Correction to the lecture of 9/29.

Lemma about canonicity of the cone in a special situtation was stated incorrectly. The correct statement reads: Lemma. Cone of a morphism f : X → Y is defined uniquely up to a unique

isomorphism provided Hom^{−1}(X, Cone(f)) = 0.

A morphism between projective resolution P_{M}^{·} → P_{N}^{·} coming from a morphism of modules M → N is an examples of this situation only if the map M → N is injective.

This Lemma is a direct consequence of the previous claim stated (and proved correctly) in class: a morphism between distinguished triangles A → B → C → A[1] and A′ → B′ → C′ → A′[1] is uniquely determined by the morphism B → B′ provided Hom^{−1}(A, C′) = 0 (apply this to A = A′ = X, B = B′ = Y, C = C′ = Cone(f)).