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

TL;DR

This paper explores the nearly Gorenstein property in Veronese subalgebras of graded algebras, introducing a new condition that ensures the property holds for large enough subalgebras, with implications for algebraic and combinatorial structures.

Contribution

It introduces a new condition $( atural)$ for Cohen--Macaulay semi-standard graded rings and proves that Veronese subalgebras inherit the nearly Gorenstein property under this condition.

Findings

Veronese subalgebras are nearly Gorenstein for large enough k under condition $( atural)$

Standard graded nearly Gorenstein algebras have nearly Gorenstein Veronese subalgebras for all k>0

The condition $( atural)$ is motivated by Ehrhart rings and extends the understanding of Gorenstein properties

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 $k>0$.