First cohomology groups of finite groups with
nontrivial irreducible coefficients have been useful in several geometric and
arithmetic contexts, including Wiles's famous paper
(1995). Internal to group theory, -cohomology plays a role in the general theory of maximal
subgroups of finite groups, as developed by Aschbacher
and Scott (1985). One can pass to the case where the group acts faithfully and
the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal
constant bounding all of the dimensions of these cohomology
groups. This paper provides the first general positive results on this
conjecture, proving that the generic 1-cohomology (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group of Lie type, with absolutely irreducible
coefficients (in the defining
characteristic of ), is bounded by a
constant depending only on the root system. In all cases, we are able to
improve this result to a bound on itself, still depending only
on the root system. The generic result, and
related results for , emerge here as a
consequence of a general study, of interest in its own right, of the
homological properties of certain rational modules , indexed by
dominant weights , for a reductive group . The modules and arise naturally
from irreducible representations of the quantum enveloping algebra (of the same type as ) at a th root of unity,
where is the characteristic of the defining field for . Finally, we
apply our -bounds, results of Bendel, Nakano, and Pillen
(2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on .