The Cohomology Ring of a Finite Abelian Group
Collin Roberts
The
cohomology
ring
of
a
finite
cyclic
group
was
explicitly
computed
by
Cartan
and
Eilenberg
in
their
1956
book
on
Homological
Algebra.
It
is
surprising
that
the
cohomology
ring
for
the
next
simplest
example,
that
of
a
finite
abelian
group,
has
still
not
been
treated
in
a
systematic
way.
The
results
that
we
do
have
are
combinatorial
in
nature
and
have
been
obtained
using
“brute
force”
computations.
In
this
thesis
we
will
give
a
systematic
method
for
computing
the
cohomology
ring
of
a
finite
abelian
group.
A
major
ingredient
in
this
treatment
will
be
the
Tate
resolution
of
a
commutative
ring
R
(with
trivial
group
action)
over
the
group
ring
RG,
for
some
finite
abelian
group
G.
Using
the
Tate
resolution
we
will
be
able
to
compute
the
cohomology
ring
for
a
finite
cyclic
group,
and
confirm
that
this
computation
agrees
with
what
is
known
from
Cartan-Eilenberg.
Then
we
will
generalize
this
technique
to
compute
the
cohomology
ring
for
a
finite
abelian
group.
The
presentation
we
will
give
is
simpler
than
what
is
in
the
literature
to
date.
We
will
then
see
that
a
straightforward
generalization
of
the
Tate
resolution
from
a
group
ring
to
an
arbitrary
ring
defined
by
monic
polynomials
will
yield
a
method
for
computing
the
Hochschild
cohomology
algebra
of
that
ring.
In
particular
we
will
re-prove
some
results
from
the
literature
in
a
much
more
unified
way
than
they
were
originally
proved.
We
will
also
be
able
to
prove
some
new
results.