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Change of rings and characteristic classes. (English) Zbl 0684.18002

Let B and Q be algebras over a commutative ring R and \(\pi\) : \(B\to Q\) a morphism of algebras. Given a right Q-module U and a left (right) B- module V, there is a change of rings spectral sequence \[ E^ 2=Tor^ Q_*(U,Tor^ B_*(Q,V))\quad \Rightarrow \quad Tor^ B_*(U,V), \]
\[ (E_ 2=Ext^*_ Q(U,Ext^*_ B(Q,V))\quad \Rightarrow \quad Ext^*_ B(U,V),\quad respectively). \] It is known that under additional hypotheses the differentials \(d^ 2\) \((d_ 2\), resp.) can be described as cup (cap) products with certain characteristic classes. The author develops a theory of characteristic classes for the change of rings spectral sequence without any additional hypotheses. More specifically, he shows that there are classes \[ V^ q_ 2\in E=Ext^ 2_ Q(Ext^ q_ B(Q,V),Ext_ B^{q-1}(Q,V)), \]
\[ (W^ q_ 2\in T=Ext^ 2_ Q(Tor^ B_ q(QV),Tor^ B_{q+1}(Q,V)),\quad resp.), \] such that up to a sign \(d_ 2=\mu (-\otimes V^ q_ 2): E_ 2^{p,q}\to E_ 2^{p+2,q-1},\) where \(\mu\) : \(E_ 2^{p,q}\otimes E\to E_ 2^{p+2,q-1}\) is a pairing \((d^ 2=\nu (W^ q_ 2\otimes - ): E^ 2_{p,q}\to E^ 2_{p-2,q+1},\) where \(\nu\) : \(T\otimes E^ 2_{p,q}\to E^ 2_{p-2,q+1}\) is a pairing, resp.).
Reviewer: R.Fröberg

MSC:

18G40 Spectral sequences, hypercohomology
Full Text: DOI

References:

[1] Wall, Proc. Cambridge Philos. Soc. 57 pp 251– (1961)
[2] Zassenhaus, Lehrbuch der Gruppentheorie (1937)
[3] DOI: 10.1016/0021-8693(77)90360-X · Zbl 0352.18021 · doi:10.1016/0021-8693(77)90360-X
[4] DOI: 10.1016/0021-8693(81)90296-9 · Zbl 0443.18018 · doi:10.1016/0021-8693(81)90296-9
[5] DOI: 10.1016/0021-8693(74)90103-3 · Zbl 0285.20045 · doi:10.1016/0021-8693(74)90103-3
[6] DOI: 10.2307/1969736 · doi:10.2307/1969736
[7] DOI: 10.2307/1994820 · Zbl 0192.34301 · doi:10.2307/1994820
[8] DOI: 10.2307/1994432 · Zbl 0163.27401 · doi:10.2307/1994432
[9] Cartan, Homological Algebra (1956)
[10] Berikashvili, Bull. Acad. Sci. Georgian 51 pp 9– (1968)
[11] Andr?, C. R. Acad. Sci. 260 pp 3820– (1965)
[12] Andr?, C. R. Acad. Sci. 260 pp 2669– (1965)
[13] DOI: 10.1016/0021-8693(74)90099-4 · Zbl 0277.20071 · doi:10.1016/0021-8693(74)90099-4
[14] MacLane, Homology 114 (1963) · doi:10.1007/978-3-642-62029-4
[15] Legrand, Homotopie des Espaces de Sections 941 (1982) · Zbl 0535.55001 · doi:10.1007/BFb0094692
[16] DOI: 10.1073/pnas.43.2.241 · Zbl 0142.21802 · doi:10.1073/pnas.43.2.241
[17] DOI: 10.1073/pnas.41.11.961 · Zbl 0067.16001 · doi:10.1073/pnas.41.11.961
[18] Huebschmann, J. Pure Appl. Algebra
[19] Huebschmann, J. Algebra
[20] Huebschmann, J. Algebra
[21] Shih, Homologie des Espaces Fibr?s (1962) · Zbl 0105.16903
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