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

We studies the nearly Gorenstein property for Veronese subalgebras of (semi-)standard graded algebras. We introduce a condition~$(\mathrm{♮})$ 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~$(\mathrm{♮})$, 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 $k>0$.