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.
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3$\begingroup$ The category of quasi-coherent sheaves is abelian on any scheme. The category of coherent sheaves, on the other hand, is only abelian on a locally Noetherian (or at best a locally coherent) scheme, I would think. E.g., consider the case of an affine scheme, which is the spectrum of an arbitrary ring. The category of finitely presented modules over such a ring is not abelian. What is "the abelian category of coherent sheaves" over such a scheme? $\endgroup$– Leonid PositselskiCommented Apr 13, 2019 at 13:41
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$\begingroup$ @LeonidPositselski The category of coherent sheaves is always abelian. $\endgroup$– user20948Commented Jan 6, 2021 at 11:41
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1 Answer
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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.
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$\begingroup$ No, the category of coherent sheaves over an infinite disjoint union of reduced points is semisimple abelian, in fact. It is equivalent to the infinite Cartesian product of the categories of finite-dimensional vector spaces over the related fields. Every object in it is naturally the direct sum of its components sitting at the points, and at the same time it is the infinite product of the same components. $\endgroup$ Commented Apr 13, 2019 at 15:19
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$\begingroup$ ... So, in particular, the structure sheaf over such a scheme $X$ is the infinite direct sum, and at the same time the infinite product, of the one-dimensional (skyscraper) sheaves $k_x$ sitting at the points $x\in X$. These skyscraper sheaves are simple objects. $\endgroup$ Commented Apr 13, 2019 at 15:24
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6$\begingroup$ The issue is maybe the correct definition of "semisimple". If I look at ncatlab.org/nlab/show/semisimple+category or en.wikipedia.org/wiki/Semi-simplicity , the definition is that every object is a direct sum of finitely many simple objects. If we remove the condition "finitely many", I agree with your comments. $\endgroup$ Commented Apr 13, 2019 at 15:38
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$\begingroup$ Oh, yes. Then you are right. $\endgroup$ Commented Apr 13, 2019 at 15:40