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I read the book "Algebraic Geometry" by U. Görtz and whenever limits are involved I struggle for an understanding. The application of limits is mostly very basic, though; but I'm new to the concept of limits.

My example (page 60 in the book): Let $A$ be an integral domain. The structure sheaf $O_X$ on $X = \text{Spec}A$ is given by $O_X(D(f)) = A_f$ ($f\in A$) and for any $U\subseteq X$ by

\begin{align} O_X(U) &= \varprojlim_{D(f)\subseteq U} O_X(D(f)) \\ &:= \{ (s_{D(f)})_{D(f)\subseteq U} \in \prod_{D(f)\subseteq U} O_X(D(f)) \mid \text{for all } D(g) \subseteq D(f) \subseteq U: s_{D(f)\big|D(g)} = s_{D(g)}\} \\ &= \bigcap_{D(f)\subseteq U} A_f. \end{align}

I simply don't understand the last equality: In my naive understanding the elements of the last set are "fractions" and the elements of the Inverse Limit are "families of fractions".

Any hint is appreciated.

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    Since A is an integral domain, all its localisations embed into its field of fractions. Moreover, let $D(h) \subset D(g)$ and $a, b \in A_g$. Then $a = b$ iff their images in $Frac(A)$ are the same, iff their images in $A_h$ are the same. Hence any element in your "family of fractions" can actually be identified with an element of $Frac(A)$ - and the last line tells you just which fractions you get.2012-12-07
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    I think it is a general fact that if you have an inverse system of "sufficiently nice" objects $\left\{A_{i}\right\}_{i\in I}$ in which all the structure maps $\phi_{ij}: A_{j}\to A_{i}$ are injections, then the inverse limit is just the intersection $\bigcap_{i \in I} A_{i}$.2012-12-07
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    @TomBachmann -- I feel dumb, but I still don't get it. Let $(s_{D(f)})_{D(f)\subseteq U} \in O_X(U)$. Then each $s_{D(f)}$ of this family can be identified with an element of $\text{Frac}(A)$. But how can the whole family $(s_{D(f)})_{D(f)\subseteq U} \in O_X(U)$ be identified with an element in $\bigcap_{D(f)\subseteq U} A_f$? (The $s_{D(f)}$ of the family don't need to define all the same element in $\text{Frac}(A)$ as the $D(f)$ don't need to be subsets of each other, right?)2012-12-08
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    Yes they do define the same element. This is essentially because your scheme is irreducible, and so all (non-empty) opens meet. In particular, while $D(f)$ and $D(g)$ need not be subsets of each other, $D(fg)$ is a subset of both (and $fg$ is not zero since this is an integral domain).2012-12-09
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    @TomBachmann -- Thank you very much, this is solved. But I can't mark your comment as accepted answer...2012-12-09
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    Don't worry about it. Didn't seem worth a full answer to me.2012-12-09
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    Compute some filtrant limits to get intuition. I think Atiyah and Macdonald´s book has some useful exercises. And for basic sheaf theory, I recommend Tenninson´s book. For a more detailed study of categories and sheaves, pick Kashiwara and Schapira´s book "Categories and Sheaves".2013-05-21

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