# Simple objects in Quasi-coherent sheaves are isomorphic to structure sheaves of a closed point

Need to prove that simple objects in Quasi-coherent sheaves are isomorphic to structure sheaves of a closed point.

"Simple object" means that there is no non-trivial subobject.

I encountered this problem in Vakil's FOAG(April 1, 2023), 6.5.L. I have proved that if the simple sheaf is a simple module on every affine open set, we have the result. The approach is to notice that, under this assumption, the simple qcoh sheaf is supported at a closed point of the scheme X on every affine open set, then we can construct a skyscraper subsheaf.

But I don't know how to continue.

My attempt to prove the "fact" (the simple qcoh sheaf is a simple module on every affine open set) is as the following.

Suppose we have M on Spec A, M is an A-module, and since M is not simple, we have a submodule of M isomorphic to A/p, where p is a maximal ideal of A, but not necessarily a closed point.

I'm thinking of constructing a skyscraper at p with A/p. But since the closure of p is not necessarily inside Spec A, I can't do the zero extension and get an injection from the skyscraper into the quasi-coherent sheaf.

PS: the scheme X may not have any closed point. I think maybe there is a more subtle way of proving using the sheaf morphisms which I have not learned systematically.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Aug 11, 2023 at 17:12

As you observe, it suffice to classify simple objects in $$\mathrm{QCoh}(\mathrm{Spec} \ A)\simeq A-\mathrm{mod}$$. Let $$M\ne0$$ be a simple $$A$$-module. Then for any $$x\ne0\in M$$, there is a sub-module $$0\ne Ax\subset M$$, so by simplicity, $$M=Ax\simeq A/I$$, where $$I=\ker(x)=\{a\in A:ax=0\}$$.
Now, $$I$$ must be maximal, since otherwise there is an ideal $$A\supsetneq J\supsetneq I$$, and $$J/I\subsetneq A/I$$ is a non-trivial sub-module. But $$A/I$$ with $$I$$ maximal is exactly the structure sheaf of the closed point $$I\in \mathrm{Spec} \ A$$.
For an arbitrary scheme $$X$$, let $$\mathcal F\ne0\in\mathrm{QCoh}(X)$$ be a simple object, so for some $$x\in X$$, $$\mathcal F|_{\{x\}}\ne0$$. Then for any closed sub-scheme $$i_x\colon \{x\}\hookrightarrow X$$, there is a surjection $$\mathcal F\to i_{x*}\mathcal F|_{\{x\}}$$, which must be an isomorphism. Then the same argument as in the affine case applies.