Let $k$ be a field. Let $A=k[X,Y]/(Y^2)$ be the quotient of polynomial ring $k[X,Y]$. Let $\mathcal{C}$ be the category of finite-dimensional $A$-modules $M$ with the action of $X$ nilpotent, and of finite projective dimension. In fact, for any $M\in\mathcal{C}$, we have $\mathrm{proj.dim}_A M\leq 1$. Then $\mathcal{C}$ is an exact category.

Let $U=k[Y]/(Y^2)$ be the quotient of $A$ by $(X)$. Then $U$ can be viewed as an $A$-module, and $U$ has finite projective dimension and is a simple object in $\mathcal{C}$. Does $\mathcal{C}$ have any other simple object? What is the Grothendieck group of $\mathcal{C}$?