Simple objects with isomorphic projective covers
Let $X$ and $Y$ be two simple objects of an abelian category. Assume that they have projective covers $P(X)$ and $P(Y)$.
Question: If $P(X)$ and $P(Y)$ are isomorphic, is it true that $X$ and $Y$ are also isomorphic?
Solution 1:
Yes.
Suppose $X$ and $Y$ are nonisomorphic simple objects which both have $P$ as a projective cover. Let $K$ be the kernel of $P\to X$. Then $K$ is not in the kernel of $P\to Y$, so $K\to Y$ is epi, since $Y$ is simple. So the epimorphism $P\to Y$ is not essential.