An Introduction to Homological Algebra by Tomi Pannila

By Tomi Pannila

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I) f : A Ñ B is the zero morphism if and only if f paq “ 0 for all a P˚ A. (ii) f : A Ñ B is a monomorphism if and only if f paq “˚ f pa1 q implies a “˚ a1 for all a, a1 P˚ A. (iii) f : A Ñ B is an epimorphism if and only if for all b P˚ B there exist a P˚ A such that f paq “˚ b. (iv) A short exact sequence A f g B C ˚ is exact if and only if for every a P A we have gpf paqq “ 0 and for every b P˚ B with gpbq “ 0 there exists a pseudo-element a1 P˚ A such that f paq “˚ b. (v) For f : A Ñ B and a, a1 P˚ A with f paq “˚ f pa1 q there exists a2 P˚ A such that f pa2 q “ 0 and for all g : A Ñ C condition gpa1 q “ 0 implies gpa2 q “˚ gpaq.

Proof. By duality, it suffices to show that pushout of a monomorphism is a monomorphism, because pushout squares correspond to pullback squares and monomorphisms to epimorphisms in the opposite category. Let f : X Ñ Y be a monomorphism and g : X Ñ Z a morphism. Consider the following commutative diagram f X Y i2 g´i1 f i1 Y ‘Z g i2 c Z W where c : Y ‘ Z Ñ W is the cokernel of i2 g ´ i1 f . 7. Let us show that i2 g ´ i1 f is a monomorphism. Let x : D Ñ X be a morphism such that pi2 g ´ i1 f qx “ 0.

12 m1 “ kerpcoker m1 q “ ker φi2 , so there exists a unique morphism h such that m1 h “ φi1 a1 h “ φi1 ψ. 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.

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