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 $\mathcal{C}$?



Simple object of $k[X,Y]/(Y^2)$
For any $\lambda\in k$, $A/(X-\lambda Y)$ is another example, I think.


