By Tomi Pannila
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Additional info for An Introduction to Homological Algebra
Every morphism f : X Ñ Y in A can be factorized uniquely, up to unique isomorphism, as f “ me, where e is an epimorphism and m is a monomorphism. In particular, we have f “ ip, where i “ kerpcoker f q and e “ cokerpker f q. Moreover, if we have the following commutative diagram X1 e1 f Y1 X2 m1 g h e2 Y2 X3 m2 Y3 where e1 and e2 are epimorphisms and m1 and m2 are monomorphisms, then there exists a unique morphism h which makes the diagram commutative. 25 Proof. 2) coker f Since f f1 “ 0, there exists a unique morphism φ such that f “ φf2 .
This shows that f paq “˚ b. Conversely, suppose that for every pseudoelement b P˚ B there exists a pseudo-element a P˚ A such that for some epimorphisms p and q we have f ap “ bq. Let x be a morphism such that xf “ 0. Since IdB is a pseudo-element of B we have f a1 p “ q for some pseudo-element a1 of A and some epimorphisms p and q. This implies that f a1 p is an epimorphism and hence f a1 is an epimorphism. Now xf a1 “ 0, so x “ 0. Thus f is an epimorphism. 10. Since gf “ 0, we have gf paq “ 0 for all pseudo-elements a of A.
Now φi1 is a monomorphism, so a1 h “ ψ. The morphism h is unique because a1 is a monomorphism. This shows that a1 is the kernel of φi2 φi1 . Similarly one shows that b1 is the cokernel of φi2 φi1 . 10 we get coker ai “ coker a1 “ cokerpker φi2 φi1 q – kerpcoker φi2 φi1 q “ ker b1 “ ker bi . This shows that cohomology is well-defined. We prove an alternative characterization for cohomology which will be used to obtain a long exact sequence from a short exact sequence of complexes. 3. 5) X i`1 is commutative.