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KK-Theory As a Triangulated Category Notes from the Lectures by Ralf Meyer

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KK-Theory As a Triangulated Category Notes from the Lectures by Ralf Meyer KK-theory as a triangulated category Notes from the lectures by Ralf Meyer Focused Semester on KK-Theory and its Applications Munster¨ 2009 1 Lecture 1 Triangulated categories formalize the properties needed to do homotopy theory in a category, mainly the properties needed to manipulate long exact sequences. In addition, localization of functors allows the construction of interesting new functors. This is closely related to the Baum-Connes assembly map. 1.1 What additional structure does the category KK have? −n Suspension automorphism Define A[n] = C0(R ;A) for all n ≤ 0. Note that by Bott periodicity A[−2] = A, so it makes sense to extend the definition of A[n] to Z by defining A[n] = A[−n] for n > 0. Exact triangles Given an extension I / / E / / Q with a completely posi- tive contractive section, let δE 2 KK1(Q; I) be the class of the extension. The diagram I / E ^> >> O δE >> > Q where O / denotes a degree one map is called an extension triangle. An alternate notation is δ Q[−1] E / I / E / Q: An exact triangle is a diagram in KK isomorphic to an extension triangle. Roughly speaking, exact triangles are the sources of long exact sequences of KK-groups. Remark 1.1. There are many other sources of exact triangles besides extensions. Definition 1.2. A triangulated category is an additive category with a suspension automorphism and a class of exact triangles satisfying the axioms (TR0), (TR1), (TR2), (TR3), and (TR4). The definition of these axiom will appear in due course. Example 1.3. The category KK with suspension and exact triangles as above is triangulated. ♦ Warning 1.4. The Z=2-graded KK-category is not a triangulated category. 1 1.2 Cones and cylinders Definition 1.5. Let f : A ! B be a ∗-homomorphism. Then the cone of f is n o cone(f) = (a; b) 2 A ⊕ C0 (0; 1];B b(1) = f(a) , while the cylinder of f is n o cyl(f) = (a; b) 2 A⊕C0 [0; 1];B b(1) = f(a) . Note that any ∗-homomorphism f : E ! Q gives rise to an extension triangle Q[−1] / cone(f) / / cyl(f) / / Q and that the cone of idA : A ! A is the cone of A. Definition 1.6. Let f : A ! B be a ∗-homomorphism. Then there is an extension B[−1] / / cone(f) / / A with a completely positive contractive section. The class of this extension in KK1(A; B[−1]) turns out to be f[−1]: A[−1] ! B[−1], so this gives a extension triangle f[−1] A[−1] / B[−1] / cone(f) / A: Such an extension triangle is called an mapping cone triangle. Warning 1.7. It is possible that in Definition 1.6, the class of the extension in 0 KK1(A; B[−1]) should be −f[−1]. This sign error is due to the fact that cone(f) = n o (a; b) 2 A ⊕ C0 [0; 1);B b(0) = f(a) , is an alternate definition of the mapping cone. Lemma 1.8. A triangle is exact if and only if it is isomorphic to a mapping cone triangle. Proof. Assume I / / E / / Q is an extension with a completely positive con- tractive section, and consider the diagram f Q[−1] I / / E / / Q α f Q[−1] / cone(f) / E / Q: The proof of excision in KK shows that α exists, and that α is invertible in KK. Moreover, δE : Q[−1] ! I makes the resulting diagram commutative, so exact tri- angles are isomorphic to mapping cone triangles. For the converse consider the diagram f Q[−1] / cone(f) / E / Q Q[−1] / cone(f) / / cyl(f) / / Q and the definition of cyl(f). Remark 1.9. By the previous lemma, it is clear that one can define exact triangles in terms of mapping cone triangles. It is worth mentioning that by using this approach, one can prove excision with Puppe sequences. 2 1.3 Properties of exact triangles - The axioms (TR0), (TR1), (TR2), and (TR3) Definition 1.10 ((TR0)). Triangles isomorphic to exact ones are exact and id A A / A / 0 / A[1] is exact. The last part can be reformulated as id 0 / A A / A / 0 is exact. Definition 1.11 ((TR1)). Any morphism in the category is part of an exact tri- angle, i.e. for any f : A ! B, there is an exact triangle f B[−1] / C / A / B: KK satisfies (TR1). By using the Cuntz picture, one can represent f 2 KK0(A; B) by a ∗-homomorphism f~: qA ! B ⊗ K. Then form the diagram f~ B ⊗ K [−1] / cone(f~) / qA / B ⊗ K ∼ ∼ ∼ B[−1] / cone(f~) / A / B where ∼ denotes KK-equivalence. ∼ Another point of view is to note that f 2 KK0(A; B) = KK1 C0(R;A);B and f[−1] then look at B[−1] / A[−1] / E / B . Definition 1.12 ((TR2)). The triangle u v w A / B / C / A[1] is exact if and only if the triangle −v −w −u[1] B / C / A[1] / B[1] is exact. Remark 1.13. In the last part of (TR2), it is only required that an odd number of the maps change sing. Thus it is enough that one of maps has a change of sign. KK satisfies (TR2). Given f : A ! B, we have extension triangle f[−1] f A[−1] / B[−1] / cone(f) / A / B ∼ − B[−1] / cone(f) / cyl(f) / B /.-,()*+ where the last triangle commutes up to sign. 3 Definition 1.14 ((TR3)). Given an commutative diagram of with exact rows A / B / C / A[1] α β α[1] A0 / B0 / C0 / A0[1] there exist a morphism γ : C ! C0 such that the resulting diagram is commutative. KK satisfies (TR3). In a special case, this is the functorality of the mapping cone. For the general case one can use the Cuntz picture. 1.4 General facts about triangulated categories For any category D and A; B objects in D, one denote by D(A; B) the group of morphisms from A to B. Proposition 1.15. Let T be a triangulated category, D an object of T and A / B / C / A[1] an exact triangle. Then there is an induced long exact sequence / T(D; A[n]) / T(D; B[n]) / T(D; C[n]) / T(D; A[n + 1]) / and an induced dual long exact sequence / T(A[n + 1];D) / T(C[n];D) / T(B[n];D) / T(A[n];D) / : Proof. Since the arguments for the two long exact sequences are similar, we only show the first part. The proof of this follows a similar argument as the one for KK. Note that it suffices to show exactness at one point in the long exact sequence. By using (TR0) and (TR3) one can show that the composition of any two consecutive morphisms in an exact triangle is zero, and this shows that \im ⊂ ker" in the long exact sequence. To show \ker ⊂ im" consider −idD[1] D / 0 / D[1] / D[1] B / C / A[1] / B[1] and use (TR3). Definition 1.16. A functor from a triangulated category is called (co)-homological if it creates long exact sequences from exact triangles as in Proposition 1.15. Theorem 1.17. If A / B / C / A[1] α β γ α[1] A0 / B0 / C0 / A0[1] is a morphism of exact triangles in T with α and β invertible, then γ is invertible. 4 Proof. Observe that for all objects D in T, the morphisms T(D; α): T(D; A) ! T(D; A0) and T(D; β): T(D; B) ! T(D; B0) are invertible. By the Yoneda lemma it suffices to prove that T(D; γ) is invertible for all D in T. This follows from the 5-lemma applied to the diagram T(D; A) / T(D; B) / T(D; C) / ··· α β γ T(D; A0) / T(D; B0) / T(D; C0) / ··· where \··· " denotes the continuing long exact sequence. Corollary 1.18 (Uniquness of mapping cones). The exact triangle f A / B / C / A[1] containing f : A ! B is unique up to isomorphism. Warning 1.19. The isomorphism in Corollary 1.18 is not a natural isomorphism. A \workaround" for this is to use model categories. Exercise 1 For all objects A; B in a triangulated category the diagram ιA πB 0 A / A ⊕ B / B / A[1] is an exact triangle. u v w Exercises 2 If A / B / C / A[1] is an exact triangle with w = 0, then it is isomorphic to the exact triangle in Exercise 1. f Exercises 3 The diagram A / B / 0 / A[1] is an exact triangle if and only if f is invertible. Exercises 4 The morphism f is invertible if and only if cone(f) = 0. 1.5 Other kinds of exact sequences 1.5.1 Mayer-Vietoris Theorem 1.20. If f1; f2 have completely positive contractive section, then there is an exact triangle A1 ⊕B A2 / A1 ⊕ A2 / B / (A1 ⊕B A2)[1] A2 f2 where A1 ⊕B A2 is the pullback of . A1 / B f1 n o Proof. Let H = (a1; a2; b) 2 A1 ⊕A2 ⊕C [0; 1];B f1(a1) = b(0); f2(a2) = b(1) , i.e. H is the homotopy pullback, and use the fact that A1 ⊕B A2 ! H is an KK- equivalence. 5 1.5.2 Crossed products Let α: A ! A be an automorphism, and define the mapping torus of α by T = n o α f 2 C [0; 1];A f(1) = a f(0) .
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