# Linear resolutions over Koszul complexes and Koszul homology algebras

## TL;DR

This paper explores new notions of Koszulness for graded algebras, analyzing how these properties relate among the algebra, its Koszul complex, and its homology algebra, with a focus on strand-Koszulness.

## Contribution

It introduces two extended definitions of Koszulness for differential graded algebras and homology algebras, and characterizes their interrelations, including the stronger strand-Koszulness property.

## Key findings

- Strand-Koszulness of homology algebra implies Koszulness of the original algebra.
- Complete characterization of relationships between Koszulness properties of R, K, and H.
- Examples of algebras with strand-Koszul homology algebra.

## Abstract

Let $R$ be a standard graded commutative algebra over a field $k$, let $K$ be its Koszul complex viewed as a differential graded $k$-algebra, and let $H$ be the homology algebra of $K$. This paper studies the interplay between homological properties of the three algebras $R$, $K$, and $H$. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one which applies to $K$ (and, more generally, to any connected differential graded $k$-algebra) and the other, called strand-Koszulness, which applies to $H$. The main theoretical result is a complete description of how these Koszul properties of $R$, $K$, and $H$ are related to each other. This result shows that strand-Koszulness of $H$ is stronger than Koszulness of $R$, and we include examples of classes of algebras which have Koszul homology algebras that are strand-Koszul.