Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic Universal Koszulity in Galois Cohomology

Marina Palaisti

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.

Asymptotic Universal Koszulity in Galois Cohomology

Abstract

We introduce the notion of asymptotic universal Koszulity for graded-commutative algebras generated in degree~, 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- 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.

Paper Structure

This paper contains 9 sections, 19 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Quadratic algebras and universal Koszulity
  3. Asymptotic universal Koszulity
  4. Local finite-type structure
  5. Structural theorems
  6. Galois cohomology
  7. Examples
  8. Conditional local-global descent
  9. Questions

Key Result

Proposition 3.2

Let $(A_i)_{i\in I}$ be a filtered system of quadratic graded $k$-algebras with injective transition maps such that each $A_i$ is finite-type and universally Koszul. Then $A := \varinjlim_{i\in I} A_i$ is asymptotically universally Koszul.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • proof
  • ...and 35 more