Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 244

The maps f are (continuous) maps of sets that are required to satisfy the
given condition. Note that g 1 f ( g 2 ) means that we evaluate f ( g 2 )andobtain
an element of M , then act on this element of M by the automorphism g 1 .
The set Z is sometimes called the set of twisted homomorphisms from G
to M . It is a group under addition of maps.
We note one important case. If G acts trivially on M ,then

Z ( G, M ) = Hom( G, M )

is the set of group homomorphisms from G to M .
There is an easy way to construct elements of Z ( G, M ). Let m be a fixed
element of M and define

f m ( g )= gm − m.

Then f m gives a map from G to M .Since

f m ( g 1 g 2 )= g 1 ( g 2 m ) − m

= g 1 m − m + g 1 ( g 2 m − m )

= f m ( g 1 )+ g 1 f m ( g 2 ) ,

we have f m ∈ Z ( G, M ). Let

B ( G, M )= {f m | m ∈ M}.

Then B ( G, M ) ⊆ Z ( G, M ) is called the set of coboundaries . Define the
first cohomology group

H 1 ( G, M )= Z/B.

In the important special case where G acts trivially, B ( G, M )=0since
gm − m =0forall g, m . Therefore

H 1 ( G, M ) = Hom( G, M )

is simply the set of group homomorphisms from G to M .
A homomorphism φ : M 1 → M 2 of G -modules induces a map

φ ∗ : H j ( G, M 1 ) → H j ( G, M 2 )

of cohomology groups for j =0 , 1. For H 0 , this is simply the restriction of φ
to M 1 . Note that if gm 1 = m 1 ,then gφ ( m 1 )= φ ( gm 1 )= φ ( m 1 ), so φ maps
M 1 into M 2 .For H 1 ,weobtain φ ∗ by taking an element f ∈ Z and defining

( φ ∗ ( f ))( g )= φ ( f ( g )) .

It is easy to see that this induces a map on cohomology groups.
Themainpropertyweneedisthefollowing.