Math 232
Problem set #2
to do in class on Wednesday January 23
Suppose Gis a finite cyclic group, and Ais an abelian group on which Gacts (i.e.,
Ais a G-module). Define N:A→Aby N(a) = Pg∈Gga. Fix a generator σof G
and define
ˆ
H0(A) := AG/image(N)
ˆ
H1(A) := ker(N)/(σ−1)A.
Define the Herbrand quotient
h(A) := |ˆ
H0(A)|
|ˆ
H1(A)|
if these quantities are finite.
(1) Show that ˆ
H1(A) is independent of the choice of σ.
(2) Compute h(A) for the following G-modules A:
(a) A=Z(with trivial G-action)
(b) A=R[G] for any ring R.
(3) If Ais finite, show that h(A) = 1. Hint: check that the following sequences
are exact
0−→ ker(N)−→ AN
−−→ image(N)−→ 0
0−→ AG→Aσ−1
−−→ (σ−1)A−→ 0.
(4) Show that if 0 →A→B→C→0 is an exact sequence of G-modules, then
there is an exact hexagon
ˆ
H0(A)//ˆ
H0(B)
!!
C
C
C
C
C
C
C
C
ˆ
H1(C)
==
|
|
|
|
|
|
|
|ˆ
H0(C)
}}|
|
|
|
|
|
|
|
ˆ
H1(B)
aaC
C
C
C
C
C
C
C
ˆ
H1(A)
oo
(5) Show that if 0 →A→B→C→0 is an exact sequence of G-modules, and
if at least two of h(A), h(B), h(C) are defined, then all three are defined and
h(B) = h(A)h(C).
(6) Show that if Ahas finite index in Bthen h(A) = h(B).
(7) Find an example of a G-module Asuch that h(A) = 1, but ˆ
H0(A) and ˆ
H1(A)
are not isomorphic abelian groups.
