Galois theory and integral models of Lambda-rings

Abstract

We show that any Lambda-ring, in the sense of Riemann-Roch theory, which is finite etale over the rational numbers and has an integral model as a Lambda-ring is contained in a product of cyclotomic fields. In fact, we show that the category of them is described in a Galois-theoretic way in terms of the monoid of pro-finite integers under multiplication and the cyclotomic character. We also study the maximality of these integral models and give a more precise, integral version of the result above. These results reveal an interesting relation between Lambda-rings and class field theory.

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arXiv:0801.2352v1 [math.KT] 15 Jan 2008
GALOIS THEORY AND INTEGRAL MODELS OF Λ-RINGS
JAMES BORGER, BART DE SMIT
Abstract. We show that any Λ-ring, in the sense of Riemann–Roch theory,
which is finite ´ etale over the rational numbers and has an integral model as a
Λ-ring is contained in a product of cyclotomic fields. In fact, we show that the
category of them is described in a Galois-theoretic way in terms of the monoid
of pro-finite integers under multiplication and the cyclotomic character. We
also study the maximality of these integral models and give a more precise,
integral version of the result above. These results reveal an interesting relation
between Λ-rings and class field theory.
Introduction
According to the most common definition, a Λ-ring structure on a commutative
ring R is a sequence of set maps λ1,λ2,... from R to itself that satisfy certain
complex implicitly stated axioms. This notion was introduced by Grothendieck [5],
under the name special λ-ring, to give an abstract setting for studying the structure
on Grothendieck groups inherited from exterior power operations; and as far as we
are aware, with just one exception [2], Λ-rings have been studied in the literature
for this purpose only.
However, it seems that the study of abstract Λ-rings—those having no apparent
relation to K-theory—will have something to say about number theory. One ex-
ample of such a relationship is the likely existence of strong arithmetic restrictions
on the complexity of finitely generated rings that admit a Λ-ring structure. The
primary purpose of this paper is to investigate this issue in the zero-dimensional
case. Precisely, what finite ´ etale Λ-rings over Q are of the form Q⊗ A, where A is
a Λ-ring that is finite flat over Z?
We will consider Λ-actions only on rings whose underlying abelian group is tor-
sion free, and giving a Λ-action on such a ring R is the same as giving commuting
ring endomorphisms ψp: R → R, one for each prime p, lifting the Frobenius map
modulo p—that is, such that ψp(x) − xp∈ pR for all x ∈ R. (The equivalence of
this with Grothendieck’s original definition is proved in Wilkerson [8].) An example
that will be important here is Z[µr] = Z[z]/(zr− 1), where r is a positive integer
and ψpsends z to zp. A morphism of torsion-free Λ-rings is the same as a ring map
f that satisfies f ◦ ψp= ψp◦ f for all primes p.
Note that if R is a Q-algebra, the congruence conditions in the definition above
disappear. Also, Galois theory as interpreted by Grothendieck gives an anti-
equivalence between the category of finite ´ etale Q-algebras and the category of
finite discrete sets equipped with a continuous action of the absolute Galois group
GQwith respect to a fixed algebraic closure¯Q. Combining these two remarks, we
Date: February 2, 2008. 18:01.
Mathematics Subject Classification: 13K05 (primary); 11R37, 19L20, 16W99 (secondary).
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2 J. BORGER, B. DE SMIT
see that the category of Λ-rings that are finite ´ etale over Q is nothing more than
the category of finite discrete sets equipped with a continuous action of the monoid
GQ×N′, where N′is the monoid {1,2,...} under multiplication with the discrete
topology. This is because N′is freely generated as a commutative monoid by the
prime numbers.
It is not always true, however, that such a Λ-ring K has an integral Λ-model,
by which we mean a sub-Λ-ring A, finite over Z, such that Q ⊗ A = K.
order to formulate exactly when this happens, we writeˆZ◦for the set of profinite
integers viewed as a topological monoid under multiplication, and we consider the
continuous monoid map
GQ× N′−→ˆZ◦
given by the cyclotomic character GQ→ˆZ∗⊂ˆZ◦on the first factor and the natural
inclusion on the second. Note that this map has a dense image. Now let K be a
finite ´ etale algebra over Q, and let S be the set of ring maps from K to¯Q. Suppose
K is endowed with a Λ-ring structure, so that we have an induced monoid map
In
GQ× N′−→Map(S,S),
where Map(S,S) is the monoid of set maps from S to itself. With the discrete
topology on Map(S,S) this map is continuous.
0.1. Theorem. The Λ-ring K has an integral Λ-model if and only if the action of
GQ× N′on S factors (necessarily uniquely) throughˆZ◦; in more precise terms, if
and only if there is a continuous monoid mapˆZ◦→ Map(S,S) so that the diagram
GQ× N′
??
???
?
?
?
?
?
?
?
?
?
ˆZ◦
??
Map(S,S)
commutes.
It follows that the category of such Λ-rings is anti-equivalent to the category of
finite discrete sets with a continuous action ofˆZ◦and that every such Λ-ring is
contained in a product of cyclotomic fields. It also suggests there is an interest-
ing theory of a Λ-algebraic fundamental monoid, analogous to that of the usual
algebraic fundamental group, but we will leave this for a later date.
Another consequence is that the elements −1,0 ∈ˆZ◦give an involution ψ−1
(complex conjugation) and a idempotent endomorphism ψ0 on any K with an
integral Λ-model. In K-theory, the dual and rank also give such operators, but
here they come automatically from the Λ-ring structure. Also observe that any
element of the subset 0S ⊆ S is Galois invariant and hence corresponds to a direct
factor Q of the algebra K. Therefore K cannot be a field unless K = Q.
In the first section, we give some basic facts and also show the sufficiency of
the condition in the theorem above. In the second section, we show the necessity.
The proof combines a simple application of the Kronecker–Weber theorem and the
Chebotarev density theorem with some elementary but slightly intricate work on
actions of the monoidˆZ◦.
The third section gives a proof of the following theorem:
0.2. Theorem. The Λ-ring Z[µr] is the maximal integral Λ-model of Q ⊗ Z[µr].

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GALOIS THEORY AND INTEGRAL MODELS OF Λ-RINGS3
Of course, the non-Λ version of this statement is false—the usual maximal order
of Q⊗Z[µr] is a product of rings of integers in cyclotomic fields and strictly contains
Z[µr], if r > 1.
A direct consequence of these theorems is:
0.3. Corollary. Every Λ-ring that has finite rank as an abelian group and has no
non-zero nilpotent elements is a sub-Λ-ring of a Λ-ring of the form Z[µr]n.
We do not need to require that the ring be torsion free because any torsion
element in a Λ-ring is nilpotent, by an easy lemma attributed to G. Segal [3, p.
295]. We emphasize that while the definition of Λ-ring that we gave above does
not literally require the ring to be torsion free, it is not the correct definition in the
absence of this assumption. In particular, Segal’s lemma and the corollary above
are false if the naive definition is used; for example, take a finite field. For the
definition of Λ-ring in the general case, see [5], [8], or [1].
Finally, many of the questions answered in this paper have analogues over general
number fields. There one would use Frobenius lifts modulo prime ideals of the ring
of integers, and then general class field theory and the class group come in, as well
as the theory of complex multiplication in particular cases. Because these analogues
of Λ-rings are not objects of prior interest, we have not included anything about
them here. But it is clear that finite-rank Λ-rings, in this generalized sense or the
original, are fundamentally objects of class field theory and that they offer a slightly
different perspective on the subject. It would be interesting to explore this further.
1. Basics
The category of Λ-rings has all limits and colimits, and they agree, as rings, with
those taken in the category of rings. (E.g. [1]) We will only need to take tensor
products, intersections, and images of morphisms, and it is quite easy to show their
existence on the subcategory of torsion-free Λ-rings using the equivalent definition
given in the introduction. For any ring R, let R[µr] denote R[z]/(zr− 1). Because
R[µr] = R ⊗ Z[z]/(zr− 1), the ring R[µr] is naturally a Λ-ring if R is.
Given a Λ-ring R, it will be convenient to call a Λ-ring K equipped with a map
R → K of Λ-rings an RΛ-ring. (Compare [1, 1.13].) When we say K is flat, or
´ etale, or so on, we mean as R-algebras in the usual sense. We call a sub-Λ-ring of
a QΛ-ring a Λ-order if it is finite over Z. We do not require that it have full rank.
1.1. Proposition. Let K be a finite ´ etale QΛ-ring. Then K has a Λ-order that
contains all others.
We call this Λ-order the maximal Λ-order of K.
Proof. Because any Λ-order A is contained in the usual maximal order of K, which
is finite over Z, it is enough to show that any two Λ-orders A and B are contained
in a third. But A⊗B is a Λ-ring that is finite over Z. Since A⊗B is the coproduct
in the category of Λ-rings, the map A⊗B → K coming from the universal property
of coproducts (i.e., a ⊗ b ?→ ab) is a Λ-ring map. Therefore its image is a Λ-ring
that is finite over Z, is contained in K, and contains A and B.
?
1.2. Proposition. Let K ⊆ L be an inclusion of finite ´ etale QΛ-rings. Let A ⊆ K
and B ⊆ L be their maximal Λ-orders. Then A = K ∩ B.

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4 J. BORGER, B. DE SMIT
Proof. The intersection K ∩ B is on the one hand a sub-Λ-ring of K and, on the
other, finite over Z. It is maximal among such rings because of the maximality
of B.
?
We can now prove the sufficiency of the conditions of theorem 0.1. For any ring
R, let R◦denote R itself but viewed only as a monoid under multiplication. So the
group R∗of units is just the group of invertible elements of the monoid R◦.
1.3. Proposition. Let r be a positive integer, let S be a finite (Z/rZ)◦-set, and let
K be the corresponding finite ´ etale QΛ-ring. Then K has an integral Λ-model.
Proof. Take a set T (such as S) admitting a surjection?
sets, the left side denoting the free (Z/rZ)◦-set generated by T. Let L be the
corresponding finite ´ etale QΛ-ring. Then K is naturally a sub-Λ-ring of L. On the
other hand L is Q[µr]Tand so has a Λ-model Z[µr]T. The intersection of this with
K is then both a Λ-ring and an order of full rank in K.
T(Z/rZ)◦→ S of (Z/rZ)◦-
?
2. Necessary conditions
Let K be a finite ´ etale QΛ-ring admitting an integral Λ-model A, and let S =
Hom(K,¯Q) be the corresponding GQ× N′-set.
The purpose of this section is to show that there is an integer r > 0 such that
this action factors through the map GQ× N′→ (Z/rZ)◦given by the cyclotomic
character on the first factor and reduction modulo r on the second.
As usual, we say a prime number p is unramified in A if A/pA has no non-zero
nilpotent elements. A prime is ramified in A if and only if it divides the discriminant
of A. Therefore the set of primes that ramify in A is finite and contains the set of
primes that ramify in the usual maximal order of K.
2.1. Proposition. The endomorphism ψpof A is an automorphism if and only if p
is unramified in A. In this case, ψpis the unique lift of the Frobenius endomorphism
of A/pA.
Proof. If ψp is an automorphism, then the Frobenius endomorphism x ?→ xpof
A/pA is an automorphism, and so p is unramified.
Suppose instead that p is unramified. Then A/pA is a finite product of finite
fields, and so the Frobenius endomorphism of A/pA is an automorphism of finite
order. The category of finite ´ etale Zp-algebras is equivalent to the category of finite
´ etale Fp-algebras, by way of the functor Fp⊗Zp−. (See [4, IV (18.3.3)], say).
Thus the endomorphism 1 ⊗ ψpof Zp⊗ A is the unique Frobenius lift and it is an
automorphism of finite order. It follows that ψp: A → A is the unique Frobenius
lift to A, and is also an automorphism.
?
2.2. Proposition. There is a positive integer c, divisible only by primes that ramify
in A, such that the action of GQ on S factors through the cyclotomic character
GQ→ (Z/cZ)∗. If p is unramified in A, then p ∈ N′and (p mod c) ∈ (Z/cZ)∗act
in the same way on S.
Proof. Define the number field N to be the invariant field of the kernel of the map
GQ→ Map(S,S). Write¯G = Gal(N/Q) and let ON be the ring of integers of N.
Take any element g ∈¯G. By Chebotarev’s theorem [7, V.6] there is an unramified
prime p of N lying over a prime number p such that g(x) ≡ xpmod p for all x ∈ ON,
i.e., g is the Frobenius element of p in the extension Q ⊂ N. Since Chebotarev’s

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GALOIS THEORY AND INTEGRAL MODELS OF Λ-RINGS5
theorem provides infinitely many such p, we may also assume that A is unramified
at p.
We now claim that for all s ∈ S = Hom(A,ON) the maps s◦ψpand g◦s from A to
ONare equal. Since A is unramified at p, the map Hom(A,ON) → Hom(A,ON/p)
is injective, so it suffices to show that their compositions with the map ON→ ON/p
are equal. But this follows from 2.1 and our choice of p. Thus, g ∈¯G and p ∈ N′
act in the same way on S.
It follows that the image of GQ in Map(S,S) is contained in the image of N′,
so¯G is abelian. By the Kronecker-Weber theorem [7, III.3.8], N is contained in
a cyclotomic field Q(µc), where c is divisible only by primes that ramify in N.
Since N is the common Galois closure of the components of A ⊗ Q, such primes
are ramified in A as well. The last statement follows from the fact that for any
prime number p ∤ c the element (p mod c) ∈ (Z/cZ)∗corresponds to the Frobenius
element of any prime over p in the extension Q ⊂ Q(µc).
?
It follows that our map of topological monoids GQ× N′→ Map(S,S) factors
throughˆZ∗×N′. We will show that it factors further throughˆZ◦with the following
criterion.
2.3. Proposition. A continuous action ofˆZ∗×N′on a finite discrete set T factors
through a continuous action ofˆZ◦if and only if
(i) all but finitely many primes p ∈ N′act as automorphisms on T, and
(ii) for all d ∈ N′there exists an integer cdsuch that the action ofˆZ∗on dT
factors through (Z/cdZ)∗and for each n ∈ N′with ndT = dT we have
— n is relatively prime to cd, and
— the elements (n mod cd) ∈ (Z/cdZ)∗and n ∈ N′act on dT in the
same way.
Proof. To show the necessity of (i) and (ii), assume that the action ofˆZ∗× N′
factors through (Z/rZ)◦for some integer r > 0. Then all primes not dividing r,
when viewed as elements of N′, act as automorphisms on T. This establishes (i).
To show (ii), take any d ∈ N′and let cdbe the smallest positive integer c for which
theˆZ∗-action on dT factors through (Z/cZ)∗. Note that cddivides all c with this
property. Now suppose that p is a prime with pdT = dT. Write r = pne, with p ∤ e.
Then for any x,y ∈ (Z/rZ)◦with x ≡ y mod e and any s ∈ dT (in fact any s ∈ T),
we have
pn?xs?= (pnx)s = (pny)s = pn(ys).
Since p acts bijectively on dT, this implies that x and y act in the same way on dT,
so the action ofˆZ◦on dT factors through (Z/eZ)◦. In particular, cd| e, so p ∤ cd,
and the elements p ∈ N′and (p mod e) ∈ (Z/eZ)∗and (p mod cd) ∈ (Z/cdZ)∗all
act in the same way on dT. Since N′is generated by the primes, part (ii) follows.
For the converse, suppose (i) and (ii) hold. For every prime number p, let apbe
the smallest integer a ≥ 0 such that paT = pa+1T. By (i) we have ap= 0 for all
but finitely many p, so r0=?
nT = gcd(n,r0)T.
Now let r be any integer divisible by dcdfor every d | r0. We will show that the
action ofˆZ∗× N′on T factors through (Z/rZ)◦. To do this, we will show directly
that any two elements (a1,d1),(a2,d2) ∈ˆZ∗×N′satisfying a1d1≡ a2d2mod r act
in the same way on T.
ppapis an integer. Note that for any n ∈ N′we have