Hilbert's Nullstellensatz

Hilbert's Nullstellensatz (German: "theorem of zeros") is a theorem which makes precise a fundamental relationship between the geometric and algebraic sides of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. The theorem was first proved by David Hilbert, after whom it is named.

Formulation

Let K be an algebraically closed field (such as the complex numbers), consider the polynomial ring K[X₁,X₂,..., X_n] and let I be an ideal in this ring. The affine variety V(I) defined by this ideal consists of all n-tuples x = (x₁,...,x_n) in Kⁿ such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in K[X₁,X₂,..., X_n] which vanishes on the variety V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number r such that p^r is in I.

An immediate corollary is the "weak Nullstellensatz": if I is a proper ideal in K[X₁,X₂,..., X_n], then V(I) cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal. This is the reason for the name of the theorem, which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumption that K be algebraically closed is essential here; the elements of the proper ideal (X² + 1) in R[X] do not have a common zero.

With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as

${\hbox{I}}({\hbox{V}}(J))={\sqrt {J}}$

for every ideal J. Here, ${\sqrt {J}}$ denotes the radical of J and I(U) is the ideal of all polynomials which vanish on the set U.

In this way, we obtain an order-reversing bijective correspondence between the affine varieties in Kⁿ and the radical ideals of K[X₁,X₂,..., X_n]. In fact, more generally, one has a Galois connection between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operators.

Generalization

This generalization is due to Bourbaki, and is the most general form of the Nullstellensatz.

Let $R$ be a Jacobson ring. If $S$ is a finitely generated R-algebra, then $S$ is a Jacobson ring. Further, if ${\mathfrak {n}}\subset S$ is a maximal ideal, then ${\mathfrak {m}}:=n\cap R$ is a maximal ideal of R, and $S/{\mathfrak {n}}$ is a finite extension field of $R/{\mathfrak {m}}$.

Another generalization states that a faithfully flat morphism $f:Y\to X$ locally of finite type with X quasi-compact has a quasi-section, i.e. there exists $X'$ affine and faithfully flat and quasi-finite over X together with an X-morphism $X'\to Y$.

See also

• Stengle's Positivstellensatz
• Differential Nullstellensatz
• Combinatorial Nullstellensatz

References

• Shigeru Mukai (2003). An Introduction to Invariants and Moduli. Cambridge studies in advanced mathematics. Vol. 81. p. 82. ISBN 0-521-80906-1. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• David Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, New York : Springer-Verlag, 1999.