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.
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
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.