Persistence module and Finite type

A persistence module $\mathcal{M}=\{M^i,\varphi^i\}_{i\geq 0}$ is a family of $R$-modules $M^i$, together with homomorphisms $\varphi^i: M^i\to M^{i+1}$.

For example, the homology of a persistence complex is a persistence module, where $\varphi^i$ maps a homology class to the one that contains it.

A persistence complex $\{C_*^i, f^i\}$ (resp.\ persistence module $\{M^i, \varphi^i\}$) is of finite type if each component complex (resp.\ module) is a finitely generated $R$-module, and if the maps $f^i$ (resp.\ $\varphi^i$) are isomorphisms for $i\geq m$ for some integer $m$.

If $K$ is a finite filtered simplicial complex, then it generates a persistence complex $\mathscr{C}$ of finite type, whose homology is a persistence module $\mathcal{M}$ of finite type.