Baer invariants and cohomology of precrossed modules
The aim of this paper is to describe the derived functors of the reflector \(U: \mathcal{PCM}\to \mathcal{CM}\) from the category of precrossed modules \(\mathcal{PCM}\) to the category of crossed modules \(\mathcal{CM}\) in several different ways. Following the work of Fröhlich and Furtado-Coelho [\textit{A. Fröhlich}, Trans. Am. Math. Soc. 109, 221--244 (1963; Zbl 0122.25702); \textit{J. Furtado-Coelho}, J. Algebra 40, 596--609 (1976; Zbl 0372.20037)] and of Everaert and the reviewer [\textit{T. Everaert} and \textit{T. Van der Linden}, Theory Appl. Categ. 12, 1--33 (2004; Zbl 1065.18011); Theory Appl. Categ. 12, 195--224 (2004; Zbl 1065.18012)], the low degrees are first treated in terms of \textit{Baer invariants} and an induced exact sequence. Then a tensor product is introduced and compared with a construction in [\textit{D. Conduché} and \textit{G. J. Ellis}, J. Algebra 123, No. 2, 327--335 (1989; Zbl 0698.18008)]. The authors show that one of the Baer invariants in the exact sequence may be expressed in terms of this tensor product. Following [\textit{K. Modi}, Simplicial methods and the homology of groups. London: University of London (PhD thesis) (1976)], the final section is devoted to higher-order derived functors of the reflector \(U\). Here an exact sequence is obtained which may be used to compare the cohomology groups (in the sense of [\textit{D. Arias} et al., Ill. J. Math. 46, No. 3, 739--754 (2002; Zbl 1033.18010)]) of a precrossed module \((M,P,\mu)\) with the cohomology groups (in the sense of [\textit{P. Carrasco} et al., J. Pure Appl. Algebra 168, No. 2--3, 147--176 (2002; Zbl 0996.18010)]) of its reflection \(U(M,P,\mu)\).
scientific article; zbMATH DE number 6307134
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