A homological exact sequence associated with a family of normal subgroups

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

A ^NTONIO G. R ^ODICIO

A homological exact sequence associated with a family of normal subgroups

Cahiers de topologie et géométrie différentielle catégoriques, tome 28, n

^o

1 (1987), p. 79-82

< http://www.numdam.org/item?id=CTGDC_1987__28_1_79_0 >

© Andrée C. Ehresmann et les auteurs, 1987, tous droits réservés.

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Article numérisé dans le cadre du programme

Numérisation de documents anciens mathématiques

A HOMOLOGICAL EXACT

SEQUENCE ASSOCIA TED

^WITH A FAMILY OF NORMAL

SUBGROUPS

by

Antonio G. RODICIO CAHIERS DE TOPOLOGIE

ET GÉOMÉTRIE

DIFFÉRENTIELLE CATEGORI QUES

Vo1 . XXVIII-1

(1987)

Résumé.

^{R. Brown et J.-L.}

Loday

^ont

^obtenu,

par ^{des métho-} des

topologiques,

^une suite exacte de 8 termes

qui

^{relie 1’homo-}

logie

^d’un groupe ^{G avec}

1’homologie

^{de deux de ses} sous-groupes

normaux M et N tels que MN = G. Ici on demontre que ^les

cinq premiers

^termes de cette suite

peuvent

6tre obtenus pour ^une famille de sous-groupes {M1 ^{1 1 ; i ((} n ) telle que

Introduction.

In this paper ^we prove ^the

f ollowing

^{theorems :}

THEOREK 1. Let A be a

ring

^{(with unit) and I a} two--sided ideal of Aft Let {J, I 1 1 n ) be a finite

family

^{of two-sided ideals of}

A such tbat

Then there exists an exact and natural sequence

THEOREMN 2. Let G

be a group and

^{{M t I 1 i n} a fini te}

familly

o f normal

subgroups

^of

G ,

^{such tha t}

Then there exists an exact eind natural sequence

where t

I.,,tA ^Mi,

^{Mkl i 1 v} A n&#x3E; is the

subgroup of by

tbe

subgroups

¹

O t#k M j, M*]

^{1 k n.}

If J, = I, 3 I n, then Theorem 1

yields

^{Theorem 1 of C2J. On}

the other hand, ^I if M..

G,

^{3 ; i} n, then sequence ⁽²⁾

yields

^the

five first terms of the

eight-term

^exact sequence ^{obtained in [1]}

by topological

^methods.

Proof of the

theorems.

We will

systematically

^{use the}

following

result: If N is a sub- module of an

A-module M,

^{then the} sequence

N@AM+M@AN ^A MOAM ^-) M/N®AM/N ^-) 0

is exact.

For each k, 1 ; K n,, ^we have a short exact sequence

which

yields

^a

right

^exact sequence

Therefore we obtain the

right

^exact sequence

We now consider the sequence

where

and

It is clear

that p

^is

injective

^{and Ker e = Im} _03BC. ^{Moreover E is sur--}

jective

^{as a} consequence ^{of I =} E.k=1 n(ni*k Ji) . Then the sequence ⁽⁴⁾ induces the

right

^exact sequence

Let us consider the

diagram

where the first and second columns are the sequences ^{(5) and (3)}

respectively, B

^is the canonical

homomorphism

^{and I is the}

adequate epimorphism making

^the upper square commutative. Then it follows that the induced

homomorphism

is

surjective.

From that we obtain the

right

^{exactness of the}

top

^row

in the

diagram

in which the

homomorphisms

^are the usual ones. Now

1m ø

^is

On the other

hand,

there exist exact sequences

and

Therefore, application

^of the ker-coker Lemma to the

diagram yields

the sequence ^(1).

in order To show tne exactness 01 tne sequence ^{(2) let A = ZG} be the

integral

group

ring

^of

G,

I = IG the

augmentation

ideal of G and J i ⁼ IM i the kernel of the

ring homomorphism

^ZG -&#x3E;-&#x3E; Z(G/Mi). It is clear that

Moreover we have an

isomorphism

^{of abelian} groups

which is obtained as in the case n = 2 (see [2]).

Therefore sequence ⁽¹⁾

yields

sequence ^(2),

References,

1. R,

BROWN &#x26;

J.-L. LODAY,

^Excision

homotopique

^en basse

dimension, C.R.A.S.

Paris

298, Ser,

^I

(1984), 353-356,

2, A,G,

₁₄

RODICIO,

^{On some} five-term exact _sequences in

homology, Comm,

ⁱⁿ

Algebra, (1986), 1357-1364

UnlverSldad de

Santiago

Facultad de Matematicas

SANTIAGO,

SPAIN