Hilbert's Nullstellensatz and the real Nullstellensatz for modules and sheaves

Source abstract

We prove the geometric halves of Hilbert's Nullstellensatz and of the real Nullstellensatz for submodules of finitely generated modules and for coherent sheaves, complementing our earlier result that semiprime (resp. real) submodules are intersections of prime (resp. real prime) submodules. First, we pass from prime to maximal submodules: over a Jacobson ring, a prime submodule $P$ of $M$ is an intersection of maximal submodules whenever $M/P$ is a torsionless $R/(P:M)$-module, in particular whenever $M/P$ is finitely generated (the latter case is due to Maani Shirazi and Sharif). Consequently, if $R$ is a finitely generated algebra over an algebraically closed field and $M$ is finitely generated, the smallest semiprime submodule containing a submodule $N$ of $M$ consists of all $f\in M$ such that, for every point $a$, the value $f(a)$ (i.e. the image of $f$ in the fibre $M/\mathfrak{m}_aM$) lies in $N(a)$ (i.e. the image of $N$ in $M/\mathfrak{m}_aM$). We prove the real analogues over real Jacobson rings; for finitely generated algebras over a real closed field they are determined by the real points. The module classifications and directional vanishing formulas yield both Nullstellensätze for coherent sheaves on schemes of finite type over a field. We also prove compatibility of semiprimeness, reality and both radicals with arbitrary localization, and the resulting stalk criteria.