An Introduction to Homological Algebra by Tomi Pannila

By Tomi Pannila

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If this is true, we have the following short exact sequence 0 ker g m1 Y e2 kerpcoker gq 0 . Proof. 6. 16 (Split short exact sequence). Let A be an abelian category. We say that a short exact sequence 0 X f Y g Z 0 splits, or that it is a split short exact sequence, if there exist morphisms i : Y Ñ X and j : Z Ñ Y such that if “ IdX and gj “ IdZ . 17. Not every short exact sequence splits. Consider the following short exact sequence f 0 Z{2Z g Z{4Z 0 Z{2Z in the category Ab, where f p1q “ 2 and gp1q “ 1.

Note that pseudo-elements cannot be used to prove equality of morphisms in general. Let a P˚ A be a pseudo-element not pseudo-equal to 0. Then a “˚ ´a, but as a morphism a ` a need not be equal to 0. Indeed, 2 ´2 the morphism Z Ñ Z in the category of Z-modules is not equal to Z Ñ Z. 5 Category of complexes In this section we construct a new category from an additive category, the category of complexes. The objects of this category are sequences of objects of the underlying category connected with differentials.

This shows that f3 is a monomorphism. To give a proof of the snake lemma, we need the following lemma. 2. Let A be an abelian category. Consider a pullback (resp. pushout) diagram ¨ ˛ f1 f1 l A B A B L ˚ ‹ 1 ˚resp. g1 ‹ k1 g g l g1 ˝ ‚ K k C f D C f D of f and g (resp. f 1 and g 1 ), where k “ ker f (resp. l “ coker f 1 ). Then k “ g 1 k 1 (resp. l “ l1 g), where k 1 “ ker f 1 (resp. l1 “ coker f ). Proof. By duality it is enough to prove the claim about the pullback diagram. Consider the morphisms k : K Ñ C and 0 : K Ñ B.

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