Remark

It follows from the proof of Corollary 1 that Corollaries 1 and 2 (resp. Corollary 3) are still valid when, instead of assuming that $\mathbf{M}$ (resp. $\mathbf{N}$) is finitely generated, the following hypothesis is made: there exists a subset $\Delta^+$ of $\Delta$ satisfying the following conditions:

(1) for $\lambda \notin \Delta^+$, $M_\lambda = \{0\}$ (resp. $N_\lambda = \{0\}$); (2) every non-empty subset of $\Delta^+$ has a least element.

This will be the case if $\Delta = \mathbf{Z}$ and $\mathbf{M}$ (resp. $\mathbf{N}$) is a graded module with positive degrees.