On comparing the cohomology of algebraic groups, finite Chevalley groups 
and Frobenius kernels 
Christopher P. Bendel, D.K. Nakano, and C. Pillen

Let G be a reductive algebraic group scheme and k be an algebraically closed 
field of characterstic p > 0. Let F: G ---> G^{(1)}  be the Frobenius map. 
For a fixed r \ge 1, let G_r be the kernel of the rth iteration of 
the Frobenius map. Furthermore, let G(F_q) where q = p^r be the finite 
Chevalley group consisting of the F_q-rational points of G. For over thirty 
years, there has been a considerable amount of effort in relating the 
representation theory of reductive algebraic groups, Frobenius kernels and 
finite groups of Lie type in the defining characteristic. In this paper we 
will be primarily interested in the following three questions. 

(1) Can all extensions between Gr-modules be found via an extension theory 
for G? 

(2) Can all extensions between G(Fq)-modules be found via an extension 
theory for G? 

(3) Given two rational G-modules M and N, is there a relationship between 
Ext*_{G_r}(M,N) and 
Ext*_G(F_q)(M,N)? 

In 1987, H.H. Andersen first posed question (2) for Ext1 between simple 
G(F_q)-modules. He was able to answer this question affirmatively for 
simple G(F_q)-modules in several generic cases. 

The methods employed in this paper will be functorial and different from the 
ideas used in the past. In order to answer questions (1) and (2) we compare 
the category of G(F_q)-modules and G_r-modules with truncated categories of 
G-modules which contain enough projective objects. This will be accomplished 
by constructing two Grothendieck spectral sequences.  Our procedure will also 
demonstrate why it is easier to provide an answer to (1) as opposed to (2) 
for simple modules.  Specifically, one obtains a formula which allows one to 
compute Ext^1 between two simple modules in mod(G_r) by knowing Ext^1 between 
simples in mod(G). We demonstrate that the situation for finite Chevalley 
groups is not as straightforward as in the Frobenius kernel 
case. However, with our functorial approach one can obtain results slightly 
stronger than Andersen's. 

We also provide some qualitative results involving the relationship between 
the cohomology of G(F_q) and G_r 
which partially answers question (3). We first give a criterion for 
projectivity of modules over G(F_q). A criterion for G_r was given earlier 
in work of C.~Bendel and D.~Nakano. This criterion involves induced modules, 
Weyl modules for G and principal series modules for G(F_q). We also 
investigate sufficient 
conditions to insure the vanishing of Ext^n_G(F_p)(M,N) for M,N in mod(G). 
This is given in terms of the vanishing of certain weight spaces in the 
cohomology for U_1 where U is the unipotent radical of the Borel subgroup 
B of G.