Semisimplicity of the category of coherent sheaves? 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. 
 A: 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. 
