4 - Translations and Standard Triangles

Summary

We talk about the translation (or shift or suspension) functor and standard triangles in the DG category C(A,M). The translation T(M) of a DG module M is the usual one. A calculation shows that T is a DG functor from C(A,M) to itself. We introduce the degree -⁠1 morphism tM : M → T(M), called the little t operator, which facilitates many calculations.

A morphism φ : M → N in Cstr(A,M) gives rise to the standard Cone(φ) = N ⊕ T(M) , whose differential is a matrix involving the degree 1 morphism φ ◦ (tM)-1. The standard cone sits inside the standard triangle associated to φ.A DG functor F : C(A,M) → C(B,N) gives rise to a T-additive functor F : Cstr(A,M) → Cstr(B,N), and it sends standard triangles in Cstr(A,M) to standard triangles in Cstr(B,N).