Dominant and codominant dimensions for quiver representations

Let $M$ be a module category and $Q$ a rooted quiver. In this paper, we study the dominant (resp. codominant) dimension of the category $\mathrm{Rep}(Q,M)$ of $M$-valued representations of $Q$. To do this, we first study injective envelopes and projective covers that play important roles in homological algebra and give explicit formulas for them in the category $\mathrm{Rep}(Q,M)$, whose origins go back to the classical representation theory of a finite quiver over a field. Then, by using such descriptions, we compute the dominant (resp. codominant) dimension of $\mathrm{Rep}(Q,M)$.

We show that the dominant dimension of $\mathrm{Rep}(Q,M)$ is at most one for every nonzero module category $M$ and any right rooted quiver with at least one arrow.