Let $(X, \mathcal{O}_{X})$ be a ringed-space with sheaves of modules $\mathcal{F}$ and $\mathcal{G}$. I would like to show that for any point $p \in X$, $$ \mathcal{F}_{p} \otimes_{\mathcal{O}_{X, p}} \mathcal{G}_{p} \simeq (\mathcal{F} \otimes_{\mathcal{O}_{X}} \mathcal{G})_{p} $$ using the fact that left adjoints preserve colimits. I have shown an adjunction between the sheaf tensor and the internal hom functor. The problem is, this seems to only give me that the functor, $$ - \otimes_{\mathcal{O}_{X}} \mathcal{F} $$ preserves colimits. Similarly I have the tensor-hom adjunction for modules over a ring, so I would ideally show this for the tensor product presheaf. But this still only gives me that $$ - \otimes_{\mathcal{O}_{X}(U)} \mathcal{F}(U) $$ preserves colimits. But of course if I want to take a directed limit over open sets containing a point $p$, the ring $\mathcal{O}_{X}(U)$ will change as $U$ changes. Similarly I would also need the result for the bifunctor $$ - \otimes_{\mathcal{O}_{X}} - $$ to get the tensor product of both stalks. Is there any way to use this method to show that the tensor product sheaf commutes with taking stalks, or do I just need to do it manually?

  • $\begingroup$ What would doing it manually involve? I'm fairly certain that at some point you'll have to move some kind of colimit inside some kind of tensor $\endgroup$ – leibnewtz Mar 7 '18 at 9:02
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    $\begingroup$ It is not a clear duplicate, but it basically answers your question: math.stackexchange.com/questions/222869/… $\endgroup$ – MooS Mar 7 '18 at 9:04
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    $\begingroup$ Use the answer linked in @MooS comments with the fact that $(\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G})_p=\varinjlim_{U\ni p}(\mathcal{F}(U)\otimes_{\mathcal{O}_X(U)}\mathcal{G}(U))$. You can prove this equality using that taking stalks commute with the sheafification functor. $\endgroup$ – Roland Mar 7 '18 at 10:17

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