The category of coherent sheaves on a locally Noetherian scheme is abelian. Are there some geometric conditions on the scheme that imply that the category of coherent sheaves is semisimple? 


Edited in response to posic's comments. 



Semisimplicity of the category of coherent sheaves?
Let $X$ be a locally Noetherian scheme. Then the abelian category of coherent sheaves on $X$ is semisimple if and only if $X$ is the disjoint union of finitely many reduced points.


The if direction is clear: the category of coherent sheaves on a finite union of reduced points is a direct sum of categories of finite dimensional vector spaces (over fields), so semisimple.


Only if direction. If the category of coherent sheaves is semisimple, then all $Ext^1$ vanish, in particular, for every closed point $x$ of $X$, we have $Ext^1(k\_x,k\_x)=0$, where $k\_x$ is the skyscraper sheaf at $x$. But $Ext^1(k\_x,k\_x)$ is the Zariski tangent space at $X$ (e.g. see <https://math.stackexchange.com/questions/75673/tangent-space-in-a-point-and-first-ext-group> ). As $X$ is locally Noetherian, the local ring at $x$ is Noetherian and the vanishing of the Zariski tangent space at $x$ implies by Nakayama lemma that the local ring at $x$ is a field. Using the fact that in a locally Noetherian scheme, every point specializes to a closed point (e.g. see <https://stacks.math.columbia.edu/tag/01OU>), it follows that $X$ is a disjoint union of reduced points. 
If this union is infinite, then the category of coherent sheaves is not semisimple (the structure sheaf is not a finite direct sum of simple objects). So $X$ has to be a finite disjoint union of reduced points. 


