Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.08008 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918078453907456 |
|---|---|
| author | Kleppe, Jan O. Miró-Roig, Rosa M. |
| author_facet | Kleppe, Jan O. Miró-Roig, Rosa M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08008 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Schur powers of the cokernel of a graded morphism Kleppe, Jan O. Miró-Roig, Rosa M. Algebraic Geometry 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. |
| title | Schur powers of the cokernel of a graded morphism |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2311.08008 |