  Admissible pairs $((\widetilde S, \widetilde L), \widetilde E)$ consisting of
an $N$-dimensional projective scheme~$\widetilde S$ of certain class with a
special ample invertible sheaf $\widetilde L$ and a locally free ${\cal
O}_{\widetilde S}$-sheaf $\widetilde E$ are considered. An admissible pair can
be produced in the procedure of a transformation (which is called a resolution)
of a torsion-free coherent sheaf $E$ on a nonsingular $N$-dimensional
projective algebraic variety $S$ to a locally free sheaf $\widetilde E$ on some
projective scheme $\widetilde S$. Notions of stability and semistability of
admissible pairs and a moduli functor for semistable admissible pairs are
introduced. Also the relation of the stability and semistability for admissible
pairs to the classical stability and semistability for coherent sheaves under
the resolution is examined. Morphisms between the moduli functor of admissible
semistable pairs and the Gieseker--Maruyama moduli functor (of semistable
coherent torsion-free sheaves) with the same Hilbert polynomial on a
nonsingular $N$-dimensional projective algebraic variety are constructed.
Examining these morphisms it is shown that the moduli scheme for semistable
admissible pairs $((\widetilde S, \widetilde L), \widetilde E)$ is isomorphic
to the Gieseker--Maruyama moduli scheme for coherent sheaves. The
considerations involve all the existing components of these moduli functors and
of their corresponding moduli schemes.
  Bibliography: 25 items.
