Truncation Functors

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_≤nA• be the truncated complex

··· → Aⁿ⁻² → Aⁿ⁻¹ → ker(Aⁿ → Aⁿ⁺¹)→ 0 → 0 → ··· , and dually we let τ_≥nA be the complex

··· → 0 → 0 → coker(Aⁿ⁻¹ → Aⁿ) → Aⁿ⁺¹ → Aⁿ⁺² → ···

Note that

H^m(τ_≤nA•) = H^m(A•) if m ≤ n

= 0 if m > n

and that

H^m(τ_≥nA•) = 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

iⁿ_A : τ_≤n A• → A•

jⁿ_A : A• → τ_≥n A•

The preceding iⁿ_A , jⁿ_A induce functorial isomorphisms

Hom_D≤n (B•,τ_≤nA•) →^~ Hom_D(B•, A•) (B• ∈ D_≤n) —– (1)

Hom_D≥n (τ_≥nA•,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 := iⁿ_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_≤nφ ◦ 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_≤nφ

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

Since τ_≤ns and τ_≤ns′′ are quasi-isomorphisms, we have

τ_≤nφ = τ_≤n f/τ_≤ns = 0/τ_≤ns′′ = 0

as desired.

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