Lifting formulas, Moyal product, and Feigin spectral sequence

It is shown, that each Lifting cocycle ${\mathrm{\Psi }}_{2n+1},{\mathrm{\Psi }}_{2n+3},{\mathrm{\Psi }}_{2n+5},\mathrm{.}\mathrm{.}\mathrm{.}$ ([Sh1], [Sh2]) on the Lie algebra ${\text{Dif}}_n$ of polynomial differential operators on an $n$-dimensional complex vector space is the sum of two cocycles, its even and odd part. We study in more details the first case $n=1$. It is shown, that any nontrivial linear combination of two 3-cocycles on the Lie algebra ${\text{Dif}}_1$, arising from the 3-cocycle~${\mathrm{\Psi }}_3$, is not cohomologous to zero, in a contradiction with the Feigin conjecture~[F]. The new conjecture on the cohomology $H_{\text{Lie}}^{\text{ndot}}({\text{Dif}}_1;\text{C})$ is made.