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I have a question about the construction of the presheaf which induces the associated sheaf $\mathcal{K}_X ^* / \mathcal{O}_X ^*$ used for the definition of cartier divisors. By definition of quotient presheaf of $\mathcal{O}_X ^* \subset \mathcal{K}_X ^*$ is defined as follows: $U \to \mathcal{K}_X ^*(U) / \mathcal{O}_X ^*(U)$ for every open $U \subset X$.

I have two questions: Firstly: Why $\mathcal{O}_X ^* \subset \mathcal{K}_X ^*$ holds? (therefore $\mathcal{O}_X ^*(U) \subset \mathcal{K}_X ^*(U)$ for every U)?

And secondly: What kind of quotient structure is $\mathcal{K}_X ^*(U) / \mathcal{O}_X ^*(U)$? Quotient of what? Abelian groups?

Here the definitions of used symbols: enter image description here

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For (1), Definition 1.13 points out that $\mathcal O_X \subset \mathcal K_X$. Passing to sheaves of groups of units (which we can do on the level of presheaves if that's worrisome) is just a restriction of the domain of each localization map, so preserves injectivity.

For (2), yes; given that the objects involved have no further structure this is the only reasonable definition. Given that you asked this question in the first place, it's probably good to recall that the notion of exactness in abelian categories (which off the top of my head I'm fairly sure determines the notion of quotient in that category) is quite often reduced to "a sequence of blah blah blahs is exact if and only if the underlying sequence of abelian groups is exact."

Obviously if $\mathcal C$ lacks a forgetful functor to $\mathcal Ab$ you may need to be more careful, but even then you can often do things like one does for sheaves, which we might describe as a variation where rather than characterize exactness of a sequence in terms of its image under a forgetful functor to $\mathcal Ab$, we instead talk about its image under a (typically) infinite collection of less trivial functors to $\mathcal Ab$ (the stalk functors $\mathcal Sh_X \to \mathcal Ab$ which are indexed by the points of $X$), and require that the induced sequence of stalks be exact at every point.

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