Holomorphy rings and the Nullstellensatz over large fields

Abstract

For an infinite field $K$ and an ideal $I\subseteq K[X_1,\dots,X_n]$, let $J(Z_K(I))$ be the ideal of polynomials vanishing on the $K$-rational zero locus of $I$. A Nullstellensatz over $K$ is an algebraic description of this closure operator on ideals. Hilbert's classical theorem provides such a description via the radical of $I$ over any algebraically closed field. In the present article, we introduce a unifying framework for Nullstellensätze over arbitrary rings using the notion of an $A$-radical, for $A$ is a subring of $K(X)$ containing $K[X]$. We show that a single canonical holomorphy ring gives a Nullstellensatz over every infinite field, recovering Hilbert's theorem when $K$ is algebraically closed. In addition, we give a new characterization of large fields (in the sense of Pop) via a particular Nullstellensatz: a perfect field is large exactly when the \emph{rational} holomorphy ring $A_n(K)$--the intersection of all $K$-rational places of residue field exactly $K$--satisfies the corresponding Nullstellensatz. Finally we study the order structure of the class of rings $A$ for which the $A$-radical gives a Nullstellensatz. We show that over non-algebraically closed fields this class has no minimal element, showing there is no optimal Nullstellensatz over such fields.