The absolutely Koszul and Backelin-Roos properties for spaces of quadrics of small codimension

TL;DR

This paper investigates the Koszul and Backelin-Roos properties of quadratic algebras with small codimension, establishing conditions under which these algebras are absolutely Koszul and constructing relevant Golod homomorphisms.

Contribution

It introduces new criteria for absolute Koszulness in quadratic algebras of small codimension and constructs explicit Golod homomorphisms linking these algebras to complete intersections.

Findings

Rings with $ ext{dim}_ extbf{k} R_2 extless 4$ are absolutely Koszul iff they are Koszul.

Constructs a Golod homomorphism from a complete intersection to the algebra.

Characterizes when such algebras are not trivial fiber extensions.

Abstract

Let $\kk$ be a field, $R$ a standard graded quadratic $\kk$-algebra with $\dim_{\kk}R_2\le 3$, and let $\ov\kk$ denote an algebraic closure of $\kk$. We construct a graded surjective Golod homomorphism $\varphi \colon P\to R\otimes_{\kk}\ov{\kk}$ such that $P$ is a complete intersection of codimension at most $3$. Furthermore, we show that $R$ is absolutely Koszul (that is, every finitely generated $R$-module has finite linearity defect) if and only if $R$ is Koszul if and only if $R$ is not a trivial fiber extension of a standard graded $\kk$-algebra with Hilbert series $(1+2t-2t^3)(1-t)^{-1}$. In particular, we recover earlier results on the Koszul property of Backelin, Conca and D'Al\`i.