The space of homogeneous preserving semistar operations on graded domains

TL;DR

This paper investigates the structure of homogeneous preserving semistar operations on graded domains, establishing their properties, topological relationships with valuation overrings, and the spectral nature of finite type operations.

Contribution

It demonstrates that the set of finite type, homogeneous preserving semistar operations forms a spectral space and relates valuation overrings to Kronecker function rings.

Findings

Homogeneous preserving semistar operations are closed under certain transformations.

The set of valuation overrings of the Kronecker function ring is homeomorphic to the set of gr-valuation overrings.

The space of finite type, homogeneous preserving semistar operations is spectral.

Abstract

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain. In this paper we study the space of homogeneous preserving semistar operations on $R$. We show if $\star$ is a homogeneous preserving semistar operation on $R$, then $\star_a$ is also homogeneous preserving. Let $KR(R,b)$ be the homogeneous Kronecker function ring of $R$ with respect to the $b$-operation. It is shown that the set of valuation overrings of $KR(R,b)$, endowed with the Zariski topology, is homeomorphism to $Zar_h(R)$, the set of gr-valuation overrings of $R$, endowed with the Zariski topology. We also show that the set $SStar_{f,hp}(R)$ of finite type, homogeneous preserving semistar operations on $R$, endowed with the Zariski topology, is a spectral space.