Rodicio, Antonio G. A homological exact sequence associated with a family of normal subgroups. (English) Zbl 0632.20033 Cah. Topologie Géom. Différ. Catég. 28, No. 1, 79-82 (1987). 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 Cited in 1 Document MSC: 20J05 Homological methods in group theory 16S34 Group rings 16Exx Homological methods in associative algebras Keywords:homology of family of normal subgroups; ring with unit; family of two- sided ideals; augmentation ideal; exact sequence; homology of groups Citations:Zbl 0573.55011 PDFBibTeX XMLCite \textit{A. G. Rodicio}, Cah. Topologie Géom. Différ. Catégoriques 28, No. 1, 79--82 (1987; Zbl 0632.20033) Full Text: Numdam EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.