  Let $M$ be a module over a Noetherian local ring $A$. We study
$M$-independent sequences of elements of $\mathfrak{m}_A$ 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.
