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There are a lot of computation problems in Bhargav Bhatt's Problem Set 1 of Perverse Sheaves 2015, which trouble me a lot.

Basically, suppose $F\colon\mathcal C\to\mathcal D$ is a functor between categories and $\mathcal S$ is the class of morphisms in $\mathcal C$ whose image in $\mathcal D$ is an isomorphism, then $F$ induces a functor $\tilde F\colon\mathcal S^{-1}\mathcal C\to\mathcal D$.

I want to know how to compute $\mathcal S^{-1}\mathcal C$ generally so I think maybe properties of $\tilde F$ will be helpful, especially discovering the conditions under which $\tilde F$ is fully faithful.

I'm looking forward to general strategies, strong enough to solve many questions there: For example, compute $\mathcal S^{-1}\mathcal C$ when $\mathcal C$ is the category of abelian groups and $F=-\otimes_{\mathbb Z}\mathbb Q$. (In general, torsion free abelian groups aren't free, this confuses me a lot), or when $ \mathcal C$ is the category of coherent $\mathscr O_X$ modules over a projective variety $X$ and $F$ is the global section functor.

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    So what's the question? – 2017-02-28
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    @pepa.dvorak General techniques to compute $\mathcal S^{-1}\mathcal C$. – 2017-02-28
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    If you are lucky, and $\mathcal S^{-1}\mathcal C$ is a reflective localisation, you can look for a full subcategory of $\mathcal C$ that satisfy the definition of the [localisation](https://ncatlab.org/nlab/show/localization). For example, the functor $F=-\otimes_{\mathbb Z}\mathbb Q:\mathbf{Ab}\to \mathbf{Ab}$ sends the actions of $\mathbb Z$ on abelian groups to invertible morphisms, which implies that the localisation is the category of $\mathbb Q$-modules, a full subcategory of $\mathbf{Ab}$. There are other settings, where the localisation is relatively easy to describe. – 2017-02-28
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    However, in general, all what one has is the category of fractions. You may would like to see [Calculus of Fractions and Homotopy Theory](http://doi.org/10.1007/978-3-642-85844-4). – 2017-02-28
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    In the case of $\mathbf{Ab}$ tensoring by $\mathbb{Q}$ yields rational vector spaces and it is maybe interesting to see, what happens to some classes of groups: bounded groups are annihilated (isomorphic to $0$) and torsion-free groups of finite rank $n$ (with possibly some bounded direct summand) are all isomorphic to $\mathbb{Q}^n$. – 2017-03-01
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    When one considers a full subcategory $\mathcal{A}$ of $\textbf{Ab}$ of finite-rank groups then tensoring only the $Hom$-sets (and leaving objects untouched) is equivalent to considering the factor-category $\mathcal{A} /\mathcal{B}$ where $\mathcal{B}$ denotes the subcategory of bounded groups. – 2017-03-01
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    @user337830 Thank you. I stupidly missed the simple fact that $\mathbb Q\otimes_{\mathbb Z}\mathbb Q\cong\mathbb Q$! So in general, if $F$ admits a section (precisely, a fully faithful adjoint), then $F$ could be realized as a localization? I don't know whether we need more technical conditions. – 2017-03-01

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