Duality and syzygies for semimodules over numerical semigroups

Abstract

Let $\Gamma =\langle \alpha , \beta \rangle$ be a numerical semigroup. In this article we consider the dual $\Delta ^*$ of a $\Gamma$-semimodule $\Delta$; in particular we deduce a formula that expresses the minimal set of generators of $\Delta ^*$ in terms of the generators of $\Delta$. As applications we compute the minimal graded free resolution of a graded ${\mathbb {F}}[t^{\alpha },t^{\beta }]$-submodule of ${\mathbb {F}}[t]$, and we investigate the structure of the selfdual $\Gamma$-semimodules, leading to a new way of counting them.