Locally Gorenstein algebras, a generalization of Gorenstein algebras

Abstract

For an Artin algebra $Λ$, we define the class of locally Gorenstein modules, which is a generalization of semi-Gorenstein projective modules and, therefore, a generalization of projective Gorenstein modules. For a locally Gorenstein module $X\in{\rm mod}\ Λ$ we construct $\underline{χ}_X$ a subcategory of $\underline{\rm mod}$ $Λ$ and we prove that the syzygy functor is faithful and full on $\underline{χ}_X$. Using that, we prove that the Igusa-Todorov functions $ϕ$ and $ψ$ vanish on the family of locally Gorenstein modules. Finally, using locally Gorenstein modules, we define the locally Gorenstein algebras, a generalization of Gorenstein algebras, and we prove that their finitistic, and the $ϕ$ and $ψ$ dimensions are finite.