Composita Stability Theorems for Enhanced Koszul Properties in Galois Cohomology

TL;DR

This paper establishes conditions under which the enhanced Koszul properties of Galois cohomology are preserved under composita of fields, with applications to specific classes of Pythagorean fields and their Galois groups.

Contribution

It formulates a composita stability theorem for Koszul properties in Galois cohomology and applies it to Pythagorean fields with specific Galois group decompositions.

Findings

Universal Koszulity is preserved under composita for certain fields.

Maximal pro-2 Galois groups of Pythagorean fields decompose as free products of Demuškin groups.

Provides obstructions for Galois groups based on cohomology properties.

Abstract

We investigate how enhanced Koszul properties of Galois cohomology behave under composita of fields. Given fields $K_1$ and $K_2$ containing $\mu_p$, with intersection $k$ and compositum $K = K_1K_2$, we formulate an abstract composita stability theorem: under a pro-$p$ amalgam decomposition $G_K \cong G_{K_1} *_{G_k} G_{K_2}$ of maximal pro-$p$ Galois groups, and natural Mayer-Vietoris compatibility assumptions on the mod-$p$ cohomology rings $H^\bullet(G_{K_1},\mathbb F_p)$, $H^\bullet(G_{K_2},\mathbb F_p)$, and $H^\bullet(G_k,\mathbb F_p)$, the quadratic presentation of $H^\bullet(G_K,\mathbb F_p)$ arises from a fiber-product construction on degree-$1$ generators and quadratic relations. Assuming stability of universal Koszulity under this quadratic gluing, we obtain that universal Koszulity of $H^\bullet(G_{K_1},\mathbb F_p)$ and $H^\bullet(G_{K_2},\mathbb F_p)$ implies universal Koszulity of $H^\bullet(G_K,\mathbb F_p)$. As a concrete application, we prove a composita stability theorem for certain Pythagorean fields whose maximal pro-$2$ Galois groups decompose as free pro-$2$ products of Demu\v{s}kin groups and free factors. For suitable composita $K = K_1K_2$ of such fields, the mod-$2$ Galois cohomology ring $H^\bullet(G_K(2),\mathbb F_2)$ remains quadratic and universally Koszul. This provides large classes of fields, built from local, global, and Pythagorean base fields by admissible extensions and composita, whose maximal pro-$p$ Galois groups have universally Koszul cohomology, and yields inverse Galois obstructions: any finitely generated pro-$p$ group with nonquadratic or non-universally Koszul mod-$p$ cohomology cannot occur as the maximal pro-$p$ Galois group of a field in these families.