Nilpotent extensions and some associated invariants (Q1911338)

The paper is concerned with certain (Schur-)Baer-(Fröhlich-)invariants associated with an augmented algebra \(A\) over the commutative ring \(k\). The corresponding functors are defined by means of nilpotent extensions. The author deduces a five term exact sequence involving some of these invariants, which generalizes the well-known five term sequence in low degrees in the homology theory of groups. (The Baer-invariants considered here and the corresponding sequence have also been obtained -- in very general circumstances including the two cases mentioned above -- by \textit{J. Furtado-Coelho} [J. Algebra 40, 596-609 (1976; Zbl 0372.20037)].) A new proof is given that one of the introduced Baer-invariants can be interpreted using homological algebra as \(\text{Tor}^A_2 (k,k)\). Various results are then deduced from the corresponding five term exact sequence; this sequence -- again in a more general form -- can be found in a paper by \textit{M.-A. Knus} [J. Algebra 9, 274-284 (1968; Zbl 0162.05103)] together with most of the consequences obtained by the author of the present paper.