We prove finite generation of the cohomology ring of any finite dimensional
pointed Hopf algebra, having abelian group of grouplike elements, under some
mild restrictions on the group order. The proof uses the recent classification
by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of
Lusztig's small quantum groups, whose cohomology was first computed explicitly
by Ginzburg and Kumar, as well as many new pointed Hopf algebras. We also show
that in general the cohomology ring of a Hopf algebra in a braided category is
braided commutative. As a consequence we obtain some further information about
the structure of the cohomology ring of a finite dimensional pointed Hopf
algebra and its related Nichols algebra.Comment: 36 pages, references adde