Motivation of $G_0$ group

Question

I want to know the motivation why we want to modulo the subgroup related to the exact sequences.

Answer 1

If $R = k$ is a field then $M', M, M''$ are $k$-vector spaces. Therefore, the short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ implies that $M = M'\oplus M''$. The operation $\oplus$ gives the set of vector spaces the structure of semigroup. Now, we can see that this semigroup can be embedded into a group $F/<[M'\oplus M''] - [M'] - [M'']>$ where $F$ is the free group generated by $\{[M]\}$. We can see that modulo by $[M'\oplus M''] - [M'] - [M'']$ is just modulo by $[M] - [M'] - [M'']$ for every s.e.s. $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$.

What you have is just a generalisation of this when $R$ doesn't need to be a field.

Answer 2

Given a "good" ring $R$ we want to say something about its "arithmetic", we want to find characteristic invariants of $R$, that are useful for instance in number theory. (If $R$ lives in this world.) It turns out that the abelian category of finitely generated $R$-modules is a good big object to associate to $R$, and then we want to study this object instead of $R$ itself. The supplementary structure is that of the short exact sequences. Using them we can understand complicated extensions hoping to start with a module $M$, involve it in a short exact sequence $$ (*) \qquad\qquad 0\to M'\to M\to M''\to 0\ , $$ then study the "pieces" $M'$ and $M''$ instead of $M$. (It is also what we do with groups, trying to mod out normal subgroups.) Now we can "destroy" o module using this strategy by taking "smaller and smaller bricks", till there are no submodules. (To insure we get to this point in a finite amount of steps we introduce the Noetherian hypothesis as a working hypothesis.) Now we have the "smallest bricks" and the one strategy to go there. But we may use an other strategy, get the same bricks or some others. How can we make order in this scissoring?

The natural way to do something is to assume that from the above exact sequence "$M$ is the bag of two bricks $M'$ and $M''$". Writing this formally, we have to introduce the free abelian group on the base $[M]$, with relations $[M]=[M']+[M'']$ for each connecting relation $(*)$.

The motivation for doing this was extracted from number theory and/or algebraic geometry, a main "simple" instance being the case of a Dedekind domain, https://en.wikipedia.org/wiki/Ideal_class_group#Properties . (The case of vector spaces over a field is trivial, we would not need to go so formal to define only the dimension of a vector space.) The fact that performing this functorial construction reproduces a main invariant from number theory, the ideal class group, encourages to apply the K-theoretic avatar on other rings, and when $K_0$ is not enough, we can also "destroy the ring" to get higher $K$-groups.