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Homological Algebra Methods in the Theory of Operator Algebras

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Homological Algebra Methods in the Theory of Operator Algebras Homological algebra methods in the theory of Operator Algebras Ryszard Nest Homological algebra methods in the theory of Homological Operator Algebras II, algebra Ideals and projectives Homological functors, derived functors, Adams I-exact complexes Projective objects spectral sequence, assembly map and BC for The phantom tower Assembly map Derived functors compact quantum groups. The ABC spectral sequence Example: Γ = Z Compact Quantum Ryszard Nest Groups University of Copenhagen 25th July 2016 Homological algebra methods in the theory of Operator Algebras Let C be an abelian category Ryszard Nest We call a covariant functor F : T → C homological if Homological algebra F (C) → F (A) → F (B) Ideals and projectives I-exact complexes Projective objects The phantom tower is exact for any distinguished triangle Assembly map Derived functors The ABC spectral ΣB → C → A → B. sequence Example: Γ = Z n Compact We define Fn(A) := F (Σ A) for n ∈ . Quantum Z Groups Similarly, we call a contravariant functor F : T → C cohomological if F (B) → F (A) → F (C) is exact for any distinguished triangle, and we define F n(A) := F (ΣnA). Homological algebra methods in the theory of Operator Algebras Ryszard Nest Homological algebra We will always be in the following situation. Ideals and projectives I-exact complexes 1 C is some abelian category equipped with a shift Projective objects The phantom tower Σ: C → C (C is stable). Assembly map Derived functors The ABC spectral 2 Our functors F : T → C are homological and stable, i. e. sequence Example: Γ = Z commute with Σ. Compact Quantum Groups Homological algebra methods in the theory of Operator Algebras We will do homological algebra relative to some ideal in T wich Ryszard Nest satisfies the following property. Homological Definition algebra Ideals and projectives An ideal I ⊆ T is called stable if the suspension isomorphisms I-exact complexes ∼ Projective objects = The phantom tower Σ: T(A, B) −→ T(ΣA, ΣB) for A, B ∈∈ T restrict to Assembly map isomorphisms Derived functors ∼ The ABC spectral = sequence Σ: I(A, B) −→ I(ΣA, ΣB). Example: Γ = Z Compact Quantum Groups Definition An ideal I ⊆ T in a triangulated category is called homological if it is the kernel of a stable homological functor. Homological algebra methods in the theory of Operator Algebras Ryszard Nest Homological Before continuing with definitions, recall the picture of algebra distinguished triangle in T. Ideals and projectives I-exact complexes Projective objects f The phantom tower A / C Assembly map _ Derived functors The ABC spectral ◦ sequence h g Example: Γ = Z C Compact Quantum Groups Homological algebra methods in the theory of Operator Algebras Definition Ryszard Nest Let T be a triangulated category and let I ⊆ T be a Homological homological ideal. Let f : A → B be a morphism in T and algebra f g h Ideals and projectives embed it in an exact triangle A → B → C → ΣA. I-exact complexes Projective objects • We call f I monic if h ∈ I. The phantom tower Assembly map • We call f I epic if g ∈ I. Derived functors The ABC spectral sequence • We call f an I equivalence if f is both I monic and I epic Example: Γ = Z or, equivalently, if g, h ∈ I. Compact Quantum • We call f an I phantom map if f ∈ I. Groups An object A ∈∈ T is called I contractible if idA ∈ I(A, A). g An exact triangle A →f B → C →h ΣA in T is called I exact if h ∈ I. Homological algebra methods in the theory of Operator Algebras Consider a chain complex C• = (Cn, dn). For each n ∈ N, we Ryszard Nest may embed the map dn in an exact triangle Homological dn fn gn algebra Cn −→ Cn−1 −→ Xn −→ ΣCn, (1.1) Ideals and projectives I-exact complexes Projective objects which is determined uniquely up to (non-canonical) The phantom tower Assembly map isomorphism of triangles. Hence the following definition does Derived functors The ABC spectral not depend on auxiliary choices: sequence Example: Γ = Z Compact Definition Quantum Groups The chain complex C• is called Iexact in degree n if the gn Σfn+1 composite map Xn −→ ΣCn −−−→ ΣXn+1 belongs to I. It is called Iexact if it is Iexact in degree n for all n ∈ Z. Homological algebra methods in the theory of Operator Algebras Ryszard Nest Homological If we recall from yesterday, in the diagram algebra Ideals and projectives I-exact complexes dn dn−1 Projective objects Cn / Cn−1 / Cn−2 The phantom tower ` c Assembly map ◦ ◦ Derived functors The ABC spectral | { sequence Xn o Xn−1 Example: Γ = Z Compact the composition given by the stipled arrow is in I. Quantum Groups Homological algebra methods in the theory of Operator Algebras Ryszard Nest Lemma Let F : T → C be a stable homological functor into some stable Homological algebra Abelian category C. Let C• be a chain complex over T. The Ideals and projectives I-exact complexes complex C• is KerF exact in degree n if and only if the Projective objects The phantom tower sequence Assembly map Derived functors The ABC spectral F (dn+1) F (dn) sequence F (Cn+1) −−−−−→ F (Cn) −−−→ F (Cn−1) Example: Γ = Z Compact Quantum in C is exact at F (Cn). Groups Later we will meet homological ideals given as intersections kernels of a family of homological ideals. Homological algebra methods in the theory of Operator Algebras Ryszard Nest Homological algebra Ideals and projectives Definition I-exact complexes Projective objects The phantom tower An object A ∈∈ T is called I-projective if the functor Assembly map T(A, ␣): T → Ab is Iexact. Derived functors The ABC spectral sequence We write PI for the class of I-projective objects in T. Example: Γ = Z Compact Quantum Groups Homological algebra methods in the theory of Operator Algebras Ryszard Nest Lemma Homological algebra An object A ∈∈ T is I-projective if and only if I(A, B) = 0 for Ideals and projectives I-exact complexes all B ∈∈ T.. Projective objects The phantom tower Assembly map Derived functors Lemma The ABC spectral sequence The class PI of I-projective objects is closed under Example: Γ = Z Compact (de)suspensions, retracts, and possibly infinite direct sums (as Quantum Groups far as they exist in T). Homological algebra methods in the theory of Operator Algebras Ryszard Nest The following will supply us with projective objects. Homological Theorem algebra Ideals and projectives I-exact complexes 1 Suppose that F is I-exact and Q ∈ T satisfies Projective objects The phantom tower Assembly map T(Q, A) = C(X, F (A)) Derived functors The ABC spectral sequence Example: Γ = Z for some object X in C. Then Q is I-projective. Compact Quantum 2 Suppose that I = Ker F . Then an object P of T is Groups projective iff F (P) is projective in C Homological algebra methods in the theory of Operator Algebras Ryszard Nest T contains enough projectives if, for any object A in T, there Homological exists an exact triangle of the form algebra Ideals and projectives I-exact complexes j Projective objects A / N . The phantom tower _ Assembly map ◦ Derived functors The ABC spectral sequence P Example: Γ = Z Compact Quantum with P projective and j ∈ I(A, N). Then we can construct Groups projective resolutions. Homological algebra methods in the theory of Operator Algebras The phantom tower. Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes AN N N N ··· Projective objects 0 1 2 3 The phantom tower Assembly map Derived functors The ABC spectral sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower. Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N N N N ··· Projective objects 0 1 2 3 The phantom tower Assembly map Derived functors The ABC spectral sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower. Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N0 N1 N2 N3 ··· Projective objects Y The phantom tower Assembly map π0 Derived functors The ABC spectral sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower. Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N0 / N1 N2 N3 ··· Projective objects Y The phantom tower Assembly map ◦ π0 Derived functors The ABC spectral Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of π0 Groups Homological algebra methods in the theory of Operator Algebras The phantom tower. Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N0 / N1 N2 N3 ··· Projective objects Y Y The phantom tower Assembly map ◦ π0 π1 Derived functors The ABC spectral Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower. Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 2 ι0 ι1 I-exact complexes A N0 / N1 / N2 N3 ··· Projective objects Y Y The phantom tower Assembly map ◦ ◦ π0 π1 Derived functors The ABC spectral Õ Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of π1 Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.
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