The space of finitely generated rings
Yves Cornulier
TL;DR
This paper explores the topological structure of the space of finitely generated rings, computing the Cantor-Bendixson rank for various cases, revealing deep insights into their complexity and classification.
Contribution
It introduces a detailed analysis of the space of finitely generated rings and modules, including explicit calculations of the Cantor-Bendixson rank for key examples.
Findings
The space of marked commutative rings is compact and metrizable.
The Cantor-Bendixson rank of the free commutative ring on n generators is omega^n.
The study extends to finitely generated modules over a commutative ring.
Abstract
The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor-Bendixson rank of any member of this space. For instance, the Cantor-Bendixson rank of the free commutative ring on n generators is omega^n, where omega is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.
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