Ok, today we are interested in finding a projective cover of a given -module, which can be done under certain circumstances. (Injective hulls, by contrast, always exist.) The setting in which we are primarily interested is the case of
for
a field. If the characteristic
doesn’t divide
, then
is semisimple and every module is projective, so this is trivial. But in modular representation theory one does not make that hypothesis. Then taking projective envelopes of simple objects gives the indecomposable projective objects.
Projective Covers
So, fix an abelian category that has enough projectives (i.e. for
there is a projective object
and an epimorphism
) where each object has finite length. Example: the category of finitely generated modules over an artinian ring.
An epimorphism is called essential if for each proper subobject
,
. A projective cover of
is a projective
with an essential map
.
Theorem 1 Each object in
has a projective cover.
Pick and write
as a quotient of some projective
, via a map
. Consider the collection of subobjects
such that
; since
has finite length, we can choose a minimal element
.
We thus have an epimorphism . I claim that
is the projective cover. To see this, choose a subobject
such that the map
is epi, and
is the minimal such subobject. There is a commutative diagram
Since is projective, we can find a lifting
making the following diagram commutative:
So if we take the kernel of
, we have a commutative diagram (note that
is epi by minimality of
):
By commutativity, we have . Since
and
is essential, so is
and consequently
essential. This implies
by the above assumptions, so
in the above commutative diagram. It now follows that since we have the monomorphism
of which
is a retraction, that
is a direct summand of
and consequently projective. We’re consequently done. There is also uniqueness:
Proposition 2 The projective cover is unique up to isomorphism.
Given two projective covers of
with essential maps
,
, we can do a lifting both ways to get a commutative diagram.
By the essentiality, the map is epi and consequently by projectivity, split. But the extra direct factor in
aside from
means that
isn’t essential after all, contradiction.
Uniqueness works in any abelian category by the above proof. Source: Jean-Pierre Serre, Linear Representations of Finite Groups, Part III.
By the way, xymatrix is definitely a more versatile package for commutative diagrams than amscd. See, e.g., James Milne’s manual.
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