Hilbert's Nullstellensatz revisited (Q1117017)

This paper deals with Hilbert's Nullstellensatz in the context of toposes. The notion of field considered here is that of a ``geometric'' field, i.e. a ring object satisfyng \((0=1)\Rightarrow \perp\) and \(\forall x(T\Rightarrow (x=0)\vee \exists y(xy=1)).\) If the topos in question is sh(X), sheaves on a topological space X, then a sheaf of rings A is a geometric field iff the stalks \(A_ x\) are fields for all \(x\in X\). The first result presented in this paper proves the Nullstellensatz for sh(X), where X is a Boolean (compact, \(T_ 2\), totally disconnected) space. The analogue of the Nullstellensatz for regular rings proved by \textit{D. Saracino} and \textit{V. Weispfenning} [Lect. Notes Math. 498, 306- 383 (1975; Zbl 0318.13032)] is obtained as a corollary. The rest of the paper contains a summary of the author's work ``On the validity of Hilbert's Nullstellensatz, Artin's theorem and related results in Grothendieck toposes'' [J. Symb. Logic 53, No.4, 1177-1187 (1988)] extending these ideas to consider analogues for the Nullstellensatz in Grothendieck, as well as elementary, toposes. This involves using techniques and results from the study of intuitionistic theories.