Editing Hilbert's Nullstellensatz (section)

== Generalizations ==
The Nullstellensatz is subsumed by a systematic development of the theory of [[Jacobson ring|Jacobson rings]], which are those rings in which every radical ideal is an intersection of maximal ideals. Given Zariski's lemma, proving the Nullstellensatz amounts to showing that if ''k'' is a field, then every finitely generated [[K-algebra|''k''-algebra]] ''R'' (necessarily of the form <math display="inline">R = k[t_1,\cdots,t_n]/I</math>) is Jacobson. More generally, one has the following theorem:

: Let <math>R</math> be a Jacobson ring. If <math>S</math> is a finitely generated [[R-algebra|''R''-algebra]], then <math>S</math> is a Jacobson ring. Furthermore, if <math>\mathfrak{n}\subseteq S</math> is a maximal ideal, then <math>\mathfrak{m} := \mathfrak{n} \cap R</math> is a maximal ideal of <math display="inline">R</math>, and <math>S/\mathfrak{n}</math> is a finite extension of <math>R/\mathfrak{m}</math>.<ref>{{Cite web |last=Emerton |first=Matthew |title=Jacobson rings |url=http://www.math.uchicago.edu/~emerton/pdffiles/jacobson.pdf |url-status=live |archive-url=https://web.archive.org/web/20220725071929/http://www.math.uchicago.edu/~emerton/pdffiles/jacobson.pdf |archive-date=2022-07-25}}</ref>

Other generalizations proceed from viewing the Nullstellensatz in [[Scheme (mathematics)|scheme-theoretic]] terms as saying that for any field ''k'' and nonzero finitely generated ''k''-algebra ''R'', the morphism <math display="inline">\mathrm{Spec} \, R \to \mathrm{Spec} \, k</math> admits a [[Section (category theory)|section]] étale-locally (equivalently, after [[Base change (scheme theory)|base change]] along <math display="inline">\mathrm{Spec} \, L \to \mathrm{Spec} \, k</math> for some finite field extension <math display="inline">L/k</math>). In this vein, one has the following theorem:
:Any [[Faithfully flat morphism|faithfully flat]] morphism of schemes <math display="inline">f: Y \to X</math> [[Glossary of algebraic geometry#finite presentation|locally of finite presentation]] admits a ''quasi-section'', in the sense that there exists a faithfully flat and locally [[Quasi-finite morphism|quasi-finite]] morphism <math display="inline">g: X' \to X</math> locally of finite presentation such that the base change <math display="inline">f': Y \times_X X' \to X'</math> of <math display="inline">f</math> along <math display="inline">g</math> admits a section.<ref>[[Éléments de géométrie algébrique|EGA]] §IV.17.16.2.</ref> Moreover, if <math display="inline">X</math> is [[Quasi-compact morphism|quasi-compact]] (resp. quasi-compact and [[Quasi-separated morphism|quasi-separated]]), then one may take <math display="inline">X'</math> to be affine (resp. <math display="inline">X'</math> affine and <math display="inline">g</math> quasi-finite), and if <math display="inline">f</math> is [[Smooth morphism|smooth]] surjective, then one may take <math display="inline">g</math> to be [[Étale morphism|étale]].<ref>EGA §IV.17.16.3(ii).</ref>
[[Serge Lang]] gave an extension of the Nullstellensatz to the case of infinitely many generators:

:Let <math display="inline">\kappa</math> be an [[infinite cardinal]] and let <math display="inline">K</math> be an algebraically closed field whose [[transcendence degree]] over its [[prime subfield]] is strictly greater than <math>\kappa</math>. Then for any set <math display="inline">S</math> of cardinality <math display="inline">\kappa</math>, the polynomial ring <math display="inline">A = K[x_i]_{i \in S}</math> satisfies the Nullstellensatz, i.e., for any ideal <math display="inline">J \sub A</math> we have that <math>\sqrt{J} = \hbox{I} (\hbox{V} (J))</math>.<ref>{{Cite journal |last=Lang |first=Serge |date=1952 |title=Hilbert's Nullstellensatz in Infinite-Dimensional Space |url=https://www.jstor.org/stable/2031893 |journal=[[Proc. Am. Math. Soc.]] |volume=3 |issue=3 |pages=407–410 |doi=10.2307/2031893 |jstor=2031893 |via=}}</ref>