# Morphisms of exact sequences

For a given module

Different choices for the generators of the kernels of those surjections (i.e., different choices of generators for the relations among those generators), lead to different exact sequences

Continuing in this way, it's very plausible to expect to deal with two very different free resolutions of the same module

There should be a way to compare these two resolutions, which means we need a way to compare exact sequences. As usual, we'll do that through the notion of a morphism of exact sequences.

A **morphism** from an exact sequence

to an exact sequence

is a collection of morphisms

As might be predicted, we can compose morphisms of exact sequences (by composing the vertical morphisms), and we can talk about isomorphisms of exact sequences (either as morphisms in which all of the vertical morphisms are isomorphisms, or equivalently as morphisms for which there exists an inverse morphism of exact sequences).

## Properties of morphisms of exact sequences

The commutativity condition on morphisms of exact sequences has some surprising consequences.

Suppose we have a morphism of short exact sequences

If

If you're wondering if there's a "Long Five Lemma", the answer is ... kind of.