ИСТИНА

We consider the group $Diff_1({ Z})$ of substitutions of formal power series $$ \varphi(t)=t+x_1t^2+x_2t^3+x_3t^4+\dots, \quad x_i \in { Z}, i=1,2,3,\dots $$ It was the fundamental observation by Buchstaber and Shokurov in 70s that the ring $S$ of left-invariant differential operators on $Diff_1({ Z})$ is isomorphic to the Landweber-Novikov algebra in the complex cobordisms theory and the tensor product $S \otimes { R}$ is isomorphic to the universal enveloping algebra $U(L_1)$ of the Lie algebra $L_1$ of formal vector fields on the real line which vanish at the zero together with their first derivative. Later Buchstaber used $Diff_1({ R})$ for his construction of the very interesting family of nilmanifolds $M^n$. The calculation of the cohomology of the corrsponding objects turns to be very important question. We consider Feigin-Fuchs-Wallach-Rocha-Carridi free resolution of the trivial one-dimensional $L_1$-module $R$. For a long time the formulas for differentials of this resolution were unknown. We present the explicit answer for this question in terms of so-called Virasoro singular vectors and discuss applications to cohomology calculations.