Fibre-Product Stability, Kummer Gluing, and Composita for Universal Koszulity

Abstract

We study universal Koszulity under algebraic gluing and field composita. For quadratic fibre products, orthogonally split pullbacks reduce to direct sums, while Stanley--Reisner pullbacks arise from graph gluing preserving universal Koszulity under a dominating-clique hypothesis. For composita, Kummer theory and the norm-residue theorem yield a graph criterion from restriction images and degree-two cup-product relations, verified for iterated Laurent-series fields with nonzero mixed symbols. In a nonabelian RAAG family, every finite \(p\)-extension has explicitly determined universally Koszul cohomology, with \[ G_L(p)\cong F_{1+q(d-1)}\times\mathbb Z_p^s.\]