TITLE: Persistent homology over the integers
QUESTION [9 upvotes]: Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)?
Currently, persistent homology is usually done over a field (like $\mathbb{Z}/2$), as the algorithms for producing the barcode only work for a field.
However, working over a field loses a lot of information. Also, there are algorithms that can compute the persistent groups over a PID (http://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf Section 5).
Thanks for any help. References will be very welcome.

REPLY [10 votes]: As mentioned in Carlsson and Zomorodian's paper (to which you have linked), the problem of computing persistence barcodes with coefficients in a ring $R$ relies essentially on classifying graded modules over the polynomial ring $R[t]$. If (and only if!) $R$ is a field, $R[t]$ is a principal ideal domain and isomorphism classes of modules over it are easy to describe. When $R = \mathbb{Z}$, the modules are well-known to be hideous. Similar problems have plagued multi-dimensional persistence, since $R[s,t]$-modules are also pretty hard to classify (even when $R$ is a field).
But if you're willing to invest a bit in the Grothendieck construction, there might be hope: https://arxiv.org/abs/1601.03107