Jordan–Hölder, modularity and distributivity in non-commutative algebra

Abstract

A study of lattices of subgroups or subrings adequate for non-commutative homological algebra can be pursued in a setting of weakly exact categories, which extend the Puppe-exact ones [D. Puppe, Korrespondenzen in abelschen Kategorien, Math. Ann. 148 (1962) 1–30; B. Mitchell, Theory of Categories, Academic Press, New York, 1965; P. Freyd, A. Scedrov, Categories, Allegories, North-Holland Publishing Co., Amsterdam, 1990] and the semi-abelian ones [G. Janelidze, L. Márki, W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002) 367–386; F. Borceux, A survey of semi-abelian categories, in: Galois Theory, Hopf Algebras, and Semiabelian Categories, in: Fields Inst. Commun., vol. 43, Amer. Math. Soc., Providence, RI, 2004, pp. 27–60; F. Borceux, D. Bourn, Mal’cev, Protomodular, homological and semi-abelian categories, in: Mathematics and its Applications, vol. 566, Kluwer Academic Publishers, Dordrecht, 2004], and are essentially based on a notion of γ -category introduced by Burgin [M.S. Burgin, Categories with involution and correspondences in γ -categories, Tr. Mosk. Mat. Obs. 22 (1970) 161–228; Trans. Moscow Math. Soc. 22 (1970) 181–257]. In this context, subobjects form w-modular w-lattices, equipped with a normality relation. The free w-modular w-lattice generated by two chains with normality conditions is determined and proved to be weakly distributive, by a construction inspired by the well-known Birkhoff theorem for free modular lattices [G. Birkhoff, Lattice Theory, 3rd ed., in: Amer. Math. Soc. Coll. Publ., vol. 25, 1973]. We show that this theorem is relevant for the study of double filtrations, much in the same way as the Birkhoff theorem in the commutative case; similarly, it should be of use in the study of spectral sequences.