  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.
