THE GOLOD-SHAFAREVICH INEQUALITY
1
The Golod-Shafarevich Inequality
By Jason Preszler
Abstract
These are notes for a presentation made on the Golod-Shafarevich
inequality and it’s application to class field towers. It is assumed the
reader is familiar with the basic theory of number fields and is acquainted
with commutative and homological algebra. A very minimal introduction
to the requisite class field theory is given.
1. Introduction
In the 1920’s abelian class field theory was put in a more or less finished
state by the work of Hilbert, E. Artin, Takagi and Hasse, among others. Class
field theory provides a framework for determining and classifying the abelian
extensions of an algebraic number field using the structure of the ground field,
something that is of great interest to number theorists. Classic results in early
class field theory are the quadratic reciprocity law of Gauss and the KroneckerWeber theorem. A fundamental part of global abelian class field theory is the
construction of the Hilbert class field, which is the topic of the next section.
Furtwangler, in 1924 was studying the Hilbert Class Field (HCF) of a
number field k. He began building towers by starting with k1 = k, then
forming k2 = HCF (k1 ), k3 = HCF (k2 ), etc. to obtain the following:
..
.
k2 = HCF (k1 )
k1
Q.
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JASON PRESZLER
Furtwangler, after calculating a number of examples, was lead to conjecture
that all such towers are finite, i.e. there exists an N such that km = HCF (km ) ∀ m ≥
N . In 1964, Golod and Shafarevich proved an inequality regarding the dimension of the Galois cohomology groups H 1 (k, Z/pZ) and H 2 (k, Z/pZ), a result
that can be phrased entirely within the context of group cohomology. They
then used this inequality to construct a counter-example to Furtwangler’s class
field tower conjecture.
After a brief introduction to abelian global class field theory (in particular we define the Hilbert Class Field), we will prove the Golod-Shafarevich
inequality and use this to produce infinite class field towers. Surprisingly, one
can find infinite class field towers of imaginary quadratic fields (but to Furtwangler’s credit the discriminant must be divisible by at least 5 distinct odd
primes, as we shall see).
2. The Hilbert Class Field
The main purpose of class field theory is to describe how primes in a field
k decompose in an extension L/k. We will deal only with that case when k
and L and algebraic number fields, i.e. finite extensions of Q. Hence these are
global fields (the integer rings Ok and OL will not be local rings). For very deep
reasons, an accurate description of this prime decomposition is only available if
the extension L/k is abelian, it is the goal of the Langlands Program to extend
class field theory to non-abelian extensions.
It is well known that for any prime p of k, there is an absolute value (and
corresponding valuation) | · |p which is called the p-adic norm. In Q there is
another absolute value given by the usual Euclidean distance. In some ways
this is more natural, but is part of a more general theory by considering a
“prime at infinity” which will then give rise to the Euclidean distance. In a
number field k, we know that the primes p of Q can split into new primes pi
in k. This also holds for the prime ∞ of Q.
Because of the correspondence between primes and valuations, it is simpler to view the above phenomena in terms of places, which are equivalence classes of valuations. The finite primes, or finite places, will correspond
to non-Archimedean valuations. The infinite places will correspond to the
Archimedean valuations. To further complicate matter, infinite primes come
in two flavors: real and complex. Real infinite primes of k are those given by
embeddings k ,→ R, while the complex places arise from embeddings k ,→ C.
We can now define the Hilbert Class Field of k.
Definition 2.1. A field L is the Hilbert Class Field of k iff L is the
maximal unramified abelian extension in which all real places of k remain real
in L.
THE GOLOD-SHAFAREVICH INEQUALITY
3
If Clk is the ideal class group of k, then a fundamental property of HCF (k)
is the following.
Theorem 2.2. If k is a number field and L = HCF (k) then
Gal(L/k) ' Clk .
Furthermore, every prime ideal of k becomes principal in HCF (k).
The last statement is simply the principal ideal theorem.
3. The Golod-Shafarevich Inequality
The purpose of this section is to prove the following:
Theorem 3.1 (Golod-Shafarevich). If G is a non-trivial finite p-group,
and if d(G) = dim H 1 (G, Z/pZ), r(G) = dim H 2 (G, Z/pZ) then
(d(G))2
< r(G).
4
It is possible to phrase this theorem in terms of the Tor dimension of the
residue field of a local ring that is not necessarily commutative. For this more
general setting see Serre[?serreGC].
Instead, our proof will demonstrate how the Golod-Shafarevich inequality
is really a theorem in non-commutative geometry. First we must prove the
following lemma.
Lemma 3.2. Let G be a finite p-group and R = Fp [G]. For any finite
G-module A such that pA = 0, there is a resolution
/ Rb0
/A
0
∂
/ R b1
∂
/ ...
such that bn = dimZ/pZ H n (G, A) and ∂((Rbn )G ) = 0.
Proof. We have a canonical isomorphism RG ' Fp . If we let a0 , . . . , ab0
be a basis of AG = H 0 (G, A) as an Fp -vector space, then we easily see that
AG ' (Rb0 )G . This isomorphism extends to an injective G-homomorphism
j : A ,→ Rb0 . The map j is injective since HomG (A, Rb0 ) → HomG (AG , (Rb0 )G )
is surjective. Thus we can lift the above isomorphism to a G-homomorphism
that must be injective by the left exactness of the invariants functor.
Furthermore, since G is a finite group and R = Fp ⊗ Z[G] we see that
R and hence Rb0 , is an induced G-module. If we consider the short exact
sequence
0
/A
j
/ R b0
/B
/0
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JASON PRESZLER
then the induced long exact sequence in cohomology yields
0
/ AG
/ (Rb0 )G f
/ BG
/ H 1 (G, A)
/0
/ H 1 (G, B)
Since AG ' (Rb0 )G we know that f must factor through zero. Thus, H i (G, B) '
H i+1 (G, A) ∀ i ≥ 0. Therefore, dim H 0 (G, B) = dim H 1 (G, A) = b1 and we
can repeat the above process on B in place of A to obtain
/B
R b0
/ R b1
where ∂ is the composition of the two maps. From this we see that ∂((Rbn )G ) =
0 and repeating the argument yields the desired resolution.
Let us recall the “ascending central series”
0 ⊆ A0 ⊆ A1 ⊆ A2 ⊆ · · · ⊆ Am = A
which exists for any G-module A. In this series A0 = 0, A1 = AG and
An+1 /An = (A/An )G . Thus, An 6= An+1 unless An = A. We will also let
cn (A) = dim(An+1 /An ). Furthermore, recall that if A, B are G-modules and
h : A ,→ B is an injective G-homomorphism, then An = h−1 (Bn ).
Additionally, the proof will require use of the Poincaré Polynomial PA (t) =
Pn
P
n
i=0 ci (A) = dim(An+1 ) and 0 < t < 1 is real then
n cn (A)t . If sn (A) =
PA (t)
X
1
=
sn (A)tn .
1−t
n
3.1. Proof of Golod-Shafarevich
Let Λ = Fp [G], and A = Fp then the above lemma gives maps ∂ and a resolution
/A
0
/Λ
/ Λr
/ Λd
/ ...
where d = dim H 1 (G, A) and r = dim H 2 (G, A). If we let E = Λ/ΛG = Λ/A,
D = Λd and R = Λr then the above resolution becomes
0
/E
∂
/D
∂
/R.
If one computes the ascending central series of each E, D, R then the
maps ∂ are defined so that ∂(Dn ) ⊆ Rn−1 and since ∂ : E → D is injective
En = ∂ −1 (Dn ). Thus, we obtain exact sequences
0
/ En
∂
/ Dn
∂
/ Rn−1
n ≥ 1.
The existence of these exact sequences force sn (D) ≤ sn (E) + sn−1 (R).
/ H 2 (G, A)
/0.
THE GOLOD-SHAFAREVICH INEQUALITY
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By considering the Poincaré polynomials and if PΛ (t) = P (t), we see that
PE (t) = P (t)−1
, PD (t) = dP (t), and PR (t) = rP (t). Thus, the above inequality
t
for 0 < t < 1 becomes
1
1
t
≤ PE (t)
+ PR (t)
1−t
1−t
1−t
P (t) − 1
dP (t) ≤
+ rtP (t)
t
0 ≤ P (t) − 1 + rt2 P (t) − dtP (t)
PD (t)
1 ≤ P (t)(rt2 − dt + 1).
Since P (t) has positive coefficients, we have
0 < (rt2 − dt + 1).
Furthermore, since
0
/Z
p
/Z
/ Z/pZ
/0
is a short exact sequence, the induced long exact sequence shows that d ≤ r <
d
to obtain
2r. Thus, we can substitute t 7→ 2r
d2
d2
−
+1
4r 2r
0 < −d2 + 4r
0<
d2
<r
4
Thus, we have obtained the Golod-Shafarevich inequality.
3.2. Remarks
The inequality above is a slight improvement (due to Gaschutz and Vinberg)
over the original bound of Golod-Shafarevich. One can obtain the same result
using homology, as seen in Shatz[?shatz], the proof is dual to the one above
which comes from [?cohomNF]. When using homology it is slightly more obvious
that one is using Rim’s lemma (a non-commutative analog of Nakayama’s
lemma) which we can avoid directly because invariants are easier to use than
co-invariants. Also, the Poincaré polynomial is a non-commutative analog of
the Hilbert-Samuel polynomial. Lastly, the very idea of looking at generators
and relations is akin to the idea behind the Hilbert Syzygy theorem. Thus, if
our group was abelian we would just be doing some basic commutative algebra
(this is the idea behind Serre’s proof of this result, study modules over a not
necessarily commutative local Artinian ring).
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JASON PRESZLER
4. Infinite Class Field Towers
In this section we bring together the previous topics in these notes to
see how the Golod-Shafarevich inequality helps us control class field towers. We continue using the notation from above, namely k is a number field,
d = dim H 1 (k, Z/pZ) = dim H 1 (Gal(k/k), Z/pZ), and r = dim H 2 (k, Z/pZ).
We will employ the following theorem from class field theory, whose proof is
not terribly difficult but would require the introduction of too many unrelated
items for our purpose. Recall that any number field has real and complex
embeddings, we will let r1 denote the number of real embeddings and r2 the
number of pairs of complex embeddings. In elementary algebraic number theory, one learns that [k : Q] = r1 + 2r2 and the Dirichlet unit theorem shows
that the order of the unit group is r1 + r2 . This motivates the following result,
due to K. Iwasawa.
Theorem 4.1 (Iwasawa). If k is a number field and L/k is a finite unramified p-extension (the extension is Galois and the Galois group is a p-group)
that contains no unramified, cyclic extension of degree p. Then
dim H 2 (Gal(L/k), Z/pZ) − dim H 1 (Gal(L/k), Z/pZ) ≤ r1 + r2 .
We will now show that infinite class field towers exist. Let Ω/k be the
maximal unramified extension of k. Then if k p is the maximal unramified
p-extension of k, we know that Ω/k is infinite if k p /k is infinite. Let G =
Gal(Ω/k p ), which is a p-group. To show that k has an infinite class field tower,
we must only show that k p is not finite degree over k. First we prove that this
can happen, then we use the proof to construct some explicit examples.
Theorem 4.2 (Shafarevich). There exist number fields k with infinite
class field towers.
Proof. Suppose that the theorem is false, then every possible G is a finite
p-group and by Iwasawa’s theorem above we know that
r(G) − d(G) ≤ r1 + r2 .
Suppose that p = 2 and k is totally imaginary (i.e. r1 = 0). We can
√
choose N distinct odd primes pi and let k = Q( −p1 · · · pN ) be an imaginary
√
quadratic field with non-trivial Hilbert Class Field. Define Ki = k( ±pi )
where the sign is determined by the Legendre symbol (i.e. the residue of
pi mod (4)). Clearly, each Ki is linearly independent and thus d(G) ≥ N .
However, r2 = 1 ≥ r(G) − d(G). But if N is large, d(G) is large and the
Golod-Shafarevich inequality force the difference r(G) − d(G) to go to infinity.
Thus, we have a contradiction.
THE GOLOD-SHAFAREVICH INEQUALITY
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4.1. Examples of Infinite Class Field Towers
In the above proof, we know that r > d2 /4 and r − d ≤ 1. So we must find
the number of distinct odd primes needed to get a contradiction. Thus, we
must find when d2 − 4d − 4 is positive, which first happens when d = 5 = N .
Therefore, we must use at least 5 distinct odd primes. For computational
purposes, we would also like the product to be congruent to 1 modulo 4, such as
−3·5·7·11·13 ≡ 1√mod (4) as is −3·5·7·11·13·17. Thus, the counter-examples
arise from k = Q( −D) where D is divisible by at least 5 odd, distinct primes.
The classic that Shafarevich originally used was D = 3 · 5 · 7 · 11 · 13 · 17 · 19.
It must be stressed that the 5 distinct odd primes is reliant on the fact
that we are looking for an imaginary quadratic field with infinite class field
tower. Other kinds of number fields will require different bounds, these will
generally be larger. Additionally,
even with the simplest counter-example
of
√
√
D = −15015, HCF (Q( D)) is a 96 degree extension of Q( D) so computations become difficult.
We should also mention a direct consequence of the existence of infinite
class field towers. If k is a number field with an infinite class field tower, then
none of the Hilbert class fields in the tower can have class number one, or else
the tower would be finite. Thus, an example of an infinite class field tower
also provides on example of a field k that doesn’t embed into a field with class
number one, hence:
Corollary 4.3. There exist algebraic number fields that can not be embedded inside a number field with class number one.
5. References
[1]
Jürgen Neukirch, Algebraic Number Theory, Springer-Verlag, 1999.
[2]
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of Number Fields,
Springer-Verlag, 2000.
[3]
Jean-Pierre Serre, Galois Cohomology, Springer-Verlag, 2002.
[4]
Stephen S. Shatz, Profinite Groups, Arithmetic, and Geometry, Annals of Mathemtaics
Studies, vol. 67, Princeton University Press, 1972.