A-schemes and Zariski-Riemann spaces

TL;DR

This paper explores properties of A-schemes, demonstrating their completeness, and introduces a universal compactification called the Zariski-Riemann space, comparing it with the classical version.

Contribution

It establishes the completeness and co-completeness of A-schemes and characterizes the Zariski-Riemann space as a universal compactification within this category.

Findings

A-schemes have properties similar to coherent schemes.

The category of A-schemes is both complete and co-complete.

The Zariski-Riemann space is identified as a universal compactification.

Abstract

In this paper, we will investigate further properties of A-schemes. The category of A-schemes possesses many properties of the category of coherent schemes, and in addition, it is co-complete and complete. There is the universal compactification, namely, the Zariski-Riemann space in the category of A-schemes. We compare it with the conventional Zariski-Riemann space, and characterize the latter by a left adjoint.