Let $A$ be a finite dimensional $\mathbb{K}$-algebra, where $\mathbb{K}$ is an algebraically closed field.
How does one compute (homologically) the number of projective-injective $A$-modules?
Maybe an easier question would be the following : Given a quiver $Q$ and an admissible ideal $I$, what sort of combinatorial properties are we looking for to assure the existence of at least one projective-injective $\mathbb{K}Q/I$ module?
For example, let $Q$ be the quiver $$Q:\;\;\;1\xleftarrow[]{\alpha} 2\xleftarrow[]{\beta}3.$$ Then, $\mathbb{K}Q$ admits only one projective-injective module while $\mathbb{K}Q/\langle\beta\alpha\rangle$ admits two. This shows the dependency on the ideal $I$. It is quite easy to find examples of algebras which admit no projective-injective modules but which have (non-semisimple) quotients admitting some.
It is also possible to reformulate the question asking about the injective dimension of projective modules.