Undergraduate-Level Background
Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A$. I am interested in computing the dimensions of the hom spaces in $C$.
If $A$ is semisimple, then every object in $C$ is isomorphic to a direct sum of some simple objects $i, j, \ldots$. The $k$-dimension of $Hom_C(i,j)$ is the dirac delta $\delta_{ij}$ (by Schur's lemma for algebraically closed fields). This provides a full answer to the problem.
If $A$ is not semisimple, each simple object $i$ corresponds to a projective indecomposable module $P_i$, which is isomorphic to the projective cover of $i$. Hence, the dimension of $Hom_C(P_i, j)$ is again $\delta_{ij}$. This partially answers the problem for $Hom_C(X,Y)$, where X is projective and Y is a direct sum of simples.
Questions
Undergraduate courses usually stop here. And I wonder what's beyond this. I am aware of Auslander-Reiten theory, but I am not sure how one can compute the dimensions using the AR quiver and almost split sequences.
Let me list the remaining cases in order of difficulty.
Compute the $dim_k Hom_C(X,Y)$ where
- $X$ and $Y$ are projective.
- $X$ is projective, and $Y$ is general.
- $X$ and $Y$ are equivalent to finite direct sums of (not necessarily projective) indecomposable modules.
In particular, are there sets of modules $\{J_a\}, \{J_a^*\}$ "dual" to the set of indecomposable modules $\{I_a\}$ in the sense that
$$dim_k Hom_C(I_a, J_b) = \delta_{ab} = dim_k Hom_C(J_a^*, I_b)?$$
