Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra $hat{frak{gl}}_{infty|infty}$ and its classical subalgebras at positive integral levels. We also ob
tain Kostant-type homology formulas for the Lie algebra $ widehat{frak{gl}}_infty$ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations.
We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal subm
odule of a corresponding reducible Kac module is generated by its proper singular vector.
