Localization and sheaves of glider representations

Abstract

The notion of a glider representation of a chain of normal subgroups of a group is defined by a new structure, i.e. a fragment for a suitable filtration on the group ring. This is a special case of general glider representations defined for a positively filtered ring \(R\) with filtration \(FR\) and subring \(S = F_0R\). Nice examples appear for chains of groups, chains of Lie algebras, rings of differential operators on some variety or \(V\)-gliders for \(W\) for algebraic varieties \(V\) and \(W\). This paper aims to develop a scheme theory for glider representations via the localizations of filtered modules. With an eye to noncommutative geometry we allow schemes over noncommutative rings with particular attention to so-called almost commutative rings. We consider particular cases of \(\mathrm{Proj}~ R\) (e.g. for some P.I. ring \(R\)) in terms of prime ideals, \(R\)-tors in terms of torsion theories and \(\underline{\mathcal{W}}(R)\) in terms of a noncommutative Grothendieck topology based on words of Ore set localizations.