Ok, today we are interested in finding a projective cover of a given ${R}$-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 ${R=k[G]}$ for ${k}$ a field. If the characteristic ${k}$ doesn’t divide ${|G|}$, then ${R}$ 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 ${\mathcal{A}}$ that has enough projectives (i.e. for ${A \in \mathcal{A}}$ there is a projective object ${P}$ and an epimorphism ${P \rightarrow A}$) where each object has finite length.  Example: the category of finitely generated modules over an artinian ring.

An epimorphism ${f:A \rightarrow B}$ is called essential if for each proper subobject ${A' \subset A}$, ${f(A') \neq B}$. A projective cover of ${A}$ is a projective ${P}$ with an essential map ${P \rightarrow A}$

Theorem 1 Each object in ${\mathcal{A}}$ has a projective cover.   (more…)