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A homological exact sequence associated with a family of normal subgroups. (English) Zbl 0632.20033

Here the author proves: Theorem 1: Let A be a ring with unit, I a two- sided ideal of A and \(\{J_ i\); \(1\leq i\leq n\}^ a \)finite family of two-sided ideals of A such that \(J_ k+(\cap_{i\neq k}J_ i)=I\), \(1\leq k\leq n\). Then there exists an exact and natural sequence \[ Tor^ A_ 2(A/I,A/I)\to \oplus^{n}_{i=1}Tor_ 2^{A/J_ i}(A/I,A/I)\to (\cap^{n}_{i=1}J_ i)/\sum^{n}_{k=1}(\cap_{i\neq k}J_ i)J_ k\to \]
\[ Tor_ 1^ A(A/I,A/I)\to \oplus^{n}_{i=1}Tor_ 1^{A/J_ i}(A/I,A/I)\to 0. \] Now let G be a group, \(\{M_ i\); \(1\leq i\leq n\}^ a \)finite family of normal subgroups of G such that \(M_ k(\cap_{i\neq k}M_ i)=G\) and \(J_ i=\ker (ZG\to Z(G/M_ i))\). Then Theorem 1 applied to ZG, IG the augmentation ideal of G and \(J_ i\) gives an exact sequence which relates the homology of the group G to the homology of the family of normal subgroups \(\{M_ i\}\). This exact sequence in homology of groups has been obtained by R. Brown and J.-L. Loday for the case \(n=2\) by using topological methods [in C. R. Acad. Sci., Paris, Sér. I 298, 353-356 (1984; Zbl 0573.55011)].
Reviewer: O.Talelli

MSC:

20J05 Homological methods in group theory
16S34 Group rings
16Exx Homological methods in associative algebras

Citations:

Zbl 0573.55011
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References:

[1] 1 R. Brown & J.-L. Loday , Excision homotopique en basse dimension , C.R.A.S. Paris 298 , Ser, I ( 1984 ), 353 - 356 , MR 745014 | Zbl 0573.55011 · Zbl 0573.55011
[2] 2, A.G. Rodicio , On some five-term exact sequences in homology, Comm , in Algebra , 14 ( 1986 ), 1357 - 1364 MR 842044 | Zbl 0598.16027 · Zbl 0598.16027 · doi:10.1080/00927878608823369
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