TITLE: Indecomposable projectives correspond to irreducibles - reference
QUESTION [5 upvotes]: Hello,
We have the following assertion:

In an abelian category that has enough projectives and in which every object has finite length, indecomposable projectives correspond bijectively to irreducibles (both up to isomorphism).

I see how to construct a well defined map from irreducibles (up to iso.) to indecomposable projectives, and to show that it is injective. What I do not know is how to show the following:

Given an indecomposable projective of finite length, how to show that it has a unique irreducible quoteint?

(I have edited the question; Sorry about the initial imprecision).
Thanks,
Sasha

REPLY [4 votes]: It's useful for questions like this to go back to the basic literature where some of these ideas are developed in context.   (Serganova's lecture notes look helpful but if course rely on older sources.)   The applications I'm familiar with arise in various areas of module theory including modular representations of finite groups, but the arguments can be imitated for abelian categories satisfying reasonable finiteness conditions; one could even resort to embedding such categories in module categories (by standard theorems going back to Freyd and Mitchell).  
The two-part treatise Methods of Representation Theory (Wiley, 1981+) by Curtis and Reiner provides a lot of general background beyond what is used traditionally for finite groups.   In particular, Section 6C of Part I has a clear discussion of projective covers.   Most of the interesting results in this direction apply to modules over arbitrary artinian rings.    Once one has enough projectives, it's straightforward to show that any finitely generated module has a projective cover (unique up to isomorphism).   
These ideas for the BGG category $\mathcal{O}$ of a semisimple Lie algebra (which is artinian) come up in Section 3.9 of my 2008 AMS text but don't involve the special features of that module category.   Not only do projective covers of  simple modules in the category exist, but one sees immediately from the definition of "essential" map that any indecomposable projective in the category has a unique simple quotient.   (Modules here actually have finite length.)  To be explicit, the given projective $P$ has at least one simple quotient $L$, which in turn has a projective cover $P_L$.   Now you get maps between $P$ and $P_L$, which from the definitions are isomorphisms in both directions.
My main point is that quite a bit of systematic work (for module categories) has been done on these questions, so it's a good idea to be aware of the relevant literature when it's needed for applications.