Schur powers of the cokernel of a graded morphism

TL;DR

This paper investigates the structure of Schur powers of the cokernel of a graded morphism between free modules, providing explicit resolutions and answering an open question in the case c=3.

Contribution

It introduces new resolutions for Schur powers of cokernels of graded morphisms, extending known results and solving an open case for c=3.

Findings

Canonical module of R/I_j(φ) relates to Schur powers of cokernel

Constructs explicit free resolutions for specific cases

Answers an open question by Buchsbaum and Eisenbud for c=3

Abstract

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.