Topics in Cohomology of Groups_Serge Lang.pdf

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Theorem 3.3. Let 92, ~8 be abeIian categories. Let
Y:92 --~ C(f8)
be an exact functor to the category of complexes in ~3. Then
there exists a cohomologicaI functor H on 92 with values in ~,
such that Hr(A) = homology of the complex Y(A) in dimension
r. Given a short exact sequence in 92:
and therefore the exact sequence
0 ~ A' ---* ~ A -5 A" ~ 0
0---+ Y(A')--~ Y(A) ---, Y(A")---+ O,
the coboundary is given by the usual formula Y(u) -ldY(v) -1 .
For the applications, readers may take ~ to be the category of
abelian groups, and 92 is Mod(G) most of tke time.
Corollary 3.4. Let 921,92 be abelian categories and F a bif~nc-
tor on 921 92 with values in ~, contravariant (resp. covariant)
in the first variable and covariant in the second. Let X be a
complex in C(921) such that the functor A F(X,A) on 92 is
exact. Then there exists a cohomological functor (resp. homolog-
icaI functor) H on 92 with values in ~, obtained as in Theorem
3.3, with F(X,A) = Y(A).
Next we deal with finite groups, for which we obtain a non-trivial
cohomological functor in all dimensions, using constructions with
complexes as in the above two theorems.
Finite groups. Suppose now that G is finite, so we have the
trace homomorphism
S=Sa:A--* A
for every A E Mod(G). We omit the index G for simplicity, so the
kernel of the trace in A is denoted by As. We also write [ instead
of Ia as long as G is the only group under consideration. It is clear
t-hat IA is contained in As and the association
A ~ As/IA
is a functor from Mod(G) into Grab. We then have Tate's theo-
rem.