Matlis duality, inverse systems and classification of Artin algebras

Abstract

[en] The aim of this project is to study the classification of some families of Artin algebras. In order to do that, we will study some important results of injective modules with the objective to be able to prove Matlis duality. In particular, we will study the case of Matlis duality when $R=\mathbf{k} \llbracket x_1, \ldots, x_n \rrbracket$ (the ring of the formal series) with maximal ideal $\mathfrak{m}=\left(x_1, \ldots, x_n\right)$. With this scenario, we are talking about Macaulay's duality. Using Macaulay's correspondence, we will be able to study important results as Hilbert functions, essential in the classification of algebras. We will study Gorenstein, level and compressed algebras. Along the paper, we use Singular [DGPS22] and the library Inverse SYST.LIB by Joan Elias [Eli14], which with we will compute some examples seen through the project.