2023/11/14 by Jan O. Kleppe, Kleppe, Jan O., Rosa M. Miró‐Roig +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2311.08008
openalex publication_date 2023/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let φ: F\longrightarrow G be a graded morphism between free R-modules of rank t and t+c-1, respectively, and let Ij(φ) be the ideal generated by the j × j minors of a matrix representing φ. In this short note: (1) We show that the canonical module of R/Ij(φ) is up to twist equal to a suitable Schur power ΣI M of M=\coker (φ^*); thus equal to \wedge t+1-jM if c=2 in which case we find a minimal free R-resolution of \wedge t+1-jM 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 ≥ 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.