  This paper generalizes Huang's cohomology theory of grading-restricted vertex
algebras to meromorphic open-string vertex algebras (MOSVAs hereafter), which
are noncommutative generalizations of grading-restricted vertex algebras
introduced by Huang. The vertex operators for these algebras satisfy
associativity but do not necessarily satisfy the commutativity. Moreover, the
MOSVA and its bimodules considered in this paper do not necessarily have
finite-dimensional homogeneous subspaces, though we do require that they have
lower-bounded gradings. The construction and results in this paper will be used
in a joint paper by Huang and the author to give a cohomological criterion of
the reductivity for modules for grading-restricted vertex algebras
