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It is often asserted in commutative algebra texts that localization is important in algebraic geometry. I would appreciate some precise examples which show the utility of the concept in this context. Although I have some knowledge of algebraic geometry, I am not an expert, so I would like the examples to be relatively simple if possible.

Mike Pierce
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Rodney Coleman
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    Something basic to start of with. If $X$ is an affine scheme equal to $Spec A$ then the basic open set $D(f) = Spec A_f$ and the local ring at a point $p$ is isomorphic to $A_p$. –  Jan 16 '14 at 12:57
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    I would say you should just start reading about algebraic geometry. Localization will show up really quickly. In fact, it might well show up in definitions before it is even needed in any results. – Tobias Kildetoft Jan 16 '14 at 13:01
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    Look up the definition of the local ring at a point on a smooth algebraic curve, say. This is used to define the order of vanishing of a function at the point (or the order of a pole there). – KCd Jan 17 '14 at 16:16
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    Related – Viktor Vaughn Nov 14 '18 at 03:46

1 Answers1

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One of the philosophical points of algebraic geometry is to study not the geometric object itself, but the set of functions on it. Imagine the easiest geometric object: the line. Now, we must work over some field $K$ so the line is $K$. What is a natural ``function space'' on $K$? Well, as we are working with algebraic stuff our "functions'' are polynomials. From the algebro-geometric viewpoint the line IS the ring $K[x]$. To the point on the line one can associate an ideal of functions vanishing in that point. In our case it would be $(x-a)$. So we have the "space'' $K[x]$ and the "point" $(x-a)$. One can ask, what is the space $K[x]$ near the point $(x-a)$? I.e. how the "infinitely" small neighborhood of this point is structured? The functions on a small neighborhood of $a$ are just functions that are not zero near $a$. And of the function is not zero near $a$ then the inverse function is still "regular". So the "functions'' on this small neighborhood are exactly the ring $K[x]_{(x-a)}$.

Don't know whether this makes sense, maybe it will be easier just to properly read some introduction to algebraic geometry.

P.S. We can work not over a field, but i think that is a situation where geometric intuition can be applied.

user68061
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  • Can you suggest a readable introduction to algebraic geometry. – Rodney Coleman Jan 17 '14 at 23:18
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    There are many texts that are easy to read, e.g. Gathmann's notes. Or the book of Perrin (Algebraic geometry an introduction). There is also an absolutely wonderful course by Ivan Mircovic that gives a very good feeling of what algebraic geometry is about and what is it needed for here. – user68061 Jan 18 '14 at 04:54
  • ``The functions on a small neighborhood of $a$ are just functions that are not zero near $a$''... I don't understand, why can't functions near $a$ have zeros near $a$? I think you started with polynomial functions but at some point shifted to rational functions? . – ArB Jan 07 '25 at 17:42