# Adelic cohomology

## TL;DR

This paper introduces an adelic-style cohomological invariant for partially ordered sets, unifying various concepts in commutative algebra and stable equivariant homotopy theory through localizations and restricted products.

## Contribution

It develops a new cohomological invariant framework inspired by adeles, applicable to partially ordered sets with auxiliary structures, bridging multiple mathematical areas.

## Key findings

- Defines an adelic cohomology for posets with auxiliary data
- Unifies existing cohomological theories in algebra and topology
- Provides new tools for analyzing local-global phenomena in mathematics

## Abstract

The characteristic feature of the adeles is that they involve localizations of products (or equivalently restricted products of localizations). The point of this paper is to introduce an adelic style cohomological invariant of a partially ordered set with auxiliary structure which covers several examples of established interest in commutative algebra and stable equivariant homotopy theory.