# Homogeneous triples for homogeneous algebras with two relations

## TL;DR

This paper classifies connected s-homogeneous algebras with two relations using s-homogeneous triples, providing criteria for s-Koszulity and describing their Ext-algebras, thus advancing understanding of their structure and properties.

## Contribution

It offers a classification of such algebras via pairs (A,M), characterizes their s-Koszulity, and describes their Ext-algebras, extending previous work with new criteria and realizations.

## Key findings

- Classification of pairs (A,M) for these algebras
- Criteria for checking s-Koszulity via Hilbert series or Koszul complex
- Description of Ext-algebras for most such algebras

## Abstract

In our preceding paper we have introduced the notion of an $s$-homogeneous triple. In this paper we use this technique to study connected $s$-homogeneous algebras with two relations. For such algebras, we describe all possible pairs $(A,M)$, where $A$ is the $s$-Veronese ring and $M$ is the $(s,1)$-Veronese bimodule of the $s$-homogeneous dual algebra. For each such a pair we give an intrinsic characterization of algebras corresponding to it. Due to results of our previous work many pairs determine the algebra uniquely up to isomorphism. Using our partial classification, we show that, to check the $s$-Koszulity of a connected $s$-homogeneous algebras with two relations, it is enough to verify an equality for Hilbert series or to check the exactness of the generalized Koszul complex in the second term. For each pair $(A,M)$ not belonging to one specific series of pairs, we check if there exists an $s$-Koszulity algebra corresponding to it. Thus, we describe a class of possible ${\rm Ext}$-algebras of $s$-Koszul connected algebras with two relations and realize all of them except a finite number of specific algebras as ${\rm Ext}$-algebras. Another result that follows from our classification is that an $s$-homogeneous algebra with two dimensional $s$-th component cannot be $s$-Koszul for $s>2$.