Local characterization of block-decomposability for 3-parameter 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 3-parameter setting, and prove a corresponding structure theorem. Our result holds in the generality of pointwise finite-dimensional modules over products of arbitrary totally ordered sets. Our approach builds upon the one by Botnan and Crawley-Boevey, with new ingredients at places where the original proof was tied to the 2-parameter setting. This gives hope for future generalizations to arbitrary numbers of parameters.