  The paper is devoted to the proof of the following conjecture due to B.
Feigin. Let $\frak u_\ell$ be the small quantum group a the primitive $\ell$-th
root of unity. Then it is known that the usual $Ext$ algebra of the trivial
$\frak u_\ell$-module is isomorphic to the algebra of regular functions on the
nilpotent variety $\cal N$ in the corresponding simple Lie algebra $\frak g$
(see [GK]). Consider semiinfinite cohomology of the trivial $\frak
u_\ell$-module introduced in [Ar1], [Ar2]. It was shown in [Ar2] that the $Ext$
algebra of the trivial $\frak u_\ell$-module acts naturally on semiinfinite
cohomology. Moreover semiinfinite cohomology space of the trivial
$\frak_\ell$-module is equipped with a natural $\frak g$-module structure. B.
Feigin conjectured that the described $\frak g$-module and $F(\cal N)$-module
structures coincide with the ones on the space of local cohomology of the
structure sheaf on $\cal N$ with support in the standard positive nilpotent
subalgebra $\frak n^+\subset\cal N\subset \frak g$. We give a detailed proof of
the conjecture. Moreover we generalize the statement and describe semiinfinite
cohomology of contragradient Weyl modules in terms of local cohomology of
certain coherent sheaves on the nilpotent cone and on its desingularization
$T^*(G/B)$. To do this we construct a certain specialization of the quantum BGG
resolution of the simple module $L(\lambda)$ defined for generic values of the
quantizing parameter into the root of unity called the quasi-BGG complex. The
complex conists of direct sums of quasi-Verma modules and provides
conjecturally a resolution of the Weyl module $W(\lambda)$.
