  We study the cohomology with trivial coefficients of Lie algebras L_k of the
polynomial vector fields on the line with zero $k$-jet, (k>=1), and the
cohomology of the similar subalgebras {L}_k of the polynomial loops algebra
$\widetilde{sl}_2$. In both cases we construct the special bases (filtering
bases) in the external complexes of these algebras. A spectral sequence based
on this construction allows to completely find the cohomology of L_k and {L}_k.
We also apply the filtering bases to find the spectral resolution of the
Laplace operators for algebras L_1 and L_0, and obtain explicit formulas for
the representing cycles of homologies for algebras L_k and {L}_k by means of
the Schur polynomials.
