how to get the injective envelope and projective cover of a given module

Question

Given a bound quiver $(Q, I)$ and a representation $M$ of $Q$, how to get the injective envelope and projective cover of $M$? how to give the corresponding essential monomorphism and superfluous epimorphism? Is there a general or specific method?

Answer 1

I assume you know the notations $S(i), P(i), I(i)$ for the simple, projective, and injective module (respectively) associated to the vertex $i$. Each of these modules is easily definable so any text on the module theory of path algebras and their quotients will have their definitions.

I will also assume you know how to find the radical and the socle of any given module. Again this is something standard that you'll find in any text on this stuff. You are given a module as a representation of the quiver and you write down a new representation according to a few basic rules.

Now some facts:

• The projective cover and injective envelope of $S(i)$ are $P(i)$ and $I(i)$ respectively.
• If $M$ and $N$ have projective covers $P(M)$ and $P(N)$ respectively, then $M \oplus N$ has projective cover $P(M) \oplus P(N)$. The same is true for injective envelopes.
• The projective cover of $M$ is isomorphic to the projective cover of the top, $M/\operatorname{rad}M$, of $M$.
• The injective envelope of $M$ is isomorphic to the injective envelope of the socle of $M$.

And that's all you need to know. So for example if you want the essential surjection from the projective cover of $M$ to $M$ you do the following:

1. Find the top: $\operatorname{top}M = M/\operatorname{rad}M$.
2. The top is a semi-simple module so decompose it into simples: $\operatorname{top}M = \bigoplus_i S(i)^{a_i}$
3. The projective cover of $M$ is $P(M) = \bigoplus_i P(i)^{a_i}$. Each $P(i)$ has an obvious map $P(i) \to S(i)$ so we get a surjection $P(M) \to \operatorname{top}M$.
4. As $P(M)$ is projective the map $P(M) \to \operatorname{top}M$ factors through the surjection $M \to \operatorname{top}M$. The map $P(M) \to M$ that you get from factoring is the essential surjection of the projective cover of $M$ onto $M$.

To get injective envelopes it's the exact same set of steps, just dualized so that you're computing injections instead of surjections.