SUMIOKA, TAKESHI

Sumioka, Takeshi

Osaka University Knowledge Archive

A note on the Grothendieck group of a finite abelian group

Osaka City University Repository

Sumioka, T.
Osaka J. Math.
10 (1973), 21-24
A NOTE ON THE GROTHENDIECK GROUP OF
A FINITE ABELIAN GROUP
TAKESHI SUMIOKA
(Received June 6, 1972)
Introduction
For any ring A, by G(Λ) we denote the Grothendieck group of left
^4-modules which are finitely generated. Let R be the ring of integers of an
algebraic number field K, and let Rπ and Kπ be the group rings of a finite
group π over R and K, respectively. If O is a maximal J?-order in Kπ which
contains Rπ, then by regarding a module over O as one over Rπ, we get a homo-
morphism
of Grothendieck groups. Swan [4] proved that ψ is an epimorphism, and
Heller and Reiner [2] described the structure of ker ^  by using a map which
depends on an ideal theory of the center of O and the modular representations of
π. The following theorem is an immediate consequence from the description.
Theorem 1. Let R{ be maximal orders in the center of the simple con-
stituents A{ (/=!, •••, s) of Kπ. If any prime ideal of R{ which divides the order
of π is contained in the ray J(R{) modulo the real archίmedian primes ramified in
Aiy then ^r is an isomorphism.
The purpose of this note is to show that under certain assumptions the con-
verse of this theorem is also true.
Theorem 2. Let π be a finite abelian group of order n and K be a cyclotomic
field. Then ^r is an isomorphism if and only if any prime ideal in O which divides
n is principal.
In this case /(/?/) is the group of all principal ideals of R£ and O is the direct
sum of the Rf. Hence the if part of this theorem is a special case of Theorem 1.
Our proof of this theorem is based on a method using the conducter from O to
Rπ, which owes to Swan ([4]).
Throughout this note, modules are assumed to be left modules which are
finitely generated and by K, R and [M] we denote an algebraic number field,
22 T. SUMIOKA
the ring of integers of K and an element of a Grothendieck group which is repre-
sented by a module M, respectively.
1. Let A be a central simple algebra over K and let O be a maximal Λ-order
in A. By I(R) and /(Λ) we denote the multiplicative group of Λ-ideals in K
and the subgroup of I(R) consisting of elements xR (x^K), respectively, where
x are positive at each real archimedian prime of K which ramifies in A.
If S is a simple O-module, there exists a unique prime ideal p of R such
that pS=0. Hence, when M is a torsion O-module and S19 9Sk are the
O-composition factors of M, we define the reduced order ideal Or(M) of M as
Or(M)=p1 !pΛ, where each & is a unique prime ideal scuh that p/S, =0.
We note that G(O) is isomorphic to the Grothendieck group of torsion-
free O-modules. Now let L be a torsion-free O-module such that K®L is a simple
R
yl-module and fix it. Let M be a torsion-free O-module. Then K®M is
R
isomorphic to a direct sum (K®L)r of r-copies of K®L and there exists a
submodule N of M such that ΛΓ^ZΛ Hence [M]=r[L] + [M/N], and Af/ΛΓ
is torsion. Set a=Or[MjN]. By setting ι?([Λf])=(r, α), a map
77: G(D)-+Z®I(R)IJ(R)
is an isomorphism, where α is an element of I(R)/J(R) represented by α and Z
is the ring of rational integers, (see Swan [5] or Heller and Reiner [2]).
For any (non-zero) ideal SI of O, by Z)(5Ϊ) we denote the set of prime ideals
of R which divide Or(O/Sί). Then by the definition of η, we have an immediate
consequence.
Lemma 3. Let 51 be an ideal of O and let M be an Ό-module which anni-
hilated by 21. If any element of D($L) is contained inJ(R), then [M]—0 in G(O).
2. Let π be a finite group and let O be a maximal 7?-order in Kπ which con-
tains Rπ. Now we consider the epimorphism
ψ: G(O) -> G(Rπ).
Let Kπ=Al@ ®As be the decomposition of Kπ into the simple constituents.
We denote by Kf the center of A£ and by R{ the ring of integers of K{. Since
O is a maximal jR-order in Kπ, there is a decomposition O=O10 ©OS of O,
where each Ot is a maximal J?rorder in A{. By K we denote the conductor
from O to Rπ (the largest O-ideal contained in Rπ), and we write K=@;
ι
0 0
(£,,, where each (£,- is an ideal of Of . It is known that K divides n the order of
Proposition 4. Suppose any element of O(SZ ) is contained in J(Rt) for
GROTHENDIECK GROUP OF A FINITE ABELIAN GROUP 23
each ί. Then the map ΨΊ G(ζ))-*G(Rπ) is an isomorphism.
Proof. We shall give a left inverse map φ of -ψ1. At first, we define a homo-
morphism
φ,: G(Λr) -* G(0,) .
Any element of G(Rπ) is represented by [M] — [ΛΓ|, where M and TV are
torsion-free ifo-modules. It therefore suffices to consider torsion-free Rπ-
modules. Now let M be a torsion-free Λτr-module. Since (£t is contained in Rπ,
we can consider an JRτr-module E,-M. We define an operation of O, for Q^M as
x(^tctmt)=^t(xct)mt (Λ?eO, , Σ^^e& M). To see that this is well defined, let
Σtctmt=Q. Since nx is contained in <£,•, n^t(xct)mt=^(nx)^tctmt=0. However
Σt(xct)mt^M, and Mis torsion-free, so we have Σt(xct)mt=Q. Hence we can
regard SZ M as an 0,-module.
We now define a homomorphism
φ,: G(Λr) -* G(0,) ,
by φ/([M])=[efJkf] It is well defined as follows. Let
be an exact sequence of torsion-free jRτr-modules. Then it induces a sequence
0 -> e,M' 5 g,M £ e,M" - 0
of O-modules, which is exact up to middle. We easily see that ker/3/imα is
annihilated by (£, , but by the assumption, any element of D((£f ) is contained in
J(Ri). So by Lemma 3, [ker/3/imα] is zero in G(O, ). This implies that
= [e/M/] + [eίΛί//], which showes that φg. is well defined.
Since G(D)=ΣφG(Of), we define a homomorphism φ with all φt ,
φ-Σφ,: G(Λτr)^G(0).
Any O-module M is regarded as an O-module by setting OyΛf=0
Then φ
ί
ψ ([M])=[e
ί
M] and φyψ ([ΛfD=[®yΛfl=0. However [M] = [e,.M] in
by Lemma 3. Therefore φψ^l, and we complete the proof.
3. Hereafter let π be abelian. Then for each ίy Af and O, coincide with
Kf and jR, , respectively. Moreover /(O,-) is the group of principal ideals of O, ,
and Z)(Sf ) is the set of prime ideals of O, dividing K,-.
Now we assume that K is a cyclotomic field or any prime rational integer
dividing n the order of π is unramified in R. Let pf is a map JRτr->Oz induced
by the projection from Kπ onto each constituent A{. Then p, is an epimorphism
(see Swan [5]). Set 9ϊ
ί
 =Ker/o> . Since pf is an epimorphism, p/(?ly) is an
24 T. SUMIOKA
ideal of Of- for each / and^'. Any Ormodule M is regarded as an O-module by
setting ζ)jM=0 for jΦ/, and moreover this is also regarded as an l?7r-module by
restriction of the operation. But such an operation of Rπ for M coincides with
one induced by pf .
Theorem 2''. Let π be an abelian group of order n. Assume that either K
is a cyclotomic field or R has a property that any prime rational integer dividing n
is unramified in R. Then the map ^r is an isomorphism if and only if any prime
ideal dividing (£ is principal in O.
Proof. Assume that any prime ideal dividing (£ is principal in O which
is equivalent to saying that for each / any prime ideal dividing (£,- is principal
in O, , i.e. D((£f )c /(O, ). Then -ψ1 is an isomorphism by Proposition 4.
Conversely, let ψ be an isomorphism. Suppose that there exists a non-prin-
cipal prime ideal *β of O, dividing (£t for some i. Set M=O, /$β. Since Sβ
is a non-principal prime ideal, [M] is not zero in G(O, ).
On the other hand Π^yC Γl 9ϊy— (£,- (see Bass Γll). Hence, for some /=M,j Φ/ yΦf
p, (5Ry) is contained in *β. If the 0,-module M is regarded as an jRτr-module (by
the way mentioned above), SlyM=p
ί
(Sly)Mc?βΛί=0. Consequently, M is an-
nihilated by 5Ry, so that from the isomorphism /?τr/5ίiy^Oy, M is also regarded
as an O/-module. Then the new Λτr-module M obtained above coincides with
the given Ot-module. It implies that in G(Rπ)y [M] is contained in the image
of G(O, ) as well as of G(Oy), which contradicts injectivity of ψ\ This proves
Theorem 2'.
Now, in particular let K be an arbitrary cyclotomic field. Then a prime
ideal β^ of O, divides (£,- if and only if β^ divides n (see Bass[l]). Thus Theorem
2 is an immediate consequence of Theorem 2'.
OSAKA CITY UNIVERSITY
References
[1] H. Bass: Algebraic K-Theory, Benjamin, New York, 1968.
[2] A. Heller and I. Reiner: Grothendieck groups of integral group rings, Illinois J.
Math. 9 (1965), 349-360.
[3] R.G. Swan: Induced representations and protective modules, Ann. of Math. 71
(1960), 552-578.
[4] R.G. Swan: The Grothendieck ring of a finite group, Topology 2 (1963), 85-110.
[5] R.G. Swan: K-Theory of Finite Groups and Orders, Lecture Note in Math.
149, Springer, Berlin, 1970.


A NOTE ON THE GROTHENDIECK GROUP OF A FINITE ABELIAN GROUP

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A NOTE ON THE GROTHENDIECK GROUP OF A FINITE ABELIAN GROUP

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