Useful constructions Sample Clauses

Useful constructions. ‌ We now record two important and new ways in which admissible complexes give rise to new families of admissible complexes. (see §1.9 for the direction of shifting of complexes) (a) an object C of Da(A), and (b) a finitely generated projective A-module P and a homomorphism of A-modules θi : P → Hi(C) for i = 1 and i = 2 where θ1 is both injective and such that cok(θ1) is R-torsion-free. Then there is a exact triangle in D(A) of the form P [−1] ⊕ P [−2] →−θ C → D → P [0] ⊕ P [−1] (2.4) in which θ is the unique morphism with Hi(θ) = θi for i = 1, 2 and D belongs to Da(A).