Algebraic Geometry

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By Walker R. J.

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11. D. When we have functors Φ : N → SymMon, we will see that N Φ and sometimes Ns Ψ give small models for the spectrum homotopy colimits of l Spt · Φ and Spt · Ψ. For Φ : N → SymMon, we define N Φ −→ N Φ F i as follows. 12, we are required to produce functors Φ(i) −→ N Φ and natural transformations Ni from Fi to Fi+1 · Φ(i → ı + 1). Fi will be Bounded K–theory and the Assembly Map in Algebraic K–theory 27 the functor from Φ(i) → N Φ given by ξ → 1[(i, ξ)] and Ni (ξ) will be the morphism 1[(i, ξ)] → 1[(i + 1, Φ(i < i + 1)(ξ))] corresponding to the identity map of Φ(i < i + 1)(ξ).

HΓ are fibrant cosimplicial spaces. )Γ → (X . )hΓ is a weak  Γ  hΓ equivalence, hence by (a), holim F | C  → holim F | C  ←− C ←− C is a weak equivalence. Proof: (a) is direct, entirely analogous to the proof that the cosimplicial space defining holim is fibrant if F (x) is a Kan complex for all x ∈ C. We ←− C leave it to the reader. To prove (b), we must show that if C is discrete category with free Γ–action, and Γ C → s–sets is a functor, then the Γ  hΓ  natural map holim F | C  → holim F | C  ←− C ←− C is a weak equivalence.

Let A be any spectrum; then we have a diagram of spectra h f (U ∩ V, A) −→   h f (U, A) h f (V, A)   −→ h f (X, A) Bounded K–theory and the Assembly Map in Algebraic K–theory 41 Let P(U, V, A) denote the homotopy pushout of the diagram below. h f (U ∩ V, A) −→ h f (V, A)   h f (U, A) α Then there is a natural map P(U, V, A) −→ h f (X, A). 15. α is a weak equivalence of spectra. Proof: We first deal with the case A = K (G, 0), where G is an Abelian group. 9 that these long exact sequences are identified, and that the map πi (P(U, V, A)) → πi (h f (X, A)) is identified with the map Hi (Cˆ∗U (X; G)) → Hi (Cˆ∗ (X; G)).

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