I am confused about notation concerning tensor products of sheaves of modules. I know that given a ringed space $X$ and $\mathcal{O}_X$-Modules $\mathcal{F}$ and $\mathcal{G}$ their tensor product is defined to be the associated sheaf to the presheaf $U \mapsto \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{G}(U)$. Despite the sheafification process I have seen authors writing sections in the form $a \otimes b$.

In the case I am dealing with at the moment, a morphism of $\mathcal{O}_X$-Algebras $\mathcal{O}_X[T, T^{-1}] \to \mathcal{O}_X[T, T^{-1}] \otimes \mathcal{O}_X[T, T^{-1}]$ is defined by $T \mapsto T \otimes T$ and I cannot see why this is meaningful, to be precise: why $T \otimes T$ represents a global section of the target. Is there some kind of convention or abuse of notation I am missing out on? Any help will be appreciated.

  • $\begingroup$ There's a natural morphism from a presheaf to its sheafification, so writing down a section of a presheaf means you've written down a section of its sheafification. $\endgroup$ – Qiaochu Yuan Mar 2 '15 at 20:15
  • $\begingroup$ Now that you say it, it seems rather obvious. Thank you for helping me out. $\endgroup$ – user220439 Mar 3 '15 at 9:57

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