Left-Exact Functors
We saw (in theorem 1 here) that the localization functor is exact, which gave us a whole slew of nice properties, including preservation of submodules, quotient modules, finite intersection/sum, etc. However, exactness is often too much to ask for.
Throughout this article, A and B are fixed rings and is a covariant additive functor.
Definition.
We say F is left-exact if it takes a short exact sequence of A-modules
to an exact sequence of B-modules
Immediately we can weaken the condition.
Lemma 1.
F is left-exact if and only if it takes an exact sequence of A-modules
to an exact sequence of B-modules
Proof
(⇐) is obvious. For (⇒) given an exact , we extend it to an exact
. [Recall that
is the cokernel of g.] We can split this exact sequence into short exact ones:
which gives an exact sequence as desired. ♦
Note
Left-exact functors are not as nice as exact ones but we still get some useful results out of them. For example, if is injective, so is
so for a submodule
, we can consider
as a submodule of
.
Also, for any , since
is exact, applying F gives an exact
and thus
.
Hom Functors Are Left-Exact
The main result we wish to show is
Proposition 1.
For any A-module M, the functor
is a left-exact functor.
Proof
Take the exact sequence . We need to show that
is exact. It is easy to show is injective (easy exercise). Next, we have
Conversely suppose satisfies
. Then
. Since f is injective it follows that for each
, we have
for a unique
. This map
is clearly A-linear so
. ♦
Here is one application of this result.
Corollary 1.
Let
; for each A-module, let
. Then
Proof
Indeed the functor is naturally isomorphic to
. Now apply the above. ♦
Exercise A
Find a surjective for which
is not surjective. This shows that the functor
is not exact in general.
Converse Statement
Next we have the following converse.
Proposition 2.
Suppose
and
are A-linear maps such that for any A-module M,
is exact. Then
is exact.
Proof
First we show f is injective. Let so that
is exact. By left-exactness of Hom we have an exact:
.
Since is injective,
and in particular
.
Next we show . Setting
gives:
so This gives
.
Finally we set . The following sequence:
is exact. The inclusion gives
and so
for some
. Then
. ♦
Another Left-Exactness
Now suppose is a contravariant additive functor.
Definition.
We say F is left-exact if it takes a short exact sequence of A-modules
to an exact sequence of B-modules
Again we have:
Lemma 2.
F is left-exact if and only if it takes an exact sequence of A-modules
to an exact sequence of B-modules
Proof
Exercise. ♦
Now we show, as before:
Proposition 3.
For any A-module M, the functor
is a left-exact functor.
Proof
Take the exact sequence . We need to show that
is exact for any A-module M. Injectivity of is obvious. Also
so
.
It remains to show . Pick
such that
so that
. Hence h factors through
. Since
, we have
for some
. ♦
Another Converse Statement
Finally, the reader should expect the following.
Proposition 4.
Suppose
and
are A-linear maps such that for any A-module M,
is exact. Then
is exact.
Proof
We leave it as an exercise to show: g is surjective, .
It remains to show ; let
and take the exact sequence
If is the canonical map,
so we have
for some
. Thus
. ♦
As an application, let us prove the following.
Corollary 2.
Let
be an ideal. If
is an exact sequence of A-modules, then we get an exact sequence of
-modules:
Proof
Let . It suffices to show that for any B-module Q, the sequence
is exact. But since gives the B-module induced from M, the above sequence is naturally isomorphic to:
which we know is exact because the Hom functor is left-exact. ♦
This leads to the following definition.
Definition.
Suppose
is a covariant additive functor. We say F is right-exact if it takes a short exact sequence of A-modules
to an exact sequence of B-modules
Exercise B
Prove that F is right-exact if and only if it takes an exact sequence of A-modules to an exact sequence of B-modules
Thus, we have shown that the functor
is right-exact. This will be generalized in future chapters.
Exercise C
- Prove that the functor
is not exact in general.
- Use a direct proof to show
is right-exact.
In summary, we have covered the following concepts: