# Algebraic geometry over the residue field of the infinite place

## TL;DR

This paper develops an elementary algebraic framework for modules and algebras over the residue field at the infinite place, establishing foundational properties and prime decomposition in this context.

## Contribution

It introduces an algebraic approach to the residue field at the infinite place, including prime congruences and Krull dimension results, advancing Arakelov geometry theory.

## Key findings

- Polynomial ring of n variables has Krull dimension n
- Prime decomposition theorem for prime congruences
- Elementary algebraic approach to modules over the residue field

## Abstract

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.