Let M be a module over a Noetherian local ring A. We study M-independent sequences of elements of m A in the sense of Lech and Hanes. The main tool is a new characterization of the M-independence of a sequence in terms of the associated Koszul complex. As applications, we give a result in linkage theory, a freeness criterion for M in terms of the existence of a strongly M-independent sequence of length edim(A), and another freeness criterion inspired from the patching method of Calegari and Geraghty for balanced modules in their 2018 paper.