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, although the proof presented in this conference version is restricted to the finite poset case for simplicity. 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.