minimal free resolution differentials land in maximal ideal

The condition that an image lie in $\mathfrak{m}{F}_{i-1}$ is exactly that all matrix entries lie in $\mathfrak{m}$, and is basis-independent. Since the complement of $\mathfrak{m}$ consists of units, failure supplies a unit entry. This is the minimality convention of the existence lemma.

Move that entry to the first position, scale it to $1$, and clear its row and column by elementary basis changes. The matrix becomes $\mathrm{diag}(1,D)$. The identities $d_{i-1}{d}_i=d_i{d}_{i+1}=0$ force adjacent maps to vanish on or into the isolated coordinates; for $i=1$ the augmentation also vanishes there. Hence these coordinates form the direct summand $0\to R\stackrel{1}{\to }R\to 0$. Deleting it preserves exactness. Conversely an identity summand cannot have zero residue differential, while every matrix with entries in $\mathfrak{m}$ does.