Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ_{≤n}A• be the truncated complex

··· → A^{n−2} → A^{n−1} → ker(A^{n} → A^{n+1})→ 0 → 0 → ··· , and dually we let τ_{≥n}A be the complex

··· → 0 → 0 → coker(A^{n−1} → A^{n}) → A^{n+1} → A^{n+2} → ···

Note that

H^{m}(τ_{≤n}A•) = H^{m}(A•) if m ≤ n

= 0 if m > n

and that

H^{m}(τ_{≥n}A•) = H^{m}(A•) if m ≥ n

= 0 if m < n

One checks that τ_{≥n} (respectively τ_{≤n}) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D. In fact if D_{≤n} (respectively D_{≥n}) is the full subcategory of D whose objects are the complexes A• such that H^{m}(A•) = 0 for m > n (respectively m < n) then we have additive functors

τ_{≤n }: D → D_{≤n} ⊂ D

τ_{≥n }: D → D_{≥n} ⊂ D

together with obvious functorial maps

i^{n}_{A} : τ_{≤n} A• → A•

j^{n}_{A} : A• → τ_{≥n} A•

The preceding i^{n}_{A} , j^{n}_{A} induce functorial isomorphisms

Hom_{D≤n} (B•,τ_{≤n}A•) →^{~} Hom_{D}(B•, A•) (B• ∈ D_{≤n}) —– (1)

Hom_{D≥n} (τ_{≥n}A•,C•) →^{~} Hom_{D}(A•,C• ) (C• ∈ D_{≥n}) —– (2)

Bijectivity of (1) means that any map φ : B• → A• (in D) with B• ∈ D_{≤n} factors uniquely via i_{A} := i^{n}_{A}

Given φ, we have a commutative diagram

and since B• ∈ D_{≤n}, therefore i_{B} is an isomorphism in D, so we can write

φ = i ◦ (τ_{≤n}φ ◦ i^{−1}_{B}),

and thus (1) is surjective.

To prove that (1) is also injective, we assume that i_{A} ◦ τ_{≤n} φ = 0 and deduce that τ_{≤n} φ = 0. The assumption means that there is a commutative diagram in K(A)

where s and s′′ are quasi-isomorphisms, and f/s = τ_{≤n}φ

Applying the (idempotent) functor τ_{≥n}, we get a commutative diagram

Since τ_{≤n}s and τ_{≤n}s′′ are quasi-isomorphisms, we have

τ_{≤n}φ = τ_{≤n }f/τ_{≤n}s = 0/τ_{≤n}s′′ = 0

as desired.

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