  We study decoupling limits and S-dualities for noncommutative open string/
Yang-Mills theory in a gravity setup by considering an $SL(2,Z)$ invariant
supergravity solution of the form ((F, D1), D3) bound state of type IIB string
theory. This configuration can be regarded as D3-branes with both electric and
magnetic fields turned on along one of the spatial directions of the brane and
preserves half of the space-time supersymmetries of the string theory. Our
study indicates that there exists a decoupling limit for which the resulting
theory is an open string theory defined in a geometry with noncommutativity in
both space-time and space-space directions. We study S-duality of this
noncommutative open string (NCOS) and find that the same decoupling limit in
the S-dual description gives rise to a space-space noncommutative Yang-Mills
theory (NCYM). We also discuss independently the decoupling limit for NCYM in
this D3 brane background. Here we find that S-duality of NCYM theory does not
always give a NCOS theory. Instead, it can give an ordinary Yang-Mills with a
singular metric and an infinitely large coupling. We also find that the open
string coupling relation between the two S-duality related theories is modified
such that S-duality of a strongly coupled open-string/Yang-Mills theory does
not necessarily give a weakly coupled theory. The relevant gravity dual
descriptions of NCOS/NCYM are also given.
