Classifying Persistence Modules over a PID via Structure Directed Acyclic Graphs

This manuscript develops a complete classification of one-parameter persistencemodules over a principal ideal domain via structure directed acyclic graphs (DAGs). Three main contributions: (1) Classification: The structure DAG is a complete isomorphism invariant; therealizability theorem establishes a bijection between DAG isomorphism classesand persistence module isomorphism classes. The connected components of the DAGfurnish a canonical SNF-adapted decomposition into indecomposables withoutrestriction on cokernels (classical Krull--Schmidt uniqueness does not hold inthis category; the DAG canonical decomposition is the strongest decompositiontheorem available). The category with fixed parameters is representation-finite. (2) Computational framework: The local--global principle reduces the isomorphismproblem over Z to a finite family of barcode computations over small Artinianprincipal ideal rings, realizable with existing persistent homology software. DAG construction and isomorphism testing run in polynomial time. (3) Applications: The ephemeral dependency graph Γ links graph topology tohomological invariants of coker (η) ᵉph: decomposition into summands indexed by (physical node, prime) pairs, sources as minimal generators, in-degree counts ofindependent thread morphisms, and reconstruction from Γ. A conjecture relatesthe longest directed path in Γ to the Ext-nilpotency index of the endomorphismalgebra. The framework unifies Gabriel's barcode, the Luo--Henselman-Petrusekinterval decomposition, and the Bockstein spectral sequence as degenerationlimits and direct corollaries.