An Important Isomorphism

Algebraic Geometry
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Tsakanikas Nickos
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An Important Isomorphism

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Post by Tsakanikas Nickos »

Let \( \left( X , \mathcal{O}_{X} \right) \) be a ringed space and let \( \mathcal{F} , \mathcal{G} \in Mod \left( X , \mathcal{O}_{X} \right) \). Fix \( x \in X \). Show that there exists a natural isomorphism \[ \displaystyle \left( \mathcal{F} \otimes_{\mathcal{O}_{X}} \mathcal{G} \right)_{x} \cong \mathcal{F}_{x} \otimes_{\mathcal{O}_{X,x}} \mathcal{G}_{x} \]
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