On graded going-down domains, II

TL;DR

We study the graded going-down property for extensions of $\Gamma$-graded domains, i.e., the $gGD$ property, using pullback constructions to extend the classical going-down framework. In a pullback of type $\bigtriangleup$, with $T=\bigoplus_{\alpha\in\Gamma}T_{\alpha}$, $M$ a maximal homogeneous ideal, $k=T/M$, $D=\bigoplus_{\alpha\in\Gamma}D_{\alpha}$ a graded subring of $k$, and $R=\varphi^{-1}(D)$, we prove that $R$ is $gGD$ if and only if both $T$ and $D$ are $gGD$. The $gGD$ property is local, so testing at maximal homogeneous ideals suffices, and the results yield corollaries for graded CPI-extensions $R(P)$. The paper also provides explicit pullback-based examples of $gGD$ domains, including cases that are not graded-Prüfer, and discusses how gr-Noetherian assumptions and $h$-dimension interact with the $gGD$ property, highlighting the utility of pullbacks in graded multiplicative ideal theory.

Abstract

In this paper we consider the graded going-down property of graded integral domains in pullbacks. It then enables us to give original examples of these domains.

Theorems & Definitions (26)

• Theorem 1.1
• Proposition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 2.6