . Hence P and P have I X K -projective resolutions of length 1. So we arein the situation of (3.1) and Theorem 3.2 computes the obstruction class from theparity-reversing part of the map ϕ in (5.1). This computation is, however, quitenon-trivial. We must first recall how the natural transformations in the filtratedK-theory diagram look like. Going beyond the results of [12], we then build anexplicit I X fil -projective resolution. Next, we observe which parts of the map ϕ areparity-reversing. This gives a class in Ext A X . To translate it into the setting ofTheorem 4.9, we still have to compare the resolution used there with the one comingfrom (5.1). This requires most of the work.We first recall the description of the Z / N T ∗ ([ a, b ] , [ c, d ])of natural transformations K ∗ ( A ([ a, b ])) → K ∗ ( A ([ c, d ])) in [12]. Let 1 ≤ a ≤ b ≤ n .The functor KK X → Ab Z / , A K ∗ ( A ([ a, b ])), is represented by an object R [ a,b ] of KK X , which is described in [12]. Let 1 ≤ a ≤ b ≤ n and 1 ≤ c ≤ d ≤ n . By theYoneda Lemma, N T ∗ ([ a, b ] , [ c, d ]) ∼ = KK X ∗ ( R [ c,d ] , R [ a,b ] ) ∼ = K ∗ ( R [ a,b ] ([ c, d ])) . These groups are computed in [12, Equation (3.1)]:(5.2)
N T ∗ ([ a, b ] , [ c, d ]) ∼ = Z + if c ≤ a ≤ d ≤ b, Z − if a + 1 ≤ c ≤ b + 1 ≤ d, a, b ] → [ c, d ] in the first two cases, that is, when there is a non-zeronatural transformation in N T ∗ ([ a, b ] , [ c, d ]). The groups N T ∗ ([ a, b ] , [ c, d ]) form a Z / N T . The filtrated K-theory of a separable C ∗ -algebra over X is a Z / F K ( A ).The computations in [12] interpret the elements of N T ∗ ([ a, b ] , [ c, d ]) as follows.Let A be a separable C ∗ -algebra over X and let M := F K ( A ). If a < b ≤ c , then[ b, c ] is relatively open in [ a, c ] with complement [ a, b − · · · → M [ b, c ] i −→ M [ a, c ] r −→ M [ a, b − δ −−→ odd M [ b, c ] → · · · where the maps i, r preserve the Z / δ reverses it. Any naturaltransformation M [ a, b ] → M [ c, d ] is an integer multiple of a product of the maps i, r, δ above. More precisely, if c ≤ a ≤ d ≤ b , then there is a commuting square(5.4) M [ a, b ] M [ c, b ] M [ a, d ] M [ c, d ] , ir τ [ c,d ][ a,b ] ri and its diagonal map τ [ c,d ][ a,b ] generates N T ([ a, b ] , [ c, d ]) ∼ = Z . And if a + 1 ≤ c ≤ b + 1 ≤ d , then there is a commuting square(5.5) M [ a, b ] M [ b + 1 , d ] M [ a, c − M [ c, d ] , δr τ [ c,d ][ a,b ] iδ and its diagonal map τ [ c,d ][ a,b ] generates N T ([ a, b ]) , [ c, d ]) ∼ = Z . We have defineda generator τ [ c,d ][ a,b ] for N T ∗ ([ a, b ] , [ c, d ]) whenever [ a, b ] → [ c, d ], that is, when-ever N T ∗ ([ a, b ]) , [ c, d ]) = 0 by (5.2). It is convenient to define τ [ c,d ][ a,b ] = 0 if N T ∗ ([ a, b ]) , [ c, d ]) = 0. By the Yoneda Lemma, the natural transformations τ [ c,d ][ a,b ] correspond to arrows (cid:0) τ [ c,d ][ a,b ] (cid:1) ∗ : R [ c,d ] → R [ a,b ] . Remark . An N T -module is called exact if the sequences (5.3) are exact for all a < b ≤ c . The exact N T -modules form a stable exact subcategory of the stableAbelian category of all
N T -modules, and the filtrated K-theory of any separableC ∗ -algebra over X is exact as an N T -module. The results in [12] imply that anyexact
N T -module has a projective resolution of length 1. Hence it lifts to an objectof the bootstrap class B X . So the image of the filtrated K-theory functor is equalto the class of exact, countable N T -modules. And the filtrated K-theory functor,viewed as a functor to the subcategory of exact, countable
N T -modules, is theuniversal I X fil -exact functor to an exact category. So I X fil has the property that itsuniversal exact functors to an Abelian and to an exact category are different. OMPUTING OBSTRUCTION CLASSES 23
Now we study the multiplication in
N T . We begin with decomposing thegenerators in
N T further. We may rewrite the natural transformations τ [ c,d ][ a,b ] definedabove as products of the special natural transformations i = τ [ a,b ][ a +1 ,b ] : [ a + 1 , b ] → [ a, b ] , a + 1 ≤ b ≤ n,r = τ [ a,b ][ a,b +1] : [ a, b + 1] → [ a, b ] , a ≤ b ≤ n − ,δ = τ [ a,n ][1 ,a − : [1 , a − → [ a, n ] , ≤ a ≤ n. This is clear in the even case. In the odd case, we use the naturality of boundary mapsto rewrite the boundary map δ : [ a, b ] → [ b + 1 , d ] as the product of i : [ a, b ] → [1 , b ],the boundary map δ : [1 , b ] → [ b + 1 , n ], and r : [ b + 1 , n ] → [ b + 1 , d ]. The generatingnatural transformations defined above form a commuting diagram as in Figure 1.The last column and the first row in the diagram are the same. So the diagramrepeats when we put a reflected copy of it next to it. Figure 2 shows the full diagramfor n = 3. We claim that all relations among the generating natural transformationsare given by this extended commuting diagram. In particular, a composite of i, r, δ vanishes if and only if it factors through one of the objects 0 on the boundary ofthe extended diagram.Let 1 ≤ a ≤ b ≤ n and 1 ≤ c ≤ d ≤ n . A product of the generators of type i, r from [ a, b ] to [ c, d ] exists if and only if c ≤ a and d ≤ b . Figure 2 shows that all suchproducts are equal. Equation (5.2) shows that this product is 0 unless a ≤ d , so that c ≤ a ≤ d ≤ b . Then it is equal to τ [ c,d ][ a,b ] by the definition in (5.4). As a consequence, τ [ c,d ][ e,g ] · τ [ e,g ][ a,b ] = τ [ c,d ][ a,b ] if c ≤ e ≤ d ≤ g and e ≤ a ≤ g ≤ b ; this is non-zero if and onlyif also a ≤ d or, equivalently, c ≤ e ≤ a ≤ d ≤ g ≤ b .Now consider a product of i, r, δ going from [ a, b ] to [ c, d ] and containing exactlyone factor of δ . Using the diagram in Figure 1, we may rearrange this product insuch a way that we first go right and then go down in the extended diagram as inFigure 2. If this goes through the zeros outside the drawn region, the product is 0.If not, we may combine consecutive i and consecutive r to bring the product intothe following form:[ a, b ] i −→ [1 , b ] δ −→ [ b + 1 , n ] r −→ [ b + 1 , d ] i −→ [ c, d ] . The combination r ◦ δ ◦ i in the beginning is the boundary map δ : [ a, b ] → [ b + 1 , d ].So we get τ [ c,d ][ a,b ] if c + 1 ≤ a ≤ d + 1 ≤ b and 0 otherwise by (5.5) and (5.2). Sincewe may rewrite all even τ [ c,d ][ a,b ] in terms of i, r , we can now compute τ [ c,d ][ e,g ] · τ [ e,g ][ a,b ] if one of the transformations τ [ c,d ][ e,g ] and τ [ e,g ][ a,b ] is even and the other one is odd.Namely, the product is τ [ c,d ][ a,b ] if a + 1 ≤ c ≤ b + 1 ≤ d , and 0 otherwise. In moredetail, the assumption that exactly one of the transformations τ [ c,d ][ e,g ] and τ [ e,g ][ a,b ] iseven means that e + 1 ≤ c ≤ g + 1 ≤ d and e ≤ a ≤ g ≤ b , or c ≤ e ≤ d ≤ g and a + 1 ≤ e ≤ b + 1 ≤ g . The assumption a + 1 ≤ c ≤ b + 1 ≤ d becomes e ≤ a < c ≤ g + 1 ≤ b + 1 ≤ d or a < c ≤ e ≤ b + 1 ≤ d ≤ g in these two cases,respectively.Finally, any product with more than two factors δ vanishes because it may bedeformed in the extended diagram in Figure 2 so as to factor through one of thezeros on the boundary. We sum up our results about the multiplication in N T :(5.7) τ [ c,d ][ e,g ] · τ [ e,g ][ a,b ] = τ [ c,d ][ a,b ] = 0 ⇐⇒ c ≤ e ≤ a ≤ d ≤ g ≤ b,e ≤ a ≤ c − ≤ g ≤ b < d,a < c ≤ e ≤ b + 1 ≤ d ≤ g, Figure 1.
Natural transformations on filtrated K-theory for general n M [ n , n ] M [ n − , n ] M [ n − , n ] ··· M [ , n ] M [ , n ] M [ n − , n − ] M [ n − , n − ] ··· M [ , n − ] M [ , n − ] M [ n , n ] M [ n − , n − ] ··· M [ , n − ] M [ , n − ] M [ n − , n ] ............ M [ , ] M [ , ] M [ , n ] M [ , ] M [ , n ] M [ , n ] ii r i r i r i rr ii r i r i r δ r i ii r i r δ r i rr i i δ r i δ i and τ [ c,d ][ e,g ] · τ [ e,g ][ a,b ] = 0 otherwise. We write[ a, b ] → [ e, g ] → [ c, d ] ⇐⇒ τ [ c,d ][ e,g ] · τ [ e,g ][ a,b ] = τ [ c,d ][ a,b ] = 0 . It can happen that [ a, b ] → [ e, g ], [ e, g ] → [ c, d ] and [ a, b ] → [ c, d ], but not [ a, b ] → [ e, g ] → [ c, d ]; that is, τ [ c,d ][ e,g ] · τ [ e,g ][ a,b ] = 0 although τ [ c,d ][ a,b ] , τ [ e,g ][ a,b ] , τ [ c,d ][ e,g ] = 0.Given 1 ≤ a ≤ b ≤ n , there are one or two proper natural transformationsto M [ a, b ] that are shortest in the sense that all others factor through them. If a < b < n , these are i : M [ a + 1 , b ] → M [ a, b ] and r : M [ a, b + 1] → M [ a, b ]. If1 < a < b = n , then r above is replaced by δ : M [1 , a − → M [ a, n ]. One of thesemaps is missing if a = b or if ( a, b ) = (1 , n ), that is, on the two outer diagonals OMPUTING OBSTRUCTION CLASSES 25
Figure 2.
Complete diagram of natural transformations on fil-trated K-theory for n = 3 M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] M [ , ] ii rr i δ r i δ i r i rr in the diagram in Figure 2 (now for general n ). We define M [ a + 1 , a ] := 0 for0 ≤ a ≤ n to make this a special case of the generic case.Now we build an I X fil -projective resolution of A of length 1. This has not yetbeen done in [12], where only the existence of such a resolution is proven. For1 ≤ a ≤ b ≤ n , let M [ a, b ] ss be the quotient of M [ a, b ] by the images of all propernatural transformations to M [ a, b ] or, equivalently, by the images of the two shortestnatural transformations: M [ a, b ] ss := ( M [ a, b ] (cid:14) ( i ( M [ a + 1 , b ]) + r ( M [ a, b + 1])) if b < n,M [ a, b ] (cid:14) ( i ( M [ a + 1 , b ]) + δ ( M [1 , a − b = n (compare [12, Definition 3.7 and Lemma 3.8]). Choose a resolution(5.8) Q [ a, b ] Q [ a, b ] M [ a, b ] ss d d of M [ a, b ] ss by countable Z / i = 0 ,
1, letˆ Q i [ a, b ] := R [ a,b ] ⊗ Z Q i [ a, b ] , where the tensor product is defined as in (2.4). Since R [ a,b ] ⊗ Z Q i [ a, b ] is a directsum of copies of suspensions of R [ a,b ] , the definition of R [ a,b ] as a representing objectimplies(5.9) KK X ( ˆ Q i [ a, b ] , B ) ∼ = Hom (cid:0) Q i [ a, b ] , K ∗ ( B [ a, b ]) (cid:1) for all C ∗ -algebras B over X ; here Hom means grading-preserving group homomor-phisms. This property characterises ˆ Q i [ a, b ] uniquely up to isomorphism in KK X .Equation (5.9) implies a similar description of the Z / KK X ∗ ( ˆ Q i [ a, b ] , B ), replacing Hom by group homomorphisms that need not respectthe grading. Given a group homomorphism g : Q i [ a, b ] → K ∗ ( B [ a, b ]), let g denotethe corresponding element of KK X ∗ ( ˆ Q i [ a, b ] , B ).Since the Z / Q [ a, b ] is free, the homomorphism d in (5.8)lifts to a grading-preserving homomorphism(5.10) f [ a, b ] : Q [ a, b ] → M [ a, b ] = K ∗ ( A [ a, b ]) . Let f [ a, b ] ∈ KK X ( ˆ Q [ a, b ] , A ) correspond to f [ a, b ] by (5.9). Let P i := M ≤ a ≤ b ≤ n ˆ Q i [ a, b ]for i = 0 ,
1. The objects ˆ Q i [ a, b ] and P i for i = 0 , I X fil -projective becauseof (5.9). There is a unique element f ∈ KK X ( P , A ) that restricts to f [ a, b ] onthe summand ˆ Q i [ a, b ]. Lemma 5.11.
The map
F K ( f ) : F K ( P ) → M := F K ( A ) is surjective. Its kernelis isomorphic to F K ( P ) as an N T -module.Proof.
Let f ∗ := F K ( f ). The N T -module M ss is defined as the quotient M/ N T nil · M for a certain ideal N T nil in N T . It follows that the functor M M ss is rightexact. Even more, it is isomorphic to the tensor product functor with the right N T -module
N T ss . This follows from the extension of N T -modules
N T nil (cid:26)
N T (cid:16)
N T ss and the right exactness of tensor product functors.Right exactness implies (coker f ∗ )[ a, b ] ss = coker( f ∗ [ a, b ] ss ) for all 1 ≤ a ≤ b ≤ n .The projective N T -module
F K ( P ) has F K ( P )[ a, b ] ss ∼ = Q [ a, b ]. Hence ( f ∗ ) ss issurjective by construction of f , and the N T -module coker f ∗ satisfies (coker f ∗ ) ss = 0.This implies coker f ∗ = 0 by [12, Proposition 3.10]. That is, f ∗ is surjective. Let N := ker F K ( f ). So there is an extension N (cid:26) F K ( P ) (cid:16) M of N T -modules.Since M and F K ( P ) are exact N T -modules, they satisfy Tor N T ( N T ss , M ) = 0and Tor N T ( N T ss , F K ( P )) = 0 by [12, Lemma 3.13]. Hence N [ a, b ] ss → F K ( P )[ a, b ] ss → M [ a, b ] ss is a short exact sequence for all 1 ≤ a ≤ b ≤ n and Tor N T ( N T ss , N ) = 0. Since F K ( P )[ a, b ] ss ∼ = Q [ a, b ], this implies N [ a, b ] ss ∼ = Q [ a, b ]. Now [12, Theorem 3.12]shows that N is a projective N T -module. In fact, the proof of this theorem showsthat N ∼ = F K ( P ). More precisely, the quotient maps N [ a, b ] (cid:16) N [ a, b ] ss splitbecause N [ a, b ] ss ∼ = Q [ a, b ] is free. Let(5.12) ϕ [ a, b ] : Q [ a, b ] ∼ = N [ a, b ] ss → N [ a, b ] ⊆ F K ( P )[ a, b ]be sections. They induce an N T -module homomorphism
F K ( P ) → N by theuniversal property of the “free” N T -module
F K ( P ). And the proof of [12, Theo-rem 3.12] shows that it is an isomorphism. (cid:3) Disregarding the Z / F K ( P )[ a, b ] = M [ c,d ] → [ a,b ] Q [ c, d ] , that is, the sum runs over all 1 ≤ c ≤ d ≤ n with a ≤ c ≤ b ≤ d or c + 1 ≤ a ≤ d + 1 ≤ b . So the map ϕ [ a, b ] in (5.12) has components ϕ [ c,d ][ a,b ] : Q [ a, b ] → Q [ c, d ]for [ c, d ] → [ a, b ]. Since ϕ [ a, b ] is even, the map ϕ [ c,d ][ a,b ] has the same parity as τ [ a,b ][ c,d ] , that is, it is grading-preserving if a ≤ c ≤ b ≤ d and grading-reversing if c + 1 ≤ a ≤ d + 1 ≤ b . The image of ϕ [ a, b ] for 1 ≤ a ≤ b ≤ n is contained OMPUTING OBSTRUCTION CLASSES 27 in N = ker F K ( f ), that is, F K ( f ) ◦ ϕ [ a, b ] = 0 as a map Q [ a, b ] → K ∗ ( A [ a, b ]).Unravelling the definition of F K ( f ), this becomes(5.13) X [ c,d ] → [ a,b ] τ [ a,b ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] = 0 : Q [ a, b ] → K ∗ ( A [ a, b ]) . The objects P and P are I X fil -projective, and F K is fully faithful on I X fil -projectiveobjects. So the arrow F K ( P ) → F K ( P ) with components ϕ [ c,d ][ a,b ] for [ c, d ] → [ a, b ]lifts uniquely to an arrow ϕ ∈ KK X ( P , P ). More precisely, the map ϕ is givenby a matrix of maps R [ c,d ] ⊗ Q [ c, d ] → R [ a,b ] ⊗ Q [ a, b ] for all 1 ≤ c ≤ d ≤ n and1 ≤ a ≤ b ≤ n . The entries of this matrix are (cid:16)(cid:0) τ [ c,d ][ a,b ] (cid:1) ∗ ⊗ ϕ [ a,b ][ c,d ] (cid:17) , that is,(5.14) ϕ = (cid:16)(cid:0) τ [ c,d ][ a,b ] (cid:1) ∗ ⊗ ϕ [ a,b ][ c,d ] (cid:17) [ c,d ] → [ a,b ] : M ≤ c ≤ d ≤ n R [ c,d ] ⊗ Q [ c, d ] → M ≤ a ≤ b ≤ n R [ a,b ] ⊗ Q [ a, b ] . Here we use the convention that τ [ c,d ][ a,b ] = 0 if not [ a, b ] → [ c, d ].Lemma 5.11 says that (5.1) with f and ϕ as above is I X fil -exact, that is, F K applied to (5.1) is an exact sequence. The I X fil -exactness of (5.1) says that thefunctor B K ∗ ( B [ a, b ]) maps it to an exact sequence for each 1 ≤ a ≤ b ≤ n . Infact, this gives projective resolutions. We write them down explicitly: Lemma 5.15.
Let ≤ a ≤ b ≤ n . Then (5.16) M [ c,d ] → [ a,b ] Q [ c, d ] (cid:0) ϕ [ e,g ][ c,d ] (cid:1) [ c,d ] → [ e,g ] → [ a,b ] −−−−−−−−−−−−−−−→ M [ e,g ] → [ a,b ] Q [ e, g ] (cid:0) τ [ a,b ][ e,g ] ◦ f [ e,g ] (cid:1) −−−−−−−−−→ K ∗ ( A [ a, b ]) is a free resolution. Here (cid:0) ϕ [ e,g ][ c,d ] (cid:1) [ c,d ] → [ e,g ] → [ a,b ] means that the matrix entry is ϕ [ e,g ][ c,d ] if [ c, d ] → [ e, g ] → [ a, b ] as described in (5.7) , and otherwise. The boundary maps in (5.16) are inhomogeneous, that is, the matrix entries ofthe maps may have even or odd degree.
Proof.
Equation (5.2) computes the group K ∗ ( R [ c,d ] ([ a, b ])) ∼ = N T ∗ ([ c, d ] , [ a, b ]): itis Z in even or odd degree if [ c, d ] → [ a, b ] and 0 otherwise. Therefore, P i [ a, b ] ∼ = L [ c,d ] → [ a,b ] Q i [ c, d ] for i = 0 ,
1, disregarding the grading. The map (cid:16)(cid:0) τ [ c,d ][ e,g ] (cid:1) ∗ ⊗ ϕ [ e,g ][ c,d ] (cid:17) between the summands R [ c,d ] ⊗ Q [ c, d ] in P and R [ e,g ] ⊗ Q [ e, g ] in P induces the map ϕ [ e,g ][ c,d ] : Q [ c, d ] → Q [ e, g ] if [ c, d ] → [ e, g ] → [ a, b ], and 0 otherwise,compare (5.7). The map K ∗ ( P [ a, b ]) → K ∗ ( A [ a, b ]) corresponds to a family of mapsK ∗ ( R [ c,d ] [ a, b ]) ⊗ Q [ c, d ] ∼ = K ∗ ( R [ c,d ] [ a, b ] ⊗ Q [ c, d ]) → K ∗ ( A [ a, b ])for 1 ≤ c ≤ d ≤ n . As above, K ∗ ( R [ c,d ] [ a, b ]) = 0 only if [ c, d ] → [ a, b ], and then themap Q [ c, d ] → K ∗ ( A [ a, b ]) induced by f : P → A is τ [ a,b ][ c,d ] ◦ f [ c, d ]. (cid:3) We have reached the first milestone in the computation of the obstruction class:the I X fil -projective resolution (5.1). It is explicit enough to express the obstructionclass that comes from filtrated K-theory in the terms of Theorem 4.9, namely, asbeing represented by a family of elements in Ext (cid:0) K ∗ ( A [ e + 1 , n ]) , K ∗ ( A [ e, n ]) (cid:1) for e = 1 , . . . , n −
1. We compute these Ext-groups with the resolutions in (5.16). Sothe obstruction class corresponds to a sequence of maps δ e : M [ a,b ] → [ e +1 ,n ] Q [ a, b ] → K ∗ ( A [ e, n ]) , e = 1 , . . . , n − . In turn, each δ e is given by maps δ [ a,b ] e : Q [ a, b ] → K ∗ ( A [ e, n ]) for all 1 ≤ a ≤ b ≤ n with [ a, b ] → [ e + 1 , n ]. The following theorem computes these maps δ [ a,b ] e . It is themain result of this section. Section 5.1 is dedicated to its proof. Theorem 5.17.
Let δ [ a,b ] e := X [ c,d ] → [ a,n ] → [ a,b ] τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] if a = e and b < n, otherwise.The resulting family of maps ( δ e ) ≤ e First we examine the smallerinvariant F X K : KK X → A X . This takes the part of filtrated K-theory consisting ofK ∗ ( A [ a, n ]) for 1 ≤ a ≤ n with the maps i between them because the minimal opensubset containing a is U a = [ a, n ]. So the diagram F X K ( A ) is simply the first rowin the diagram in Figure 1. We have i a C ∼ = R [ a,n ] for 1 ≤ a ≤ n because bothobjects represent the same functor A K ∗ ( A [ a, n ]). So R [ a,n ] for 1 ≤ a ≤ n is I X K -projective, and F X K ( R [ a,n ] ) is the diagram(5.18) P [ a,n ] := F X K ( R [ a,n ] ) = (cid:16) Z = Z = · · · = Z | {z } a times ← · · · = 0 | {z } n − a times (cid:17) . in A X . If 1 ≤ a ≤ b ≤ n − 1, then F X K ( R [ a,b ] ) is the diagram(5.19) P [ a,b ] := F X K ( R [ a,b ] ) = (cid:16) · · · = 0 | {z } a times ← Z − = · · · = Z − | {z } b +1 − a times ← · · · = 0 | {z } n − − b times (cid:17) because of the formula for K ∗ ( R [ a,b ] ([ c, n ])) in (5.2). Thus the objects R [ a,b ] for1 ≤ a ≤ b ≤ n are even if b = n and odd if b < n .The formula for the obstruction class in Theorem 3.2 uses the parity-reversingpart of ϕ ∈ KK X ( P , P ). This is described by the following lemma: Lemma 5.20. The component (cid:0) τ [ a,b ][ c,d ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] of ϕ is parity-reversing if and onlyif [ c, d ] → [ a, n ] → [ a, b ] and b < n .Proof. If τ [ a,b ][ c,d ] = 0, then either a ≤ c ≤ b ≤ d or c + 1 ≤ a ≤ d + 1 ≤ b . Themap (cid:0) τ [ a,b ][ c,d ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] always belongs to KK X ( ˆ Q [ a, b ] , ˆ Q [ c, d ]). So ϕ [ c,d ][ a,b ] : Q [ a, b ] → Q [ c, d ] is parity-preserving in the first case and parity-reversing in the second case.The object R [ a,b ] is even if b = n and odd if b < n .First let a ≤ c ≤ b ≤ d . If b = n , then d = n as well, so that R [ a,b ] and R [ c,d ] have the same parity, and ϕ [ c,d ][ a,b ] preserves parity. So we get a parity-preservingcomponent of ϕ . For the same reasons, we get a parity-preserving component if a ≤ c ≤ b ≤ d < n , and a parity-reversing component if a ≤ c ≤ b < d = n .Now let c + 1 ≤ a ≤ d + 1 ≤ b , so that ϕ [ c,d ][ a,b ] reverses parity. If b = n , then d < n , so that R [ a,b ] and R [ c,d ] have opposite parity. Hence (cid:0) τ [ a,b ][ c,d ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] isparity-preserving altogether. If b < n , then d < n and so R [ a,b ] and R [ c,d ] have the OMPUTING OBSTRUCTION CLASSES 29 same parity. Thus (cid:0) τ [ a,b ][ c,d ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] is parity-reversing. Inspection shows that theparity-reversing components are exactly those for which [ c, d ] → [ a, n ] → [ a, b ] asin (5.7) and b < n . (cid:3) Let 1 ≤ a ≤ b < n . The exact triangle(5.21) Σ R [ a,n ] i ∗ −→ Σ R [ b +1 ,n ] δ ∗ −→ R [ a,b ] r ∗ −→ R [ a,n ] in KK X is I X K -exact, that is, F X K ( r ∗ ) = 0. Since Σ R [ a,n ] and Σ R [ b +1 ,n ] are I X K -projective, it is an I X K -projective resolution of R [ a,b ] of length 1. Since weallow both odd and even arrows in diagrams, we may drop the suspensions in (5.21).For 1 ≤ a ≤ b ≤ n , let M ( a, b ) := ( a if b = n,b + 1 if b < n. Then (cid:0) τ [ M ( a,b ) ,n ][ a,b ] (cid:1) ∗ : R [ M ( a,b ) ,n ] → R [ a,b ] is an I X K -epimorphism both for b = n and b < n . Its cone is R [ a,n ] if b < n and 0 if b = n . Now we write down I X K -projectiveresolutions of P i for i = 0 , 1. Let P i := M ≤ a ≤ b ≤ n R [ M ( a,b ) ,n ] ⊗ Q i [ a, b ] , (5.22) P i := M ≤ a ≤ b The image of the parity-reversing part ϕ − of ϕ in Ext I X K ( P , P ) is the map P = M ≤ a ≤ b In terms of the matrix description of ϕ , each matrix entry (cid:0) τ [ a,b ][ c,d ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] has even or odd parity and thus belongs to either Hom or Ext, respectively. ByLemma 5.20, the entry belongs to ϕ − if and only if τ [ a,b ][ c,d ] factors through r ∗ : R [ a,b ] →R [ a,n ] . In this case, it factors as (cid:16)(cid:0) τ [ a,n ][ c,d ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] (cid:17) ◦ ( r ∗ ⊗ id). Since r ∗ is theboundary map in (5.21), this exhibits a map P → P . The map from Ext A X toKK X in the UCT is defined by composing with the boundary map in the exacttriangle that contains the given resolution. So the map P → P found above isthe relevant component of ϕ . The formula in the lemma follows. (cid:3) According to the recipe in Theorem 3.2, the obstruction class in Ext I X K (Σ A, A ) isthe composite of the parity-reversing part of ϕ , viewed as an element of Ext I X K ( P , P ),with f ∈ KK X ( P , A ) and with the class of the extension (5.1) in Ext I X K ( A, P ).Composing the two extensions gives a length-2 resolution(5.26) P (cid:26) P → P (cid:16) A. The component R [ a,n ] ⊗ Q [ a, b ] → A in the composite map P → P → A is themap R [ a,n ] ⊗ Q [ a, b ] → A that corresponds to(5.27) X [ c,d ] → [ a,n ] → [ a,b ] τ [ a,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] : Q [ a, b ] → K ∗ ( A [ a, n ])under the isomorphism (5.9). Here the sum runs only over those [ c, d ] with [ c, d ] → [ a, n ] → [ a, b ] as in Lemma 5.25. In contrast, the sum over all [ c, d ] with [ c, d ] → [ a, b ]is 0 by (5.13).In a sense, we have now computed the obstruction class. The length-2 resolutionin (5.26) is, however, different from the one that is implicitly used in Theorem 4.9to compute the relevant Ext -group and the obstruction class in it. To translatethe formula for the obstruction class that we get from filtrated K-theory into thesetting of Theorem 4.9, we must compare the underlying length-2 resolutions. First,we replace the resolution in (5.26) by one that is I X K -projective.The entries P and P are already I X K -projective, and (5.24) is an I X K -projectiveresolution of P . The objects P i and P ij are all sums over 1 ≤ a ≤ b ≤ n , withsummands of the form R [ x,y ] ⊗ Q i [ a, b ] for suitable x, y depending on a, b ; thesummands in P i are 0 for b = n . Let ( τ ∗ ⊗ ϕ ) denote the map between these sumsfor i = 1 to those for i = 0 with matrix entries (cid:0) τ [ x,y ][ z,w ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] : R [ x,y ] ⊗ Q [ a, b ] → R [ z,w ] ⊗ Q [ c, d ] . As usual, τ [ x,y ][ z,w ] = 0 if N T ∗ ([ z, w ] , [ x, y ]) = 0. Lemma 5.28. There is a commuting diagram P P P P P P L τ ∗ ⊗ id( τ ∗ ⊗ ϕ ) L τ ∗ ⊗ id( τ ∗ ⊗ ϕ ) ( τ ∗ ⊗ ϕ ) L τ ∗ ⊗ id L τ ∗ ⊗ id Proof. We compare maps between direct sums by comparing their matrix coeffi-cients. For the two composite maps P → P , these are maps R [ a,n ] ⊗ Q [ a, b ] →R [ M ( c,d ) ,n ] ⊗ Q [ c, d ] for 1 ≤ a ≤ b < n and 1 ≤ c ≤ d ≤ n . The compositemap through P is (cid:0) τ [ a,n ][ M ( c,d ) ,n ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] if [ a, n ] ← [ M ( a, b ) , n ] ← [ M ( c, d ) , n ],and 0 otherwise; and the composite map through P is (cid:0) τ [ a,n ][ M ( c,d ) ,n ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] if[ a, n ] ← [ c, n ] ← [ M ( c, d ) , n ] and d < n , and 0 otherwise; the condition d < n comes OMPUTING OBSTRUCTION CLASSES 31 in because P contains only summands R [ c,n ] ⊗ Q [ c, d ] with 1 ≤ c ≤ d < n . Inboth cases, the map vanishes unless [ c, d ] → [ a, b ] because of the factor ϕ [ c,d ][ a,b ] . Weclaim that if [ c, d ] → [ a, b ] and b < n , then [ a, n ] ← [ M ( a, b ) , n ] ← [ M ( c, d ) , n ] ifand only if [ a, n ] ← [ c, n ] ← [ M ( c, d ) , n ] and d < n ; here M ( a, b ) = b + 1 because b < n . Indeed, if d = n , then M ( c, d ) = c , and [ a, n ] ← [ b + 1 , n ] ← [ c, n ] means a ≤ b + 1 ≤ c , which contradicts [ c, n ] → [ a, b ]. If d < n , then M ( c, d ) = d + 1. Then[ a, n ] ← [ c, n ] ← [ d + 1 , n ] and [ a, n ] ← [ M ( a, b ) , n ] ← [ M ( c, d ) , n ] are equivalent to a ≤ c ≤ d + 1 and a ≤ b + 1 ≤ d + 1, respectively. If [ c, d ] → [ a, b ], both conditionssay that we are in the case a ≤ c ≤ b ≤ d . The computations above show that thetwo maps P → P are equal.Now consider the two maps P → P . Its matrix coefficients are maps R [ M ( a,b ) ,n ] ⊗ Q [ a, b ] → R [ c,d ] ⊗ Q [ c, d ] , ≤ a ≤ b ≤ n, ≤ c ≤ d ≤ n. As above, the composite maps through P and P are (cid:0) τ [ M ( a,b ) ,n ][ c,d ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] or 0. Forthe maps through P and P , the former case occurs if [ M ( a, b ) , n ] ← [ a, b ] ← [ c, d ]or [ M ( a, b ) , n ] ← [ M ( c, d ) , n ] ← [ c, d ], respectively. We may assume [ a, b ] ← [ c, d ] and [ M ( a, b ) , n ] ← [ c, d ] because otherwise ϕ [ c,d ][ a,b ] = 0 or τ [ M ( a,b ) ,n ][ c,d ] = 0.Under these assumptions, [ M ( a, b ) , n ] ← [ a, b ] ← [ c, d ] always holds by (5.7). And[ c, d ] → [ M ( a, b ) , n ] implies [ M ( a, b ) , n ] ← [ M ( c, d ) , n ] ← [ c, d ] because any naturaltransformation K ∗ ( A [ c, d ]) → K ∗ ( A [ e, n ]) for some 1 ≤ e ≤ n factors through τ [ M ( c,d ) ,n ][ c,d ] . So the two maps P → P are equal as well. (cid:3) Using also the resolution (5.1) of A , we get the following I X K -projective resolutionof A : P (cid:18) − ( τ ∗ ⊗ ϕ ) L τ ∗ ⊗ id (cid:19) −−−−−−−−−−→ P ⊕ P (cid:0)L τ ∗ ⊗ id ( τ ∗ ⊗ ϕ ) (cid:1) −−−−−−−−−−−−−−−−−−→ P L f [ a,b ] ◦ ( τ ∗ ⊗ id) −−−−−−−−−−−−→ A. The computation of the obstruction class in Theorem 4.8 starts with the following I X -projective resolution of length 1 in KK X :(5.29) n − M b =1 R [ b,n ] ⊗ A [ b + 1 , n ] (cid:26) n M b =1 R [ b,n ] ⊗ A [ b, n ] (cid:16) A. Since R [ b,n ] = i b ( C ) in the notation of Section 2.2, we have R [ b,n ] ⊗ A [ b, n ] ∼ = i b ( A [ b, n ]). The restriction of the second map in (5.29) to this direct summand isthe one that corresponds to the identity map on A [ b, n ] under the isomorphismin (2.15). The first map in (5.29), restricted to the summand R [ b,n ] ⊗ A [ b + 1 , n ], isthe difference of the two maps( τ [ b,n ][ b +1 ,n ] ) ∗ ⊗ id : R [ b,n ] ⊗ A [ b + 1 , n ] → R [ b +1 ,n ] ⊗ A [ b + 1 , n ] , id ⊗ τ [ b,n ][ b +1 ,n ] : R [ b,n ] ⊗ A [ b + 1 , n ] → R [ b,n ] ⊗ A [ b, n ] , where τ [ b,n ][ b +1 ,n ] denotes the inclusion of A [ b + 1 , n ] into A [ b, n ]; we could have written i for τ [ b,n ][ b +1 ,n ] as in Figure 1. It is shown in Section 2.2 that this sequence is I X -exact.And it is easy to prove this directly.The projective resolutions of Abelian groups in (5.16) imply that there is an I X K -projective resolution M [ c,d ] → [ b +1 ,n ] R [ b,n ] ⊗ Q [ c, d ] (cid:26) M [ c,d ] → [ b +1 ,n ] R [ b,n ] ⊗ Q [ c, d ] (cid:16) n − M b =1 R [ b,n ] ⊗ A [ b +1 , n ] . Splicing it with the resolution in (5.29) gives an I X K -exact chain complex W (cid:26) W → n M b =1 R [ b,n ] ⊗ A [ b, n ] (cid:16) A with(5.30) W i := n − M b =1 M [ c,d ] → [ b +1 ,n ] R [ b,n ] ⊗ Q i [ c, d ] , i = 0 , . Next we are going to compare the two I X K -exact chain complexes built above. Weare going to build maps γ ij and δ for 0 ≤ i, j ≤ δ gives the obstruction class:(5.31) P P ⊕ P P AW W L ne =1 R [ e,n ] ⊗ A [ e, n ] AA γ ( γ γ ) γ δ We describe maps between direct sums through matrices of maps between the directsummands. Recall that W i is defined in (5.30) and that P i := M ≤ a ≤ b ≤ n R [ M ( a,b ) ,n ] ⊗ Q i [ a, b ] , P i := M ≤ a ≤ b 1. The matrix coefficients of γ are maps γ e, [ a,b ]00 : R [ M ( a,b ) ,n ] ⊗ Q [ a, b ] → R [ e,n ] ⊗ A [ e, n ]for 1 ≤ a ≤ b ≤ n and 1 ≤ e ≤ n . We let γ e, [ a,b ]00 = 0 if e = M ( a, b ). Let e = M ( a, b ).Then γ e, [ a,b ]00 corresponds to a map( γ e, [ a,b ]00 ) [ : Q [ a, b ] → K ∗ ( R [ e,n ] [ e, n ] ⊗ A [ e, n ])under the isomorphism (5.9). We have already used above that R [ e,n ] ⊗ A [ e, n ] ∼ = i e ( A [ e, n ]); so(5.32) R [ e,n ] [ e, n ] ⊗ A [ e, n ] ∼ = A [ e, n ] . Using this isomorphism implicitly, we let( γ e, [ a,b ]00 ) [ := τ [ e,n ][ a,b ] ◦ f [ a, b ] : Q [ a, b ] f [ a,b ] −−−→ K ∗ ( A [ a, b ]) τ [ e,n ][ a,b ] −−−→ K ∗ ( A [ e, n ]) . As usual, this is 0 unless [ a, b ] → [ e, n ]. The map γ : P → W is given by amatrix of maps γ e, [ c,d ] , [ a,b ]01 : R [ a,n ] ⊗ Q [ a, b ] → R [ e,n ] ⊗ Q [ c, d ]for 1 ≤ a ≤ b < n , 1 ≤ e < n , and 1 ≤ c ≤ d ≤ n with [ c, d ] → [ e + 1 , n ]. We let γ e, [ c,d ] , [ a,b ]01 := ((cid:0) τ [ a,n ][ e,n ] (cid:1) ∗ ⊗ id Q [ a,b ] if [ a, b ] = [ c, d ] , a = c and b = d < n , then [ c, d ] → [ e + 1 , n ] if and only if a ≤ e ≤ b , so that τ [ a,n ][ e,n ] = 0 in this formula.The map γ : P → W is given by a matrix of maps γ e, [ c,d ] , [ a,b ]10 : R [ M ( a,b ) ,n ] ⊗ Q [ a, b ] → R [ e,n ] ⊗ Q [ c, d ] OMPUTING OBSTRUCTION CLASSES 33 for 1 ≤ a ≤ b ≤ n , 1 ≤ e < n , and 1 ≤ c ≤ d ≤ n with [ c, d ] → [ e + 1 , n ]. We let γ e, [ c,d ] , [ a,b ]10 := ((cid:0) τ [ M ( a,b ) ,n ][ e,n ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] if M ( a, b ) ≤ e < M ( c, d ) , γ : P → W is given by a matrix of maps γ e, [ c,d ] , [ a,b ]11 : R [ a,n ] ⊗ Q [ a, b ] → R [ e,n ] ⊗ Q [ c, d ]for 1 ≤ a ≤ b < n , 1 ≤ e < n , and 1 ≤ c ≤ d ≤ n with [ c, d ] → [ e + 1 , n ]. We let γ e, [ c,d ] , [ a,b ]11 := ((cid:0) τ [ a,n ][ e,n ] (cid:1) ∗ ⊗ id Q [ a,b ] if a = c, b = d, . The map δ is given by a family of maps δ b, [ c,d ] : R [ b,n ] ⊗ Q [ c, d ] → A for 1 ≤ b < n and [ c, d ] → [ b + 1 , n ]. These correspond to maps δ [b, [ c,d ] : Q [ c, d ] → K ∗ ( A [ b, n ]) bythe isomorphism (5.9). We define δ so that the maps δ [b, [ c,d ] are the maps denotedby δ [ c,d ] b in Theorem 5.17.Now we must prove that the squares in the diagram commute. We begin on theright, comparing the two maps P → A . Its restrictions R [ M ( a,b ) ,n ] ⊗ Q [ a, b ] → A correspond to maps Q [ a, b ] → K ∗ ( A [ M ( a, b ) , n ]) under the isomorphism (5.9). Thismap is τ [ M ( a,b ) ,n ][ a,b ] ◦ f [ a, b ] both for the direct boundary map P → A and for themap through L ne =1 R [ e,n ] ⊗ A [ e, n ]. So this square commutes.Next we compare the two maps from P ⊕ P to L nb =1 R [ b,n ] ⊗ A [ b, n ]. We firstconsider the restriction to P , then to P . The matrix coefficients of the mapon P are maps R [ a,n ] ⊗ Q [ a, b ] → R [ e,n ] ⊗ A [ e, n ] for 1 ≤ a ≤ b < n and 1 ≤ e ≤ n .Such maps correspond to group homomorphisms Q [ a, b ] → K ∗ ( R [ e,n ] [ a, n ] ⊗ A [ e, n ]).Recall that R [ e,n ] [ a, n ] = C if a ≤ e and 0 otherwise. So we may assume without lossof generality that a ≤ e , and then we get corresponding maps Q [ a, b ] → K ∗ ( A [ e, n ]).The map γ picks out the summand with e = M ( a, b ) = b + 1, and (cid:0) τ [ a,n ][ M ( a,b ) ,n ] (cid:1) ∗ induces the identity map Z ∼ = K ∗ (cid:0) R [ a,n ] ([ b + 1 , n ]) (cid:1) → K ∗ (cid:0) R [ b +1 ,n ] ([ b + 1 , n ]) (cid:1) ∼ = Z .Therefore, the map in the square through γ contributes the map δ e,b +1 τ [ e,n ][ a,b ] ◦ f [ a, b ] : Q [ a, b ] → K ∗ ( A [ e, n ]) . When we map through γ instead, then we first map R [ a,n ] ⊗ Q [ a, b ] to the directsum of R [ g,n ] ⊗ Q [ a, b ] over all g ∈ [ a, b ] using τ ∗ ⊗ id and then apply the boundarymap on W . This gives a contribution in K ∗ ( A [ e, n ]) if g = e or g = e − 1, andthese two contributions cancel each other for a < e ≤ b . For e = b + 1, we get thesame term as for the map that goes through P . And we get 0 for e = a because τ [ a,n ][ a +1 ,n ] ◦ τ [ a +1 ,n ][ a,b ] = τ [ a,n ][ a,b ] = 0. So the two maps are equal on P .The matrix coefficients of the two maps on P are maps R [ M ( a,b ) ,n ] ⊗ Q [ a, b ] →R [ e,n ] ⊗ A [ e, n ] for 1 ≤ a ≤ b ≤ n and 1 ≤ e ≤ n . As above, we may assume M ( a, b ) ≤ e because otherwise any such map is zero. And then maps R [ M ( a,b ) ,n ] ⊗ Q [ a, b ] → R [ e,n ] ⊗ A [ e, n ] correspond to maps Q [ a, b ] → K ∗ ( A [ e, n ]). We shallexamine the difference of the map through P and the map through W . We firstconsider the map through P . It first applies the matrix τ ∗ ⊗ ϕ , going to the directsum of R [ M ( c,d ) ,n ] ⊗ Q [ c, d ] for 1 ≤ c ≤ d ≤ n . The map Q [ a, b ] → K ∗ ( A [ e, n ]) forthe composite map through R [ M ( c,d ) ,n ] ⊗ Q [ c, d ] is δ M ( c,d ) ,e τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] . Sowe get the sum of these terms over all 1 ≤ c ≤ d ≤ n . When we apply γ : P → W ,then we apply the maps (cid:0) τ [ M ( a,b ) ,n ][ g,n ] (cid:1) ∗ ⊗ ϕ [ c,d ][ a,b ] to the direct summands R [ g,n ] ⊗ Q [ c, d ]of W , where 1 ≤ c ≤ d ≤ n and 1 ≤ g < n are such that [ c, d ] → [ g + 1 , n ] and M ( a, b ) ≤ g < M ( c, d ). The condition [ c, d ] → [ g + 1 , n ] is equivalent to c > g if d = n and c ≤ g < d + 1 if d < n . So the set of g that are allowed is an interval [ x, y ] or empty. The upper bound is always y := M ( c, d ) − 1. The lower bound x is M ( a, b ) if d = n or the maximum of c and M ( a, b ) if d < n . By convention, weredefine x := M ( c, d ) if the lower bound is bigger than M ( c, d ). So g runs throughthe interval [ x, y ] if x ≤ y , and otherwise x = y + 1 = M ( c, d ) and the set ofpossible g is empty.We must compose γ with the boundary map on W . As above, this onlycontributes to the map R [ M ( a,b ) ,n ] ⊗ Q [ a, b ] → R [ e,n ] ⊗ A [ e, n ] if g = e or g = e − 1. And the contribution to the corresponding map Q [ a, b ] → K ∗ ( A [ e, n ]) is − τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] if g = e and + τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] if g = e − 1. The contributionsfor g = e and g = e − g inthe interval [ x, y ] above, we get − τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] if e = x , τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] if e = y + 1, and 0 otherwise. So we get the map( δ e,y +1 − δ e,x ) · τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] : Q [ a, b ] → K ∗ ( A [ e, n ]) . This formula remains correct if no g is allowed because then x = y + 1. Since y + 1 = M ( c, d ), the map involving δ e,y +1 is equal to the one that we get from themap through P . So when we take the difference of the two maps in the square,this term is cancelled. We remain with(5.33) X [ c,d ] → [ a,b ] δ e,x · τ [ e,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] , where x depends on a, b, c, d as above. Recall that we only need the case e ≤ M ( a, b ).We are going to prove that the sum in (5.33) vanishes under this assumption. Firstwe have to study the lower bound x for the different order relations among a, b, c, d .We may assume [ c, d ] → [ a, b ] because otherwise ϕ [ c,d ][ a,b ] = 0. So either a ≤ c ≤ b ≤ d or c + 1 ≤ a ≤ d + 1 ≤ b . It is also important whether b = n or d = n . First assume b = n . So M ( a, b ) = a . If a ≤ c ≤ b = d = n , then M ( c, d ) = c and so x = a . If c + 1 ≤ a ≤ d + 1 ≤ b = n , then d < n . So M ( c, d ) = d + 1 and x = a as well.Therefore, x = a and [ e, n ] = [ a, b ] whenever b = n . In this case, the sum in (5.33)vanishes because of (5.13).Now assume b < n , so M ( a, b ) = b + 1. If a ≤ c ≤ b < d = n , then M ( c, d ) = c c = M ( c, d ) and d = n . So we may assume d < n from now on. If c + 1 ≤ a , then it followsthat c + 1 ≤ a ≤ e < d + 1 ≤ b < n . Again, none of the three maps P → W give a non-zero contribution in this case. So we may assume a ≤ c . Then either a ≤ c ≤ e ≤ b ≤ d < n or a ≤ c ≤ b < e ≤ d < n . In the first case, the maps through W and P give contributions that cancel each other, and the map through P givesno contribution because e < M ( a, b ) = b + 1. In the second case, the maps through P and P give contributions that cancel each other, and the map through W vanishes because b < e . Hence we get 0 in all cases, as needed. This finishes theproof that the square of maps P → W commutes.Finally, we compute the composite map δ ◦ γ : P → A in our commutingdiagram. Consider the restriction to R [ a,n ] ⊗ Q [ a, b ] for some 1 ≤ a ≤ b < n . Thismap corresponds to a map Q [ a, b ] → K ∗ ( A [ a, n ]) by (5.9). The map δ [e, [ c,d ] = δ [ c,d ] e vanishes unless e = c , and the matrix coefficient γ c, [ c,d ] , [ a,b ]00 vanishes for [ a, b ] = [ c, d ]and is the identity map if [ a, b ] = [ c, d ]. So the composite map corresponds simplyto the map X [ c,d ] → [ a,n ] → [ a,b ] τ [ a,n ][ c,d ] ◦ f [ c, d ] ◦ ϕ [ c,d ][ a,b ] : Q [ a, b ] → K ∗ ( A [ a, n ]) . This is exactly the formula for the obstruction class in (5.27). This finishes theproof of Theorem 5.17.5.2. The case of extensions. We now specialise to the case n = 2. Then anobject of KK X is equivalent to a C ∗ -algebra extension I i (cid:26) A r (cid:16) A/I, where I = A [2], A = A [1 , 2] and A/I = A [1] and the maps are those in (5.3). Herewe abbreviate [1] = [1 , 1] and [2] = [2 , ( I ) K ( A ) K ( A/I )K ( A/I ) K ( A ) K ( I ) i ∗ r ∗ δδ r ∗ i ∗ The morphisms between the filtrated K-theory invariants are grading-preservingchain maps (morphisms of six-term exact sequences).The invariant in Theorem 2.12 is the KK-class [ i ] ∈ KK ( I, A ). The invariant inTheorem 4.8 is the induced map i ∗ : K ∗ ( I ) → K ∗ ( A ), together with the obstructionclass. To compute the latter, let i − ∈ Ext (cid:0) K ∗ ( I ) , K ∗ ( A ) (cid:1) be the parity-reversing part of [ i ] in the Universal Coefficient Theorem for KK ( I, A ). The obstructionclass is the image of i − in the cokernel of the map(5.35) Ext (cid:0) K ∗ ( I ) , K ∗ ( I ) (cid:1) ⊕ Ext (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) → Ext (cid:0) K ∗ ( I ) , K ∗ ( A ) (cid:1) , ( t I , t A ) i ◦ t I + t A ◦ i. It follows from our theory that i ∗ and the image of i − in the cokernel of (5.35)determine an object of B X uniquely up to KK X -equivalence. The cokernel comesin because there are isomorphisms of C ∗ -algebra extensions that act identicallyon K ∗ ( I ) and K ∗ ( A ), but have non-trivial components in Ext (cid:0) K ∗ ( I ) , K ∗ ( I ) (cid:1) orExt (cid:0) K ∗ ( A ) , K ∗ ( A ) (cid:1) . So the isomorphism class of an object in KK X does notdetermine i − uniquely. Only its image in the cokernel of (5.35) is unique. And ourtheory shows that isomorphism classes of pairs consisting of i ∗ ∈ Hom (cid:0) K ∗ ( I ) , K ∗ ( A ) (cid:1) and an element in the cokernel of (5.35) are in bijection with isomorphism classesof objects in the bootstrap class B X ⊆ KK X .We are going to compare this classification result with the filtrated K-theoryclassification by the long exact sequences in (5.34). The long exact sequence in (5.34)contains i ∗ and the extension(5.36) coker (cid:0) i : K ∗ ( I ) → K ∗ ( A ) (cid:1) (cid:26) K ∗ ( A/I ) (cid:16) ker (cid:0) i : K ∗ +1 ( I ) → K ∗ +1 ( A ) (cid:1) , and we may reconstruct the long exact sequence from these two pieces. Twoextensions as in (5.36) have the same class in Ext if and only if the long exactsequences associated to them (for the same i ∗ ) are isomorphic with an isomor-phism that is the identity on K ∗ ( I ) and K ∗ ( A ). Therefore, the filtrated K-theoryinvariant is equivalent to the pair consisting of i ∗ : K ∗ ( I ) → K ∗ ( A ) and a classin Ext (cid:0) ker( i ∗ ) , coker( i ∗ ) (cid:1) . Now the following proposition clarifies the relationshipbetween our different invariants: Proposition 5.37. The cokernel in (5.35) is naturally isomorphic to the group Ext A X (Σ A, A ) ∼ = Ext (cid:16) ker (cid:0) K ∗ +1 ( I ) i ∗ −→ K ∗ +1 ( A ) (cid:1) , coker (cid:0) K ∗ ( I ) i ∗ −→ K ∗ ( A ) (cid:1)(cid:17) . And the obstruction class is the class that corresponds to minus the extensionin (5.36) .Proof. Let G be a Z / (cid:0) K ∗ +1 ( A ) , G (cid:1) → Ext (cid:0) i ∗ (K ∗ +1 ( I )) , G (cid:1) is surjectiveand · · · → Ext (cid:0) i ∗ (K ∗ +1 ( I )) , G (cid:1) → Ext (cid:0) K ∗ +1 ( I ) , G (cid:1) → Ext (cid:0) ker( i ∗ : K ∗ +1 ( I ) → K ∗ +1 ( A )) , G (cid:1) → i ∗ : Ext (cid:0) K ∗ +1 ( A ) , G (cid:1) → Ext (cid:0) K ∗ +1 ( I ) , G (cid:1) is Ext(ker( i ∗ ) , G ). If we let G := K ∗ ( A ), then we may identify the cokernel of themap in (5.35) with the cokernel of the mapExt (cid:0) K ∗ +1 ( I ) , K ∗ ( I ) (cid:1) restrict −−−−→ Ext (cid:0) ker( i ∗ ) , K ∗ ( I ) (cid:1) i ∗ −→ Ext (cid:0) ker( i ∗ ) , K ∗ ( A ) (cid:1) . Since the first map is surjective, this is equal to the cokernel of i ∗ : Ext (cid:0) ker( i ∗ ) , K ∗ ( I ) (cid:1) → Ext (cid:0) ker( i ∗ ) , K ∗ ( A ) (cid:1) . A variant of the proof above for the second variable identifies this cokernel with thegroup Ext (cid:0) ker( i ∗ ) , coker( i ∗ ) (cid:1) as claimed. Given a class δ ∈ Ext (cid:0) K ∗ ( I ) , K ∗ ( A ) (cid:1) ,the map to Ext (cid:0) ker( i ∗ ) , coker( i ∗ ) (cid:1) simply applies the bifunctoriality of Ext for thequotient map K ∗ ( A ) → coker (cid:0) i ∗ : K ∗ ( I ) → K ∗ ( A ) (cid:1) OMPUTING OBSTRUCTION CLASSES 37 and the inclusion mapker (cid:0) i ∗ : K ∗ +1 ( I ) → K ∗ +1 ( A ) (cid:1) → K ∗ +1 ( I ) . It remains to compute the image of the obstruction class in Theorem 5.17 inExt (cid:0) ker( i ∗ ) , coker( i ∗ ) (cid:1) . We recall the data used in Theorem 5.17 in our special case.The semi-simple part of the filtrated K-theory consists ofK ∗ ( A [2]) ss ∼ = K ∗ ( I ) δ (K ∗− ( A/I )) ∼ = i ∗ (K ∗ ( I )) ⊆ K ∗ ( A ) , K ∗ ( A [1 , ss ∼ = K ∗ ( A ) i ∗ (K ∗ ( I )) = coker( i ∗ ) , K ∗ ( A [1]) ss ∼ = K ∗ ( A/I ) r ∗ (K ∗ ( A )) ∼ = δ ∗ (K ∗ ( A/I )) = ker( i ∗ ) ⊆ K ∗− ( I ) . Our construction is based on free resolutions of these Z / Q [1] (cid:26) Q [1] (cid:16) ker( i ∗ ). We lift theseresolutions to free resolutions Q [2] ⊕ Q [1] (cid:26) Q [2] ⊕ Q [1] (cid:16) K ∗ ( A [2]) ,Q [1 , ⊕ Q [2] (cid:26) Q [1 , ⊕ Q [2] (cid:16) K ∗ ( A [1 , ,Q [1] ⊕ Q [1 , (cid:26) Q [1] ⊕ Q [1 , (cid:16) K ∗ ( A [1]) , which contain the maps ( τ f [ a, b ]) and ϕ [ c,d ][ a,b ] . We shall need the maps in the thirdextension and put them into a larger diagram, which commutes because of theconstruction of the maps ( τ f [ a, b ]) and ϕ [ c,d ][ a,b ] :(5.38) Q [1] ⊕ Q [1 , Q [1] ⊕ Q [1 , 2] K ∗ ( A/I ) Q [1] Q [1] K ∗ ( A/I ) r ∗ (K ∗ ( A )) r ∗ (K ∗ ( A )) K ∗ ( A/I ) ker( i ∗ ) (cid:18) ϕ [1][1] ϕ [1 , ϕ [1 , , (cid:19) ( f [1] r ∗ f [1 , ϕ [1][1] d incl. δ ∗ pr pr can .f [1] ◦ ϕ [1][1] f [1] ∼ = δ The matrix coefficient δ [ a,b ] e in Theorem 5.17 is defined only if e = a ≤ b < n . Inour case n = 2, there is only one such matrix coefficient, namely, δ [1]1 : Q [1] → K ∗ ( A [1 , ∗ ( A ). The sum defining it has only one summand, which is indexedby [1 , → [1 , → [1]. So δ [1]1 = f [1 , ◦ ϕ [1 , : Q [1] → K ∗ ( A ) . We compose this map with the quotient map K ∗ ( A ) (cid:16) coker( i ∗ ). The exact se-quence (5.34) shows that r ∗ induces an isomorphism from coker( i ∗ ) onto r ∗ (K ∗ ( A )) ⊆ K ∗ ( A/I ). 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