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.

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