  The paper is concerned with cohomology of the small quantum group at a root
of unity, and of its upper triangular subalgebra, with coefficients in a
tilting module. We relate it to a certain t-structure on the derived category
of equivariant coherent sheaves on the Springer resolution, and to equivariant
coherent IC sheaves on the nil-cone. The support of these cohomology is
described in terms of cells in affine Weyl groups, and the basis in the
Grothendieck group they provide is related to the Kazhdan-Lusztig basis, as
predicted by J. Humphreys and V. Ostrik.
  The proof is based on earlier results of Arkhipov, Ginzburg and the author
which allow us to reduce the question to purity of IC sheaves on affine flag
varieties.
