# The absolutely Koszul property of Veronese subrings and Segre products

## TL;DR

This paper develops a criterion to identify non-absolutely Koszul rings and applies it to classify which Segre products of polynomial rings are absolutely Koszul, revealing new insights into their algebraic structure.

## Contribution

It introduces a new criterion for detecting non-absolutely Koszul rings and completely classifies absolutely Koszul Segre products of polynomial rings in characteristic zero.

## Key findings

- Identified large families of non-absolutely Koszul Veronese subrings and Segre products.
- Classified all absolutely Koszul Segre products of polynomial rings in characteristic zero.
- Provided a practical method combining theoretical criteria and machine computations.

## Abstract

Absolutely Koszul algebras are a class of rings over which any finite graded module has a rational Poincar\'e series. We provide a criterion to detect non-absolutely Koszul rings. Combining the criterion with machine computations, we identify large families of Veronese subrings and Segre products of polynomial rings which are not absolutely Koszul. In particular, we classify completely the absolutely Koszul algebras among Segre products of polynomial rings, at least in characteristic $0$.