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

Hm≤nA•) = Hm(A•) if m ≤ n

= 0 if m > n

and that

Hm≥nA•) = Hm(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 Hm(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

inA : τ≤n A• → A•

jnA : A• → τ≥n A•

The preceding inA , jnA induce functorial isomorphisms

HomD≤n (B•,τ≤nA•) →~ HomD(B•, A•) (B• ∈ D≤n) —– (1)

HomD≥n≥nA•,C•) →~ HomD(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 iA := inA

Given φ, we have a commutative diagram

Untitled

and since B• ∈ D≤n, therefore iB is an isomorphism in D, so we can write

φ = i ◦ (τ≤nφ ◦ i−1B),

and thus (1) is surjective.

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

Untitled

where s and s′′ are quasi-isomorphisms, and f/s = τ≤nφ

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

Untitled

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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