Introduction. If P is a ring and M a left P-module, then homological algebra attaches three dimensions to M, projective, weak, and injective(1)By taking the supremum of one of these dimensions as M ranges over all left P-modules, one obtains one of the left "global" dimensions of R. Auslander and Buchsbaum [3] and, subsequently, Serre [14], found it relevant and fruitful, in the study of commut...

Covariant differential complexes on quantum linear groups. Abstract We consider the possible covariant external algebra structures for Cartan's 1-forms (Ω) on GL q (N) and SL q (N). Our starting points are that Ω's realize an adjoint representation of quantum group, and all monomials of Ω's possess the unique ordering. For the obtained external algebras we define the differential mapping d poss...

We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial, unipotent Tg-module for all g ≥ 4 and give an explicit presentation of it as a Sym • H1(Tg,C)-module when g ≥ 6. We do this by proving that, for a finitely generated group G satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that th...

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems. Moreov...

For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M ]. This strengthens the K2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stableK2-group is substituted by the unstable ones. The proof is based on a polyhedral/combinatorial techniques, computations...