Topics in Cohomology of Groups_Serge Lang.pdf
Topics in Cohomology of Groups_Serge Lang.pdf
Topics in Cohomology of Groups_Serge Lang.pdf
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44<br />
f<strong>in</strong>ite number <strong>of</strong> generators <strong>of</strong> the complex <strong>in</strong> each idmension. S<strong>in</strong>ce<br />
Hr(G, A) is a torsion group by the preced<strong>in</strong>g corollary, it follows<br />
that it is f<strong>in</strong>ite.<br />
Corollary 1.13. Suppose G f<strong>in</strong>ite and A E Mod(G) is uniquely<br />
divisible by every <strong>in</strong>teger m 6 Z, m # O. Then Hr(G, A) = 0 for<br />
all r E Z.<br />
Proposition 1.14. Let U C G be a subgroup <strong>of</strong> f<strong>in</strong>ite <strong>in</strong>dex.<br />
Let A, B 6 Mod(G) and let f : A ---* B be a U-morphism. Then<br />
Ha(SV(f)) = trv o Hv(f) o rest,<br />
and similarly with H <strong>in</strong>stead <strong>of</strong> H when G is f<strong>in</strong>ite.<br />
Pro<strong>of</strong>. We use the fact that the assertion is immediate <strong>in</strong> dimen-<br />
sion 0, together with the technique <strong>of</strong> dimension shift<strong>in</strong>g. We also<br />
use Chapter I, Lemma 2.4, that we can take a G-morp~sm <strong>in</strong> and<br />
out <strong>of</strong> a trace, so we f<strong>in</strong>d a commutative diagram<br />
0 , A , MG(A) , XA , 0<br />
! 1 l<br />
0 , B , MG(B) , XB , 0<br />
the three vertical maps be<strong>in</strong>g sU(f),SU(M(f)) and sU(x(f)) respectively.<br />
In the hypothesis <strong>of</strong> the proposition, we replace f by<br />
X(f) : XA ----+ XB, and we suppose the proposition proved for<br />
X(f). We then have two squares which form the faces <strong>of</strong> a cube as<br />
shown:<br />
res : Hu(XA)<br />
HG(XA)<br />
HG(XB) ~ res<br />
~(A{Hc(S~(x(I)))<br />
HG(B) "<br />
tr<br />
-<br />
\<br />
Hu(A)<br />
Hu(B)<br />
The maps go<strong>in</strong>g forward are the coboundary homomorphisms, and<br />
are surjective s<strong>in</strong>ce MG erases cohomology. Thus the diagram al-<br />
lows an <strong>in</strong>duction on the dimension to conclude the pro<strong>of</strong>. In the<br />
case <strong>of</strong> the special functor H, we use the dual diagram go<strong>in</strong>g to the<br />
left for the <strong>in</strong>duction.