CUMC talk notes v3

A Model Theoretic Proof of Hilbert’s
Weak Nullstellensatz
Eeshan Wagh
University of Waterloo
July 11, 2013
Eeshan Wagh (University of Waterloo) Model Theory and the Weak Nullstellensatz July 11, 2013 1 / 16

What is Model Theory?
Model Theory is a branch of mathematical logic which studies classes of
mathematical structures using ﬁrst-order logic. For example, one
important class of objects we study in Model Theory are the Algebraically
Closed Fields (abbreviated as ACF). These have some very nice properties
(as we shall see) and form an important connection between Model Theory
and Abstract Algebra. In particular, we will use it prove Hilbert’s Weak
Nullstellensatz.

The Weak Nullstellensatz
The goal of this talk will be to prove the following statement:
Theorem
Suppose K is an algebraically closed ﬁeld and I is a proper ideal in the
polynomial ring K[X1, . . . , Xn]. Then, there exists a tuple a ∈ Kn such
that P(a) = 0 for all P ∈ I.
The statement of this theorem is purely algebraic. There is no mention of
any logic or model theory and one can prove this theorem just using
algebra. However, once we have developed the standard machinery, this
proof will follow nicely.

A brief introduction to Model Theory
Deﬁnition
A Structure M consists of a non-empty set M (called the universe of M)
together with
A set {ci : i ∈ IC } of distinguished elements of M called constants
A set {fi : Mni → M : i ∈ IF } of distinguised maps from various
cartesian powers of M to itself, called Basic Functions and
A set {Ri : Mki : i ∈ IR} of distinguished subsets of various cartesian
powers of M called Basic Relations
The constants, basic functions and relations are called the signature of M.
A structure is nothing but a set M equipped some additional structure.
For example, consider R as an additive group. Then, M = R, 0 is a
constant and the basic function is the addition of real numbers (there is no
relation in this example).

Deﬁnition
A language L a set of symbols consisting of
Constant symbols
Function symbols
Relation Symbols
An L-Structure is a structure M with a bijective correspondence, j,
between the signature of M with the language. if c ∈ L is a constant and
then we let cM := j(c), the associated element in M (and use the same
convention for functions and relations)
Example
Let L = (0L, +L,−1 ), the language of abelian groups and let M = R,
viewed as an additive group. Then, if we map the symbol 0L to the real
number 0, +L maps to + and −1(x) = −x, then R is an L-structure. In
this case we have 0M
L = 0, etc.

Deﬁnition
Suppose L is a language and M, N are L-structures. An L-embedding of
M in N is an injective map j : M → N such that
For all constants c ∈ L, have j(cM) = cN .
For all function symbols in f ∈ L, if a ∈ Mn then j(f M(a)) = f N (a).
For all relation symbols R ∈ L if a ∈ Mk, then
a ∈ RM ⇐⇒ j(a) ∈ RN .
If j is also a surjection, then we say that j is an L-isomorphism. If M ⊆ N
and j is the inclusion map, then M is a substructure of N.
Example
Let L1 = {0, 1, +, −, ×}, the language of rings, L2 = {0, +, −}, the
language of additive groups and L3 = ∅. Then, under the natural
interpretations, the L1-substructures of R are its subrings, the
L2-substructures are the additive subgroups and the L3-substructure are
the non-empty subsets.

Definition
An L-formula is somewhat technical to define so we will use an informal
definition here for our purposes. An L-formula is a logical formula
φ(x1, . . . , xn) which can use symbols from L as well as logical operators
such as ∧, ∨, ¬ and quantifiers such as ∃, ∀. I think its best to illustrate it
with an example. If L = {0, 1, +, −, ×}, then let
φ(x) := ∃y(xy = 1)
φ says that x has a multiplicative inverse. If M is an L-structure and
a ∈ M, then we say M |= φ(a), or M satisfies φ(a) if a has a
multiplicative inverse in M. The set of a such that M |= φ(a) is called
the set defined by φ in M, denoted φM. For example, in a ring φ defines
the set of units of that ring.
Note that an L-formula is only allowed to use the constant symbols in the
formula. We could not refer to a specific element of the universe. If we
wanted to be able to refer to a set A in our formulas, we call these
LA-formulas.

Deﬁnition
Suppose M and N are L-structures with universes M and N respectively
with M ⊆ N. Then we say that N is an elementary extension of M (or M
is an elementary substructure), denoted by M N, if for all L-formulas
φ(x1, . . . , xn) and all tuples a ∈ Mn, then M |= φ(a) ⇐⇒ N |= φ(a).
The property of being an elementary substructure is much stronger than
simply being a substructure.
Example
Consider (Z, 0, +, −) ⊆ (Q, 0, +, −) which is a substructure since Z is a
subgroup of Q. Let φ(x) := ∃y(y + y = x), well we have that Q |= φ(1)
but there is no z ∈ Z such that z + z = 1, so Z |= ¬φ(1). So Z is not an
elementary substructure of Q as an additive group.

Deﬁnition
An L-formula φ that has no free variables is called an L-sentence. For
example if L = (0, 1, +, −, ×), φ := ∀xy(x + y = y + x) is an L-sentence.
Deﬁnition
An L-theory is a set of L-sentences. A Model of a theory T is an
L-structure M such that M |= σ for each σ ∈ T. If T has a model then
we say that T is consistent.
Example
Let L = (0, 1, ×,−1 ) and T be the set of sentences:
∀ab∃c(ab = c)
∀abc(a(bc) = (ab)c)
∀a∃b(ab = ba = 1)
∀a(a × 1 = 1 × a)
Then T is the theory of Groups, and if G is a group then G is a model of
T.

Example
Other examples of theories we might study in Model Theory are
The theory of Algebraically Closed Fields. We will discuss properties
of this theory shortly and will use these properties to prove the
Nullstellensatz.
The theory of Abelian Groups, Torsion Free Groups
For any fixed field K, the theory of K-vector spaces
Let L = {R} the language of consisting of a single binary relation
symbol. Then, we can study the theory of posets, graphs, linear
orders, etc.
Let L = {∈}, then we can study the theory of Zermelo-Fraenkel Set
Theory as each of the axioms can be expressed as L-sentences.
The theory of compact, complex manifolds.
The theory of Differentially Closed Fields.

Our Required Machinery
And now we get to the model theoretic property which will be the crux of
the proof of the Nullstellensatz
Deﬁnition
A theory T is said to Model-Complete if whenever M, N are models of
T and M ⊆ N, then M N.
We remarked earlier that just because if M is a substructure of N does
not mean that M N. However, if our theory has Model-Completeness,
and M ⊆ N are both models of the theory then it is true. So what
theories admit Model-Completeness?
Fact
The theory of Algebraically Closed Fields of characteristic p admit
Model-Completeness.

Proving that ACFp has model-completeness requires some extra work but
is a very standard result in Model Theory. Other theories that admit
Model-Completeness
Example
Infinite Vector Spaces over a field K
Dense Linear Orders: Let L = {≤} and T the theory which says that
≤ is a linear order such that ∀xy∃z : x < z < y.
Real Closed Ordered Fields: L = {0, 1, +, −, ×, ≤} T is theory of
fields together with an axiom that says any model is not algebraically
closed but the algebraic closure is a finite field extension.

The Weak Nullstellensatz
Theorem
Let K be an algebraically closed field and I is a proper ideal in the
polynomial ring K[X1, . . . , Xn]. Then, there exists a tuple a ∈ Kn such
that P(a) = 0 for all P ∈ I.
Proof
Let I be an ideal of K[X1, . . . , Xn]. Then, I is contained in some
prime ideal (for example, by Zorn’s lemma I is contained in a maximal
ideal). So we may assume that I is prime.
By the Hilbert Basis Theorem, since K is noetherian, I is finitely
generated by P1, . . . , Pl . So it will suffice to find a common zero for
these polynomials.
let Γ := K[X1, . . . , Xn]/I, Γ is an integral domain since I is prime. Let
F be the fraction field of Γ and Falg denote the algebraic closure of F.

Proof Continued
We can view K naturally as a subfield of Γ and thus as a subfield of
Falg . By Model-Completeness since K, Falg are both algebraically
closed, K Falg .
Let π : K[X1, . . . , Xn] → Γ denote the quotient map.
For all polynomials P ∈ I, we have
P(πX1, . . . , πXn) = π(P(X1, . . . , Xn)) = 0 (since π is a
homomorphism).
Thus, ∀P ∈ I we have (πXi , . . . , πXn) ∈ L is a root of P. So let
φ := ∃x1, . . . , xn
n
i=1
Pi (x) = 0
phrasing our above observation Model-Theoretically, we have by our
definition of elementary extension that since Falg |= φ have K |= φ.
That is, there exists b ∈ Kn such that Pi (b) = 0 and so for all P ∈ I
we have P(b) = 0.

If Time Permits
In Model Theory one of the fundamental question one can ask about a
theory is, what sort of sets are definable? and one can look at the
geometry of these definable sets.
Example
Let T = ACFp, then the definable sets are just boolean combinations
(finite unions, intersections and complements) of zero-sets of polynomials.
ACFp belong to a very well-behaved class of theories known as Strongly
Minimal Theories which consist of theories whose models are such that
every definable set is either finite or co-finite. The theory of infinite vector
spaces over a field K is also strongly minimal.

Extending this idea further, given a structure M we can look at what kind
of algebraic structures we can define in M. For example, what sort of
groups can we define in M: By this we mean that the underlying set of
the group is definable and the group operations are definable functions. In
the case of algebraically closed fields, the result is quite nice
Theorem [Weil-Van den Dries-Hrushovski]
Let G be a group definable in an algebraically closed field K, then G is
definably isomorphic to an algebraic group (an algebraic group is a group
whose underlying set is a variety and the group operations are given by
regular maps )

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