Suppose I have a filtration of a simplicial complex $K$: $$ \emptyset=K_0\subseteq K_1\subseteq K_2\subseteq,...,\subseteq K_n=K $$
Suppose $\sigma_j$ is a $d$-dimensional simplex that first appears in $K_j$ and $\gamma_i$ is a $d-1$-dimensional simplex that appears first in $K_i$ where $i<j$. Additionally, there is a chain in $c\in C_d(K_j)$ (i.e the free group generated by the $d$-dimensional simplexes in $K_j$) such that $c=a_j\sigma_j+\sum a_k\sigma_k$ and $a_j\not=0$ such that the boundary of $c$ has $\gamma_i$ as a non-zero summand. Is it possible to show that the boundary of $c$ is not the boundary of any other $d$-dimensional chain in $C_d(k_h)$ where $i\leq h<j$?
The reason I ask is because in persistent homology, we use a form of gaussian elimination to calculate the birth and death barcodes of a persistent module: https://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf (Pg:8-9). So in the above situation, the homology class of boundary of $c$ is said to be born in $K_i$ but dies in $K_j$. However, I don't see why it can't be the case that there exists a chain $l$ in $K_h$, where $i\leq h<j$ such that $l$ also has the same boundary as $c$.
Can someone explain what is going on?