M ath. Res. Lett. 16 (2009), no. 5, 843–850

c International Press 2009

A NOTE ON GENERATORS OF EVEN-NUMBERED K-GROUPSOF RINGS OF INTEGERS
Wojciech Gajda and Cornelius Greither

Abstract. In this note we give explicit generators for étale even-numbered K-groupsof the ring of integers in some cyclotomic fields, using generalised symbols and standard
results in cyclotomic Iwasawa theory.

1. Introduction
In this short note we intend to give explicit generators for étale even-numbered
K-groups of the ring of integers in some cyclotomic fields. These K-groups are finite,etoften nontrivial, and interesting; for example, K2n(Z[ 1l ]), for an odd natural numbern and an odd prime l, is expected to be cyclic, and this cyclicity is equivalent to the
Iwasawa conjecture on cyclicity of the minus part of the l-Sylow subgroup of the classgroup of the cyclotomic field Q(µl ), see for instance [Ga] or [Ku].
Unpublished work of Voevodsky and Rost implies that the étale K-groups agreewith the corresponding algebraic K-groups of Quillen, in the l-part for every odd
prime l, see for instance Theorem 70 in [We]. However we prefer to stay within theétale cohomology framework throughout this paper.
The generators of the K-groups which we construct arise as images under transfer
of certain multiplicative combinations of generalised symbols. Indeed, for K2 we dohave symbols in the usual sense. The generalised symbols that we use always involve
non-units and therefore only lie in the K-group of a certain field, and the whole point
is to choose the multiplicative combinations of them in such a way that we get allelements of the K-group of the appropriate ring of integers, and nothing else.
Our final result (Theorem 3.2) does not make much sense without the constructions
which go into it, so we choose not to describe it in full detail here. For the ease of
the reader, the construction is first explained in a special setting in Section 2 (themain result of that section is Proposition 2.3). A reader just interested in the final
statement is invited to skip Section 2 completely (at least at the first reading).
The techniques should certainly work in greater generality, but we decided to keep
the setting simple and very explicit. We use some well-known results on Stickelberger
elements and a fairly simple-minded algebraic Lemma 3.1. Even though we use classgroups in our proofs, it does not seem that our methods will yield results on classgroups; they only work for higher K-groups.

Received by the editors October 31, 2008.
Key words and phrases. Class group, cyclotomic field, étale cohomology, K-groups.
843

844

WOJCIECH GAJDA AND CORNELIUS GREITHER

2. The setting, and a special case of our construction
Let l be an odd prime, let F = Q(µl ) and F + = Q(µl )+ be the cyclotomic
field and its maximal real subfield, respectively. We also consider the cyclotomic
field E = Q(µlk ), where the exponent k will be specified below. Let n be an odd
natural number. In what follows, let S denote a finite set of finite primes of F ;however, in context, this will also indicate the set of primes of E above that finite
etset. In this note we will identify the étale K-theory groups K2n(OL,S , Z/lk ) andetk
K2n+1 (OL,S , Z/l ), of the ring of S-integers of a number field L (where S contains
the primes of L over l), with the étale cohomology groups H2 (OL,S , Z/lk (n+1)) andH1 (OL,S , Z/lk (n+1)), respectively via the Dwyer-Friedlander isomorphisms, which
respect transfers for finite field extensions and preserve the multiplicative structureet(cf. [DF]). We will also use identifications: K2n(OL,S ) = H2 (OL,S , Zl (n+1)) and1
etK2n+1 (OL,S ) = H (OL,S , Zl (n+1)), which follow from the Dwyer-Friedlander isoet
morphisms by passing to the inverse limits over k. By definition K2n(OF,S )[lk ] is the2kl -torsion part of H (OF,S , Zl (n + 1)).
Consider the homomorphism×etα : OE,S⊗ Z/lk (n) −→ K2n(OF,S )[lk ]which we define using the cohomological construction of the étale K-group. There isa map α0 :×
OE,S
⊗ Z/lk (n) → H1 (OE,S , Zl (1)) ⊗ H0 (OE,S , Z/lk (n)) → H1 (OE,S , Z/lk (n + 1)),

where the first map is standard in the first tensor factor (and trivial in the other), andthe second map is the cup product. To obtain α, we then follow up this map with thecanonical epimorphism b (the Bockstein map) from the group H1 (OE,S , Z/lk (n + 1))
to H2 (OE,S , Zl (n + 1))[lk ], and then with the transfer map down to the analogous
group with E replaced by F . (Of course one could interchange the Bockstein map
and transfer, in the obvious sense.)
The main result of this section is Prop. 2.3 which shows that the étale evennumbered K-groups of Z[ 1l ] can be described by means of the map αQ defined as
α followed by transfer from F to Q. To this end we have to place αQ into a certain
commutative diagram. Before we do this, let us give a comparison result, to put themap α into context.

Lemma 2.1. The above definition of the map α gives the same map as the construction in Lemma 2.3 of [BG1] (where in loc. cit. we replace Q by F ).

GENERATORS OF EVEN-NUMBERED K-GROUPS

845

Proof: Consider the following diagram
=

×
OE,S
⊗ Z/lk (n)
αk



k

(n)

∼
=


/ H1 (OE,S , Z/lk (n+1))∼=

/ H1 (OF,S , Z/lk (n+1))

Tr

Tr

et
K2n+1
(OF,S , Z/lk )E,S ⊗ Z/l
α0

et
K2n+1
(OE,S , Z/lk )



/ O×

b

b

et
K2n
(OF,S )[lk ]∼=/ H1 (OF,S , Zl (n+1))[lk ]where the composition of vertical maps on the left hand side is the map of Lemma
2.3 in [BG1], and the composition of vertical maps on the right hand side is thecohomological counterpart of the map α. The horizontal maps in the diagram are
the Dwyer-Friedlander isomorphisms. The map αk was defined in [BG1] as the composition of α0 with the inverse of the Dwyer-Friedlander isomorphism. This makes
the upper square commute. Commutativity of the lower squares follows, since theDwyer-Friedlander isomorphisms commute with transfers and with the connecting
homomorphisms b in K-theory and étale cohomology.

Recall from the paper [BG1] that the map α is surjective if the norm map
××
NE/F : OE,S−→ OF,S

is onto, for which it suffices that the set S contains the prime over l and a set ofprimes p of degree one of OF such that the primes q above these p generate the classgroup of E.
We will need the Galois groups G = Gal(E/F ), G̃ = Gal(E/Q), and ∆ =Gal(F/Q). Note that G̃ = G × ∆.
Let us treat a particularly simple situation in this section: we assume until the endof the section that the minus part of the l-part of the class group of OF is cyclic asa Zl [∆]-module, hence the minus part of the l-part of the class group OE with k ≥ 1
is cyclic over Zl [G̃] (codescent plus Nakayama’s Lemma). Take a Galois module
generator q of Cl(OE ){l} (the l-part of the class group of OE ) and let S consist of theprime p below q and the prime above l. The standard Iwasawa module XF− is thencyclic over Λ = Zl [G][∆], and isomorphic to Λ− /(Θ∗ ), where Θ∗ is the Stickelberger
element multiplied by a generator of the principal Λ-ideal Ann(µl∞ ), and the subscript
minus at Λ means modding out by 1+j (with j meaning complex conjugation). (Note:
We will later use Θ∗ , or rather its image in Zl [G̃], as an exponent; in order for thisto make sense, one simply takes a sufficiently good approximation in Z[G̃]. All ourarguments are carried out modulo a high l-power.)
We now let αQ be α followed by the transfer from F to Q. Then αQ is again onto
by a standard argument, and this remains so if we take the plus part of its domain:×−etαQ : OE,S⊗ Z/lk (n) −→ K2n(OQ,S )[lk ].

846

WOJCIECH GAJDA AND CORNELIUS GREITHERet
We now choose k such that lk is the number of elements of the group K2n(Z[ 1l ]).
Let q be the rational prime below p and q. Then lk divides q−1, and the group
etK2n−1(Fq ) is of order q n −1, and this is exactly divisible by some power lm ≥ lk
depending only on q and n. The map val in the commutative diagram (d1) below
comes from the valuation map at every conjugate of q (note that G̃ acts freely on theset of these conjugates), and ∂ comes from the localisation sequence:

1
1
et
et
et
(Z[ ])[lk ] → K2n−1(Fq )[lk ].
0 → K2n(Z[ ])[lk ] → K2nllq
The map c is just the epimorphism Z/lk [G̃](n) → Z/lk [G̃](n)G̃ followed by the
canonical isomorphism identifying these coinvariants with Z/lk . Finally, the group
etK2n−1(Fq ){l} is identified with the Fq -rational points of µ⊗n
l∞ (note also that the suetkperscript can certainly be omitted here), and its l -torsion is visibly generated byζ ⊗n , where ζ ∈ Fq is the image of a chosen primitive lk -th root of unity ζlk . The map
β sends ζ ⊗n to −1 ∈ Z/lk by definition; it is an isomorphism.
etK2n(Z[ 1l ])[lk ]
_

×−
OE,S
⊗ Z/lk (n)

αQ


et
/ / K2n(Z[ 1 ])[lk ]
lq

(d1)

∂


et
K2n−1
(Fq )[lk ]

val


k
Z/l [G̃](n)β/ Z/lkc

The kernel of ∂ is the image of the vertical inclusion map placed above it.
Lemma 2.2. The above diagram is commutative.
Proof: In the whole argument, we will identify Z/lk (1) with µlk , where 1 maps
to the chosen root ζlk of unity. Then Z/lk (n) is generated by 1 in additive notation(twist is hidden), or by ζl⊗n
in multiplicative notation (twist is visible).
k(1) For n = 1 one can check that αQ (u ⊗ ζlk ) = T rE/Q {u, ζlk }, where T r stands fortransfer, and inside the T r term we have a Steinberg symbol in K2 . (It was shownby Tate that the Steinberg symbol corresponds to cup-product on the cohomologicalside, cf. Theorem 3.1 in [Ta]).
(2) Let v be a place of E above q, Ev the v-adic completion, and Fv the residue
field of Ov , the integers of Ev . We look at a similar diagram in this local situation:
Ov× ⊗ Z/lk (n)


αv

2n

∂v

val

Z/lk (n)/ K et (Ov )[lk ]βv−1/ K et (Fv )[lk ].
2n−1

The isomorphism βv sends ζ ⊗n to −1 ∈ Z/lk (n). We claim this diagram commutes.
Since the connecting homomorphism ∂v (which comes from the localisation sequence

GENERATORS OF EVEN-NUMBERED K-GROUPS

847

in étale cohomology) commutes with cup product, and from the definitions, we seethat we only need to deal with the case n=1. (The case n=0 is excluded!) Here theupper horizontal map is given by symbols: u ⊗ ζlk 7→ {u, ζlk }Ev , and ∂v is the tame−val(u)symbol, mapping {u, ζlk }Ev to ζlk, where val is the v-adic valuation map. From
this, the commutativity of the diagram is clear.
(3) The lemma now follows from the compatibility of α and ∂ with localisation andtransfer. To see this, the following diagram should be helpful (v always runs over theE-places above q; obvious localisation maps have no labels):et/ K et (OE,S )[lk ] T r / K2n(Z[ lq1 ])[lk ]
2n
l
lll
lll
l
l
l
Q
l
v ll
Q × 
αv Q
k
et
k
/
∂
]
v Ov ⊗ Z/l (n)v K2n (Ov )[l
RRR Q
RRR ∂v
Q
RRR
valv
RRR
R(
Q −1


βv
Q
k
/ Q K et (Fv )[lk ]/ / K et (Fq )[lk ]
Z/l
2n−1
2n−1
v
v
α

×
OE,S
⊗ Z/lk (n)∼=
β −1 cZ/lk [G̃]/ K et2n−1 (Fq )[lket
The right hand map in the third row is a norm map, more precisely on K2n−1(Fv )[lk ]etit is the canonical identification map wth K2n−1(Fq )[lk ]. The commutativity of theupper right hand pentagon can be taken from [Sou], p.276.


We now use Diagram (d1) to deduce an important property of the map αQ .
et
Proposition 2.3. The image of the map αQ is already contained in K2n(Z[ 1l ])[lk ]etand im αQ = K2n(Z[ 1l ]) is generated by αQ (x ⊗ 1), where x is a generator of qΘ∗ .
×
Proof: By Stickelberger’s theorem we have indeed qΘ∗ = (x) for some x ∈ OE,S.
k+
The left vertical map in the diagram (d1) sends x to Θ̄∗ ∈ Z/l [G̃](n) . Given
the definition of the map c, this means that the composition of the left vertical andthe lower horizontal map in Diagram (d1) sends x to the image of Θ∗ under thecomposition:Λ∼=/ Λ(n)ε/ Zl/ Z/lk .

Here ε is just the canonical map identifying the Gal(Q(ζl∞ )/Q)- invariants of Λ(n)with Zl . Let us denote the first isomorphism by an exponent (n). Then by the class(n)etnumber formula (cf. [BG2], Section 5), #K2n
(Z[ 1l ]) = [Zl : εΘ∗ ], and the left handside equals lk by assumption. Hence Θ∗ is annihilated by the map in the last display,
and therefore the composition of the left vertical and the lower horizontal maps of the
diagram (d1) sends x to zero. Since Θ∗ is the precise annihilator of the classgroup×−of E, it follows that OE,S
⊗ Z/lk (n) is generated (as a Galois module) by x ⊗ 1 andζlk ⊗ 1. (This strongly uses the cyclicity of the class group as a Galois module!) Bothof these elements map to zero in the lower right hand corner, so this shows that theet(Z[ 1l ])[lk ]. Since the map
image of αQ is contained in the kernel of ∂, which is K2n

].

848

WOJCIECH GAJDA AND CORNELIUS GREITHER

αQ was surjective to begin with, the upper vertical map in (d1) is an equality, andetetim αQ = K2n(Z[ 1l ])[lk ] = K2n(Z[ 1l ]).
It remains to see that the second generator αQ (ζlk ⊗ 1) is superfluous. In fact it
is zero. For n=1 this is a well-known property of symbols: before applying transfer,ζlk ⊗ 1 maps to the symbol {ζlk , ζlk }, which is trivial by a simple calculation, For
general n we again get the trivial element already before applying transfer, as one
××⊗ Z/lk⊗ Z/lk (n) as the cup product of ζlk ⊗ 1 ∈ OE
sees by writing ζlk ⊗ 1 ∈ OE
with 1 ∈ Z/lk (n−1).

3. The general construction
We now remove the cyclicity assumption on our class groups. Moreover we willproceed in a more general way, descending only to F + instead of all the way toQ. (This could easily have been done in the previous part already, using character
decomposition for the group ∆ = Gal(Q(ζl )/Q).) Let us begin with an elementary
algebraic lemma.
Lemma 3.1. Consider a presentation(Z/lk )s·A/C

/ (Z/lk )s/ 0,

where the matrix A ∈ Ms,s (Z/lk ) can be lifted to a matrix Ã ∈ Ms,s (Zl ) such thatd := det Ã divides lk in Zl . Then the kernel of the multiplication ·A is the image ofthe multiplication by B := d−1 lk ad(A) : (Z/lk )s −→ (Z/lk )s , where ad(A) denotes
the adjoint matrix of A.
Proof: Since A · ad(A) is d times the identity matrix, we see that AB is zero. (Thequantity d−1 lk has to be read in Zl of course.) Suppose conversely that v ∈ Zsl isa row vector, v̄∈(Z/lk )s its image, and assume that v̄ is in the kernel of ·A. Thenv Ã = lk w with w∈Zsl and v Ã = wlk d−1 ad(Ã) · Ã, and w0 = wlk d−1 is in Zsl . We may
cancel Ã and get v̄ = w0 B, so v̄ is in the image of multiplication by B.

We now prepare the stage for an application of this lemma. Let k be such thatetlk =|K2n(OF + [ 1l ])|, and recall that E = Q(ζlk ). We denote by X the minus part ofthe l-Sylow subgroup of the class group Cl(OE ). Let R denote the ring Zl [G̃]/(1+j).
Since X is of projective dimension 1 (or 0) over R (see [Sch]; the argument given there
for prime cyclotomic fields works just as well in our case), and its Fitting ideal is theprincipal ideal generated by Θ∗ , we have a presentation of X of the form
(d2)

Rs

·M

/ Rs

/X

/ 0,

where M ∈ Ms,s (R) and det(M ) = (Θ∗ ). Let R0 = Zl [∆]+ be the ring Zl [∆]/(1−j),etand let ε be the natural projection from Λ to R0 . Then K2n(OF + [ 1l ]) has the same(n)order as [R0 : ε(Θ∗ )], by the class number formula already mentioned. Thus, if A is(n)ε(M ) taken modulo lk , then A is liftable to a matrix over R0 whose determinant d(n)(n)
is ε(Θ∗ ); and since the index [R0 : ε(Θ∗ )] has the exact value lk , we obtain that ddivides lk in R0 .
Now let q1 , . . . , qs be prime ideals of absolute degree 1 representing the generatorsof X that correspond to the presentation (d2) of X. Let S consist of these primes

GENERATORS OF EVEN-NUMBERED K-GROUPS

849

and the prime above l in E. We consider the diagram:(d3)/ K et (OF + ,S )[lk ]

α

×−
OE,S
⊗ Z/lk (n)2nLsval=val1 ⊕...⊕vals∂=∂1 ⊕...⊕∂s



i=1

L

v|qi ∩F + K2n−1 (Fv )[l

k

]

β


Ls
( i=1 Z/lk [G̃](n))+

c/ Ls (Z/lk [∆])(n)+ .
i=1

A few explanations: vali corresponds to qi exactly as val corresponded to q in theprevious section. The map ∂ is the localisation map for the set of F + -places below
S with the prime above l removed; its i-th component goes to the direct sum of all
K2n−1 (Fv ) with v being conjugate to qi ∩ F + . The map β is similar as in the previous
section; note that every K2n−1 (Fv )[lk ] is again generated by ζ ⊗n , with ζ the image ofζlk . More precisely, β maps the tuple having ζ ⊗n at position qi ∩F + and 1 everywhere
else to the tuple having −1 at position i and zero everywhere else. The map c thistime is canonically induced by taking G-coinvariants. Up to obvious technical changes(the main one being that transfer is not taken all the way down to Q but only to F + )the argument of Lemma 2.2 proves that diagram (d3) is again commutative.
×−
Now OE,S
is generated up to roots of unity (which we may neglect later on, exactly
as previously) by elements x1 , . . . , xs satisfyingi1is(xi ) = qm
. . . qms ,1

where the mij are the coefficients of the matrix M in the presentation (d2). We
extend the diagram (d3) somewhat; ι maps the i-th standard basis vector to xi ⊗ 1.
Z/lk [G̃]s/ O×− ⊗ Z/lk (n)

ι

=

(d4)


Z/lk [G̃]s

α

E,S

2n

β∂

val


·M /
Z/lk [G̃](n)s,+/ K et (OF + ,S )[lk ]c/ Z/lk [∆](n)s,+
∼
=

ε


Z/lk [∆]s,+·A/ Z/lk [∆]s,+

Then val(xi ) is simply the i-th row of M read in Zl [G̃] and modulo lk . Therefore
the upper left square commutes. Since c is essentially taking coinvariants of the n-th
twist, we get that c(val(xi ) ⊗ 1) is the ith row of the matrix A, and so the lower
rectangle commutes as well.
et
We now obtain generators of K := K2n(OF + [ 1l ]) as follows:K= α(ker(c ◦ val))= αι(ker(c ◦ val ◦ ι))= αι(ker(c ◦ (·M ))).

Now the map αι factors through the map ε (which takes G-coinvariants), to give amap β. Thus K = β(ker(·A)). By Lemma 3.1, the kernel of multiplication of A (thebottom horizontal map of the diagram (d4)) agrees with the image of multiplication
by lk d−1 ad(A). Lifting this back through ε, we obtain the main result:

850

WOJCIECH GAJDA AND CORNELIUS GREITHER

Theorem 3.2. Let the matrix M ∈ Ms,s (R) arise from a presentation of the l-partof the minus class group of E as explained, and let d ∈ Zl [∆] be the image of det(M )under ε (twist n times and send G to 1). We recall that k is chosen such that theetgroup K2n
(OF + [ 1l ]) has order lk . Let B = (bij ) be the adjoint matrix of M multipliedetby lk /d ∈ R0 . Then the group K2n(OF + [ 1l ]) is generated by the set of elements
yi := α(xb1i1 · · · xbsis ⊗ 1), i = 1, . . . , s.
Remark 1: By very similar arguments we can prove that the relations between the
yi ’s are generated by the rows of A. Actually this is exactly what one expects, sinceetthe size of K2n(OF + [ 1l ]) is predicted by the determinant of A. In fact one may
etshow a priori using Iwasawa theory and codescent that the group K2n(Z[ 1l ]) has apresentation of this type. What we have done is only to give an explicit descriptionof the generators.
Remark 2: If Iwasawa’s cyclicity conjecture holds (note that this conjecture is aconsequence of Vandiver’s conjecture), then we may take s=1, and everything becomesmuch simpler. This is the situation considered in the first part of this note.
Acknowledgements
The second author gratefully acknowledges the hospitality of AMU Poznań during
two visits. The first author was partially supported by the Alexander von Humboldt
Research Fellowship and an MNiSzW grant. The first author also thanks Centre deRecerca Matematica in Bellaterra for hospitality and support during a visit in summer
2008.
References[BG1] G. Banaszak and W. Gajda, On the arithmetic of cyclotomic fields and the K-theory of Q,
Contemp. Math. 199 (1996), 7-18.
, Euler systems for higher K-theory of number fields, J. Number Th. 58 (1996), 213[BG2]252.
[DF] W. Dwyer and E. Friedlander, Étale K-theory and arithmetic, Trans. Amer. Math. Soc. 292(1985), 247-280.
[Ga] W. Gajda, On K∗ (Z) and classical conjectures in the arithmetic of cyclotomic fields, Contemp. Math. 394 (2004), 217-237.
[Ku] M. Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of Z, Compositio Math. 81 (1992), 223-236.
[Sch] R. Schoof, Minus class groups of the fields of the l-th roots of unity, Math. Comp. 58 (1998),
1225-1245.
[Sou] C. Soulé, K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent.
math. 55 (1979), 251-295.
[Ta] J. Tate, Relations between K2 and Galois cohomology, Invent. math. 36 (1976), 257-274.
[We] C. Weibel, Algebraic K-theory of integers in local and global fields, in: Handbook of K-theory
vol. I, Springer 2005, 139-190

Department of Mathematics, Adam Mickiewicz University, 61614 Poznań, Poland
E-mail address: gajda@amu.edu.pl
Fakultät Informatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany
E-mail address: cornelius.greither@unibw.de