On finite generation in magnitude (co)homology and its torsion

The aim of this paper is to apply the framework developed by Sam and Snowden to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton, and we prove that it is a finitely generated functor (on graphs of bounded genus). More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs.