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Usually, the localization of a $R$-module $M$ by a multiplicative subset $S \subseteq R$ with $1 \in S$ is categorically defined as the initial object of the full subcategory $\mathbf C$ of $M \, \backslash \, R\!-\!\mathbf{Mods}$ constituted by the objects $M \to N$ with $N$ such that $s\times \cdot \in \mathrm{Aut}_{R-\mathbf{Mods}}(N)$ for all $s\in S$.

I was wondering if the following definition is valid as well : let $\mathbf C$ be the full subcategory of $R\!-\!\mathbf{Mods}$ constituted by the objects $N$ such that $s\times \cdot \in \mathrm{Aut}_{R-\mathbf{Mods}}(N)$ for all $s\in S$ ; then the functor of localization $S^{-1}$ is the left adjoint of the functor of inclusion $$ i \colon \mathbf C \to R\!-\!\mathbf{Mods}.$$ It would have the advantage to present $S^{-1}$ directly as a (colimits preserving) functor. The canonical homomorphisms $M \to S^{-1}M$ then are just the (components of the) unit of the adjunction $S^{-1} \dashv i$.

However, I never saw such a presentation of the localization. Is my statement wrong ?


P.S. : I am aware that in either case, I need to explicitly construct the localization. My question is really about the point of view on the localization.

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  • $\begingroup$ You don't have to construct it, the existence follows from adjoint functor theorems. By the way, the second definition is standard, and in fact more common than the first one. I wonder why you see this vice versa ... $\endgroup$ Aug 18, 2013 at 21:12
  • $\begingroup$ @MartinBrandenburg I'm not very familiar with the adjoint functor theorems. As for your interrogation, Wikipedia, Algebra (Serge Lang) or the Ravi Vakil's notes bring localization of a module $M$ as a map from $M$ that factories uniquely every map from $M$ to a module $N$ in which multiplication by a element of $S$ is invertible : that is clearly the first definition I gave. (Even Stacks Project does not explicitly state the adjunction.) $\endgroup$
    – Pece
    Aug 18, 2013 at 22:13
  • $\begingroup$ Oh sorry, it was too late yesterday. You are right. $\endgroup$ Aug 19, 2013 at 7:47
  • $\begingroup$ @MartinBrandenburg I updated myself about AFT (General form for now). Here, my locally small category $\mathbf C$ as all small limits, created by the inclusion $i$ (because the $S$-inverting property is stable under small products and equalizers), and $i$ is therefore continuous. It remains to check the Solutions Set Condition to conclude ; and I'm having trouble finding the solutions set for a fixed $R$-module $M$ (well, except for the set $\{S^{-1}M\}$ that we search to avoid...). Do you have any hint ? $\endgroup$
    – Pece
    Aug 19, 2013 at 15:40

1 Answer 1

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This is an instance of a standard fact about left adjoints:

Let $U : \mathcal{C} \to \mathcal{D}$ be a functor. Then $U$ has a left adjoint if and only if each comma category $(D \downarrow U)$ has an initial object, and the value of the left adjoint at an object $D$ is an initial object in $(D \downarrow U)$.

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  • $\begingroup$ Thanks, I did not know that theorem. Actually, that question came to my mind because of the similarity (in the duality of the presentation) with the sheafification process of a presheaf over a site. Your answer explains that similarity too. $\endgroup$
    – Pece
    Aug 18, 2013 at 15:30

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