Schur powers of the cokernel of a graded morphism

Let $φ: F\longrightarrow G$ be a graded morphism between free $R$-modules of rank $t$ and $t+c-1$, respectively, and let $I_j(φ)$ be the ideal generated by the $j \times j$ minors of a matrix representing $φ$. In this short note: (1) We show that the canonical module of $R/I_j(φ)$ is up to twist equal to a suitable Schur power $Σ^I M$ of $M=\coker (φ^*)$; 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.