The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation


$[X]=[Y] + [X\setminus Y]$


for any $Y \subset X$ a closed immersion and with the product structure given by


$[X\times Y]= [X]\cdot[Y]$.


My question is related to the condition of separateness. Why do we need the schemes to be separated? I have been searching in different papers and notes and I couldn't find anything about it. Could the Grothendieck ring be extended to non separated schemes?


Thank you very much.



Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?
If I understand the setup of the question correctly, the answer is yes. Using non separated schemes makes no difference.


This is explained [here](http://arxiv.org/pdf/1002.4372v1.pdf).


In brief, whether you do the Grothendieck ring of "integral, finite type, separated" or "finite type" or even "finite type algebraic spaces" it's the same. When you pass to stacks it's a whole other mess of course.


