# Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?

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.

• Have a look at mathoverflow.net/questions/25122/… for instance. Of course, you can define the grothendieck group without the condition of separatedness. I think noetherian (or just locally noetherian) should be enough; see Definition 1.4 in math.leidenuniv.nl/scripties/MasterJavanpeykar.pdf (I dont recommend you read that text too thoroughly...) – Ariyan Javanpeykar Sep 16 '14 at 9:15
• Thank you for both comments. Yes, the question is about the Grothendieck ring, defined as isomorphims classes of varieties modulo some relations. I should have written the definition in the post, I will edit it so that it is clearer. – Manuel Mérida Angulo Sep 16 '14 at 12:42
• Sorry I misread the question. – Ariyan Javanpeykar Sep 16 '14 at 13:51
• This is not an answer to your question, but let me note that schemes need not be reduced neither. If you apply the relation $[X]=[Y]+[X-Y]$ to the inclusion $X_{red} \hookrightarrow X$, them you get $[X]=[X_{red}]$ since the underlying topological space of $X-X_{red}$ is empty. So you can erase "reduced" in your definition of the Grothendieck ring and you get the same thing. – schn93 Sep 21 '14 at 23:38