I'm reading Computational Topology (by Edelsbrunner & Harer). The authors describe (pg 180) generating the persistence diagram from a filtration of simplicial complexes. The approach is to define the birth and death times of each homology generator using the elder rule to say that a class dies at step $i$ if it merges with an older class.
My question is: how does this relate to the algebraic perspective where the persistence diagram is given by the decomposition of the persistence module as a finitely generated $\mathbb{N}$-graded module over a PID? Why is it true that the persistence diagram generated using the elder rule is the same as the diagram given by the structure theorem?
Best Answer
Maybe a simple example will help. Let $k$ be a field and consider the following filtered complex of $k$-vector spaces, starting at zero: $$k \xrightarrow{(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix})} k^2 \xrightarrow{\left(\begin{smallmatrix} 1 & 1 \end{smallmatrix}\right)} k.$$
If all you knew was that
then there isn't enough information to uniquely specify the module: it could be $k[t]/(t^3) \oplus t \cdot k[t]/(t)$ or $k[t]/(t^2) \oplus t \cdot k[t]/(t^2)$ in the language of persistence modules as $k[t]$-modules.
The extra information - which is contained in the decomposition of $k[t]$-modules but not in just the birth/death times - needed to reconstruct the module is the elder rule: which class was born earlier? This tells us among the classes that are merged at time $2$, there is an older class which survives from time $0$ to time $2$, so that the interval decomposition contains the module $k[t]/(t^3)$. So the correct decomposition is $k[t]/(t^3) \oplus t \cdot k[t]/(t)$.