Any exact sequence of injective objects splits

Question

For injective objects $I^i$, does any exact sequence $I^0 \to I^1 \to \cdots \to I^r$ split?

In lemma 7.4. of Chapter 3 of Hartshorne's Algebraic Geometry, the author says this.

Please help.

Answer 1

Let $0\to A\to B\to C \to 0$ be a short exact sequence in an abelian category. If $A$ is injective, the sequence splits. Dually, if $C$ is projective, the sequence splits. And easy particular case of this last claim is the case of free modules: if $C$ is a free module, you can easily find a section of the last map just by giving the image of a basis.

Let us prove that if $A$ is injective, the sequence splits. The lifting property of an injective object $I$ is as follows:

For every monomorphism $T\hookrightarrow S$ and every morphism $T\to I$ we may find an extension $S\to I$.

We want to show that the sequence $0\to A\to B\to C\to 0$ splits. This is equivalent to finding a retraction $B\to A$ of the first map $A\to B$. This means finding a map $B\to A$ such that the composition $A\to B\to A$ is the identity on $A$.

So just apply the defining property of injective objects to your case: consider the identity on $A$, which is a map $A\to A$, and the monomorphism $A\hookrightarrow B$. We may find an extension $B\to A$, which is precisely a retraction of $A\to B$.

PS: I am sorry for the bad style, I feel like there is too many words and this could have been solved with just one diagram, but I don't know how to draw them on MSE.