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Stream: theory: mathematics

Topic: Tensor product commute with colimits in the base ring

Jack Jia (Jan 17 2025 at 22:08):

I am wondering if there is a more categorical proof of the fact that for $\mathcal{O}_X$-modules $\mathcal{G}, \mathcal{H}$, "the stalk of the tensor product is the tensor product of the stalk", namely $\lim\limits_{\longrightarrow} (\mathcal{F}(U) \bigotimes_{\mathcal{O}_X(U)} \mathcal{G}(U))= \lim\limits_{\longrightarrow} (\mathcal{F}(U)) \bigotimes_{ \lim\limits_{\longrightarrow}(\mathcal{O}_X(U))} \lim\limits_{\longrightarrow}(\mathcal{G}(U))$, where the colimit is taken over all open sets containing the single point.

I read from this answer by Martin Brandenberg that mentions tensor product should also commute with colimits in the base ring, but I am not sure how to prove this.

Fernando Yamauti (Jan 17 2025 at 23:28):

Everything can be written using only tensor products of abelian grps. The tensor product over some base ring of two modules is the coequaliser of some tensor product of abelian groups of 3 guys going to the tensor product of ab grps of the same two modules, now, seen as abelian groups. But I don't know what adjunction he's referring to.