Reduced submodules of finite dimensional polynomial modules

2022/09/27 by Tilahun Abebaw, Nega Arega, Abebaw, Tilahun +5
Mathematics · #13E10 #16D60 #16D80 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2209.13447

openalex publication_date 2022/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a field with characteristic zero, R be the ring k[x₁, ⋯, x_n] and I be a monomial ideal of R. We study the Artinian local algebra R/I when considered as an R-module M. We show that the largest reduced submodule of M coincides with both the socle of M and the k-submodule of M generated by all outside corner elements of the Young diagram associated with M. Interpretations of different reduced modules is given in terms of Macaulay inverse systems. It is further shown that these reduced submodules are examples of modules in a torsion-torsionfree class, together with their duals; coreduced modules, exhibit symmetries in regard to Matlis duality and torsion theories. Lastly, we show that any R-module M of the kind described here satisfies the radical formula.

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