
TL;DR
This paper explores the concept of Diophantine sets, their generalizations to other rings, and proves a key surjectivity property for finitely presented schemes over rings, contributing to algebraic geometry and number theory.
Contribution
It compares various definitions of Diophantine sets across different rings and establishes a surjectivity result for finitely presented schemes over rings.
Findings
Comparison of Diophantine set definitions in different rings
Existence of affine schemes with surjective rational points maps
Advancement in understanding Diophantine properties in algebraic geometry
Abstract
Diophantine subsets of play a key role in the negative answer to Hilbert's tenth problem. The definition of diophantine set generalizes in several ways to other commutative rings. We compare these definitions. Along the way, we prove that for every finitely presented scheme over a ring , there exists an affine -scheme with a finitely presented -morphism such that is surjective for every -algebra .
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Rings, Modules, and Algebras
