I am reading the book The Structure and Stability of Persistence Modules. In chapter 2 it's told that under some conditions, a persistence module can be expressed as a direct sum of interval modules, and this decomposition is unique. Then the book gives example (2.10): image 1, image 2.
I am confused about the decomposition here. For example, I don't know at the point $c$, why $b$ must be the one that is killed rather than $a$, and why the decomposition can not be :
$H_0(X_{sub})=k[a,c)\oplus k[b,+\infty)\oplus k[d,e)$
or
$H_0(X_{sub})=k[a,c)\oplus k[b,e)\oplus k[d,+\infty)$
Thanks!
One possible answer here is the following. Note that in the persistence module $H_0(X_{sub})$, the map $H_0(X_a)\to H_0(X_f)$, i.e. from index $a$ to index $f$, has rank one. However, in either the module $k[a,c)\oplus k[b,+\infty)\oplus k[d,e)$ or the module $k[a,c)\oplus k[b,e)\oplus k[d,+\infty)$, the rank of the map from index $a$ to index $f$ is zero. Hence $H_0(X_{sub})$ cannot be isomorphic to either $k[a,c)\oplus k[b,+\infty)\oplus k[d,e)$ or $k[a,c)\oplus k[b,e)\oplus k[d,+\infty)$.
Indeed, it instead turns out that the persistence module $H_0(X_{sub})$ is isomorphic to $k[a,+\infty)\oplus k[b,c)\oplus k[d,e)$; Gabriel's theorem says it can be isomorphic to exactly one such decomposition into indecomposables (up to reordering of the indecomposable factors).