I am stuck at undderstanding a claim made in Derksen's book An Introduction to Quiver Representations. This is Lemma 2.4.3. But the part I am stuck at is purely homological algebra it seems.
Let $A$ be a finite dimensional $\mathbb{C}$-algebra and $V$ and $W$ are two modules.
Suppose we have length injective and projective resolutions as follows (this happens for all modules in our context, but that doesn't matter here):
$0 \to P_1 \to P_0 \to V \to 0$ and
$0 \to W \to I^{0} \to I^{1} \to 0$.
Using the usual right, left exactness of the various functors we get the following diagram (take $Q$ as $A$)
Now the claim that I dont understand is:
'Using the snake Lemma to the above diagram it follows that the cokernel of $Hom_{A}(P_0, W) \to Hom_{A}(P_1, W)$ is isomorphic to $Hom_{A}(V, I^{0}) \to Hom_{A}(V, I^{1})$.
I know that this will be true since both are equal to $Ext^{1}(V, W)$ from the homological algebra definition of $Ext$ in terms of derived functors.
But I want to do it directly in this situation by applying the Snake Lemma as suggested by the author but I don't see how to.
Please help.