The Yoneda algebra of a graded Ore extension

TL;DR

This paper investigates the relationship between the Yoneda algebras of a graded algebra and its Ore extension, proving a converse to a known property linking the K_2 condition of the base algebra and its extension.

Contribution

It establishes that if a graded Ore extension of an algebra satisfies the K_2 property, then the original algebra also satisfies K_2, completing the previous understanding.

Findings

Proves the converse of Cassidy and Shelton's result.

Shows the K_2 property is preserved from the extension to the base algebra.

Enhances understanding of the structure of Yoneda algebras in graded Ore extensions.

Abstract

Let A be a connected-graded algebra with trivial module k, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A) := Ext_A(k,k) to E(B). Cassidy and Shelton have shown that when A satisfies their K_2 property, B will also be K_2. We prove the converse of this result.