Localizations 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=F0R. 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 non-commutative geometry we allow schemes over non-commutative rings with particular attention to so called almost commutative rings. We consider particular cases of Proj R (e.g. for some P.I. ring R) in terms of prime ideals, R-tors in terms of torsion theories and W_(R) in terms of a non-commutative Grothendieck topology based on words of Ore set localizations.