Asymptotic Universal Koszulity in Galois Cohomology

TL;DR

The paper introduces asymptotic universal Koszulity for graded algebras, especially in Galois cohomology, providing structural properties, criteria, and principles for understanding their algebraic and cohomological behavior.

Contribution

It develops a new framework for analyzing infinite-dimensional algebras via filtered systems of finite-type Koszul subalgebras, with applications to Galois cohomology.

Findings

Proves stability of asymptotic universal Koszulity under colimits, products, and base change.

Establishes a colimit theorem for Galois cohomology rings of pro-p groups.

Identifies conditions under which Galois cohomology is asymptotically universally Koszul.

Abstract

We introduce the notion of asymptotic universal Koszulity for graded-commutative algebras generated in degree~$1$, capturing the idea that an infinite-dimensional algebra can be approximated by a filtered system of finite-type universally Koszul quadratic subalgebras. We establish basic structural properties of this class, including stability under filtered colimits, direct products, and base change, as well as a local finite-type criterion expressed in terms of finite-dimensional subspaces of the degree-one component. In the context of Galois cohomology, we prove a colimit theorem for pro-$p$ groups under mild assumptions, showing that cohomology rings arise as filtered colimits of finite quotients. This yields a general criterion under which the cohomology algebra of a profinite group is asymptotically universally Koszul. We further analyze finitely generated quotients via a finite-type capture result, identifying their cohomology with canonical quadratic subalgebras of the ambient algebra. Finally, we formulate conditional local--global and patching principles that isolate the mechanisms by which asymptotic universal Koszulity may arise in arithmetic settings. These results provide a flexible structural framework linking homological algebra, quadratic algebras, and Galois cohomology, and suggest several directions for further investigation.