Fix a field $k$. For a singular variety $X$, I understand that the Grothendieck group $K^0(X)$ of vector bundles on $X$ is not necessarily isomorphic to the Grothendieck group $K\_0(X)$ of coherent sheaves on $X$. 


I am curious to learn what is known about these two groups in one family of examples: $\mathbb P^n\_{D}$, where $D$ is the dual numbers $D=k[\epsilon]/(\epsilon^2)$. 


References would be especially appreciated, as I know very little about K-theory.



Grothendieck group for projective space over the dual numbers
If $X$ is a noetherian separated scheme and $X\_{red}$ its reduction , we have $K\_0(X)=K\_o(X\_{red})$: in other words $K\_o$ doesn't see nilpotents .  

Much more generally and profoundly, Quillen has proved that for all his $K$-theory groups, $K\_i(X)=K\_i(X\_{red})$.  

In your particular case you thus have (in the following $T$ is an indeterminate)
$$K\_0(\mathbb P^n\_D)=K\_0(\mathbb P^n\_k)=\mathbb Z[T]/(T^{n+1})$$


As for $K^0$, a special case of a theorem of Berthelot (SGA 6, Exposé VI, Théorème 1.1, page 365)
states that, for any commutative ring $A$, we have $K^0(\mathbb P^n\_A)=K^o(A)[T]/(T^{n+1})$.  

If $A=D=k[\epsilon]$, we have $K^0(D)=\mathbb Z$, since projective modules over local rings (like $D$) are free.  

So here too $$K^0(\mathbb P^n\_D)=\mathbb Z[T]/(T^{n+1})$$


**Bibliography**  

Srinivas has written [this nice book](http://books.google.com/books/about/Algebraic_K_Theory.html?id=Sr8iTL98eiAC) on $K$-theory. 


And as an homage to the recently sadly departed Daniel Quillen, let me refer to his groundbreaking paper  

 "Higher algebraic $K$-theory I", published in Springer's Lecture Notes LNM 341.


