be an abelian category. For any object
the extension group
is defined to be the derived functor of
the ``hom'' functor
We note that the
functor is a ``bifunctor''.
We may thus consider the right derived functor of
and that of
Fortunately, both coincide:
The extension group
may be calculated by using either an
injective resolution of the second variable
a projective resoltuion of the first variable
See [1, Proposition 8.4,Corollary 8.5].
Let us compute the extension groups
- We may compute them by using an injective resolution
- We may compute them by using a free resolution
Compute an extension group
of your choice.
(Please choose a non-trivial example).
In the last lecture we mentioned the notion of injective hulls.
Although they are not essential part of our lecture,
students may find it interesting to calculate some of the injective hulls
of known modules.
So we write down some definitions and results related to them.
Such an essential extension is called maximal if no module properly containing E is an essential extension of M .
is called an essential extension
if every non-zero submodule of
non-trivially. We denote this as
is injective if and only if
has no proper essential extensions.
, there exists an injective module
whichis minimal among such. The module
is unique up to a (non-unique)
in the above theorem is called the injective hull
hulls may then be used to obtain the ``minimal injective resolution''
of a module.
be a positive integer.
The injective hull of a
is equal to
Thus an injective resolution of
is given as follows.