By Tomi Pannila
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Extra info for An Introduction to Homological Algebra
We define f to be null-homotopic if there exists a family of morphisms hi : X i Ñ Y i´1 in A such that 52 i i`1 i f i “ di´1 dX ‚ for all i P Z. Two morphisms f, g : X ‚ Ñ Y ‚ in CpAq are defined to be homotopic, denoted Y‚ h `h f „ g, if f ´ g is null-homotopic. 3. Let A be an additive category. i (a) For any objects X ‚ , Y ‚ of CpAq, and any collection hi : K i Ñ Li´1 of morphisms in A, hi`1 diX ‚ ` di´1 Y‚ h : i i K Ñ L is a morphism of complexes. (b) For any complexes X ‚ and Y ‚ over A, the subset IX ‚ ,Y ‚ Ă MorA pX ‚ , Y ‚ q consisting of morphisms of the i i i i´1 form hi`1 diX ‚ ` di´1 is any family of morphisms in A, is an abelian group and Y ‚ h , where h : X Ñ Y stable under composition of morphisms.
Fix notation by the following commutative diagram X i´1 di´1 X‚ a Xi i φi1 ker diX ‚ γ H i pX ‚ q First we need to verify that the map ψ is well-defined. Let x, y P˚ X i be pseudo-elements such that diX ‚ x “ diX ‚ y “ 0 i´1 and xe1 ´ ye2 “ di´1 and epimorphisms e1 and e2 . By definition of kernel there X ‚ z for some pseudo-element z of X 1 1 i 1 i i exist unique morphisms x and y such that φ1 x “ x and φi1 y 1 “ y. Since φi1 px1 e1 ´ y 1 e2 q “ di´1 X ‚ z “ φ1 a z, and 1 1 1 1 i i x e1 ´ y e2 is the unique morphism with this property, we have x e1 ´ y e2 “ a z, because φ1 is a monomorphism.
3 (iv) it suffices to show that for a P˚ Cylpf q such that p2 a “ 0 there exists a pseudo-element of X ‚ mapping to a. Since a “ IdCylpf q a “ pi1 p1 ` i2 p2 qa “ i1 p1 a, the pseudo-element p1 a maps to a. Hence the short sequence is exact. The following lemma captures the main properties of mapping cone and mapping cylinder of a morphism. 9. Let A be an abelian category.
An Introduction to Homological Algebra by Tomi Pannila