# Reduction algorithm for Persistent Homology

The reduction algorithm (pg. 5 of http://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf) enables us to compute homology for modules over a PID.

I am curious why the reduction algorithm cannot compute persistent homology (over a PID)? Where exactly does the algorithm fail?

I am aware that over a field $R$, then $R[t]$ is a PID, and the nice structure of the persistent module described in the paper allows the algorithm to work.

Persistent homology group is defined as: $H_k^{i,p}=Z_k^i/(B_k^{i+p}\cap Z_k^i)$, which is covered more in detail in pg. 6 of the above linked paper (Zomorodian and Carlsson). Superficially, the definition is similar in form to the definition of homology, except for some changes in the denominator.

Thanks for any enlightenment.