A linear generalization of the nearly Gorenstein property, with applications to Veronese subalgebras

Abstract

We studies the nearly Gorenstein property for Veronese subalgebras of (semi-)standard graded algebras. We introduce a condition~$(\natural)$ for Cohen--Macaulay semi-standard graded rings, motivated by the study of Ehrhart rings. We show that if a semi-standard graded algebra \( R \) satisfies~$(\natural)$, then its Veronese subalgebras \( R^{(k)} \) are nearly Gorenstein for all sufficiently large \( k \). We also prove that if a standard graded algebra $R$ is nearly Gorenstein so does its Veronese subalgebra $R^{(k)}$ for all $k>0$.