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An Introduction to Homological Algebra by Tomi Pannila

By Tomi Pannila

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Now i i`1 i i i`1 i i pdi´1 dX ‚ qpaq “ pdi´1 dX ‚ qpaq “˚ pdi´1 Y‚ h `h Y ‚ h qpaq ` ph L‚ h qpaq, because phi`1 diX ‚ qpaq “ 0 by assumption on the pseudo-element a. Therefore we can choose b “ hi a. 4 (Translation). Let A be an additive category. Define a functor T : CpAq Ñ CpAq as follows. For an object X ‚ P CpAq, let T pXq, denoted also by Xr1s, to be the complex T pdiX ‚ qi “ ´di`1 X‚ . T pXqi “ X i`1 , For a morphism f : X ‚ Ñ Y ‚ , let T pf qi “ f i`1 . This is also denoted by f r1s. Clearly this defines a functor, which is an isomorphism on CpAq, where the inverse T ´1 is given by translation to other direction.

2 it follows that e1 is the cokernel of s and p1 is the kernel of s1 . Now A1 Ñ X1 Ñ B1 Ñ B2 Ñ X2 “ 0 by commutativity, so there exists a unique morphism φ : ker h Ñ X2 such that X1 Ñ ker h Ñ X2 “ X1 Ñ B1 Ñ B2 Ñ X2 . 12) By commutativity and the fact that e1 is an epimorphism ker h Ñ X2 Ñ C2 “ 0, so there exists a unique morphism δ˜ : ker h Ñ coker f such that ker h Ñ coker f Ñ X2 “ ker h Ñ X2 . 3 (iv), because gpyq is mapped to zero. ˜ We show that δpxq “˚ upzq. 3 (vi) there exists a pseudo-element ˚ x0 P X1 such that k 1 px0 q “˚ y and e1 px0 q “˚ x.

Construction of h: From di`1 X ‚ ψ1 φ1 “ dX ‚ dX ‚ “ 0 we get dX ‚ ψ1 “ 0, because φ1 is an epimorphism. i´1 i`1 i i Hence, there exists a unique morphism h : coker dX ‚ Ñ ker dX ‚ such that ψ1 “ ψ2 h. Now ψ2i φi2 “ ψ1i φi1 “ ψ2i hφi1 and φi2 “ hφi1 by the fact that ψ2i is a monomorphism. This shows that the resulting diagram is commutative. i´1 i ‚ H i pX ‚ q – ker h: Let α1 : ker h Ñ coker di´1 X ‚ be the kernel of h and α2 : H pX q Ñ coker dX ‚ the kernel of i´1 i i`1 i i i φ1 : coker dX ‚ Ñ X .

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