I'm reading Zomorodian and Carlsson's paper on computing persistent homology. The object of their algorithm is, given a filtered complex, to find a representation of the kth persistent homology group as: $$(\bigoplus_{i=0}^n\Sigma^{\alpha_i}F[t])\oplus(\bigoplus_{i=0}^n\Sigma^{\beta_i}F[t]/(t^{\gamma_i}))$$ Where $F$ is a field and $\Sigma^{\alpha_i}$ represents a shift up in the standard grading of $F[t]$ by $\alpha_i$. They represent each factor in this decomposition as a "P-Interval:" $(a,\;b)$ where $a\leq b\in\mathbb{N}\;\cup \{\infty\}$.

Let $M_k$ be the matrix representation of the kth boundary operator. The main part of the algorithm is computing the appropriate basis for $B_k$ inside $Z_k$. This is accomplished "inductively" by putting $M_k$ in column eschelon form and then doing the same thing to $M_{k+1}$ after deleting the rows of $M_{k+1}$ which correspond to non-zero columns of $M_k$ (we know that these rows must be zero since $M_k M_{k+1}=0$). If the simplices of a given dimension are sorted in reverse according to their filtration index, it can be shown that the pivot elements of the column eschelon form correspond to the diagonal elements of the normal form.

Corollary 4.1 is what I don't understand. If row i of the reduced form of $M_k$ has pivot entry $t^n$ then it contributes $(\deg\hat{e}_i,\;\deg\hat{e}_i+n)$ to then description of $H_{k-1}$ and $(\deg\;\hat{e}_i, \infty)$ if the pivot is $0$. I understand $\deg\;\hat{e}_i$ to be the filtration index of the $i$th (k-1)-simplex in the filtration. Honestly I'm not at all sure how to process this, it doesn't seem reminiscent of the calculation of simplicial/singular homology at all.


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