  Let $\varphi: F\longrightarrow G$ be a graded morphism between free
$R$-modules of rank $t$ and $t+c-1$, respectively, and let $I_j(\varphi)$ be
the ideal generated by the $j \times j$ minors of a matrix representing
$\varphi$. In this short note: (1) We show that the canonical module of
$R/I_j(\varphi)$ is up to twist equal to a suitable Schur power $\Sigma ^I M$
of $M=\coker (\varphi ^*)$; thus equal to $\wedge ^{t+1-j}M$ if $c=2$ in which
case we find a minimal free $R$-resolution of $\wedge ^{t+1-j}M$ for any $j$,
(2) For $c = 3$, we construct a free $R$-resolution of $\wedge ^2M$ which
starts almost minimally (i.e. the first three terms are minimal up to a precise
summand), and (3) For $c \ge 4$, we construct under a certain depth condition
the first three terms of a free $R$-resolution of $\wedge ^2M$ which are
minimal up to a precise summand. As a byproduct we answer the first open case
of a question posed by Buchsbaum and Eisenbud.
