LOCAL CHARACTERIZATION OF BLOCK-DECOMPOSABILITY FOR MULTIPARAMETER PERSISTENCE MODULES

Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a prominent role in levelset persistence. Here we generalize the local condition for decomposability into block modules to the n-parameter setting, and prove a corresponding structure theorem. Our result holds in the generality of pointwise finite-dimensional modules over finite products of arbitrary totally ordered sets. Our proof extends the one by Botnan and Crawley–Boevey from 2 to n parameters, which requires some crucial adaptations at places where their proof is fundamentally tied to the 2-parameter setting.