# A note on a sheaf of real regular functions

## TL;DR

This paper confirms that for real rings and integral domains, the global sections of a sheaf on the real Zariski spectrum coincide with a specific localization of the ring, answering a question posed in 1979.

## Contribution

It provides a positive answer to Coste and Roy's 1979 question about the structure sheaf on the real Zariski spectrum for real rings and integral domains.

## Key findings

- The global sections of the sheaf equal the localization of the ring at a certain multiplicative set.
- The result applies specifically to real rings and integral domains.
- It advances understanding of the structure sheaf in real algebraic geometry.

## Abstract

Coste and Roy in 1979 defined a structural sheaf on the real Zariski spectrum of a semi-real ring $A$ and asked whether the ring of the global sections is $\sum^{-1}_1 A$ where $\sum_1$ is the multiplicative subset $\{1+\sum_{i=1}^n a_i^2|a_i\in A, n\in N\}$ of $A$. We give a positive answer to this question for real rings and integral domains.