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

Abstract

Let k be a field, R a standard graded quadratic k-algebra with dimk⁡R2≤3, and let k‾ denote an algebraic closure of k. We construct a graded surjective Golod homomorphism φ:P→R⊗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 k-algebra with Hilbert series (1+2t−2t3)(1−t)−1. In particular, we recover earlier results on the Koszul property of Backelin [4], Conca [7] and D'Alì [11].