Fundamental algebraic sets and locally unit-additive rings
Neil Epstein
TL;DR
The paper develops a tight correspondence between fundamental algebraic sets and unit-additive rings, extending the Fundamental Theorem of Algebra to a broad algebro-geometric setting. It introduces local, generic, and geometric variants of unit-additivity and proves that (locally) fundamental varieties correspond exactly to (locally) unit-additive coordinate rings, using decompositions along minimal primes and base-change techniques. The work further analyzes how these properties behave under localization, scheme structure, and base change, providing both positive results and subtle counterexamples (e.g., the unit circle) to delineate the limits of geometric descent. Overall, the results unify a spectrum of algebraic and geometric notions, offering a robust dictionary between fundamentality and unit-additivity with implications for affine varieties and schemes across base fields.
Abstract
The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be fundamental, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of local fundamentality is introduced and its behavior developed. On the algebraic side, the notions of locally, geometrically, and generically unit-additive rings are introduced, thus complementing unit-additivity as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout.