On a particular subspace of homogeneous preserving star operations

TL;DR

This paper studies homogeneous star operations on a graded integral domain, showing they extend classical star operations and that the set of finite type homogeneous star operations forms a spectral space.

Contribution

It demonstrates that homogeneous star operations are restrictions of classical star operations and characterizes the space of finite type homogeneous star operations as spectral.

Findings

Homogeneous star operations extend classical star operations.

The set of finite type homogeneous star operations is a spectral space.

Homogeneous star operations are restrictions of classical star operations.

Abstract

Let $\Gamma$ be a torsionless commutative cancellative monoid, $R=\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain. In this note we show that each homogeneous star operation $\star:\mathbf{HF}(R)\to\mathbf{HF}(R)$ of $R$, is the restriction of a (classical) star operation $e(\star):F(R)\to F(R)$ of $R$. We also show that the set $HStar_f(R)$ of homogeneous star operations of finite type on $R$, endowed with the Zariski topology, is a spectral space.