We know that localized rings contains inverse of elements at which we localized our ring. I want to know why do we need to invert elements to study local behavior? I would be grateful if someone help me understand this. I know this may be very basic thing, but I am really confused.
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We can think of the polynomial ring $k[x]$ as being the ring of polynomial functions on $\Bbb{A}^1_k$ (the vector space $k$ with its Zariski topology). The Zariski topology here is equivalent to the finite complement topology. So, let's look at $U=\Bbb{A}^1_k\setminus \{0\}$. We might then ask what the algebraic functions on this open set are. They are all of the rational functions that are defined away from zero. In particular, any $r(x)=\frac{p(x)}{q(x)}$ so that $q(x)$ has no zeroes on $U$ will be an admissible rational function.
Well, we can see that this condition forces $q(x)=x^k$ for some $k\ge 0$. Note that this is exactly the localization of $k[x]$ with respect to the multiplicative subset: $\{x^k\}_{k\ge 0}=S$.
The idea is that "smaller" open sets in an algebraic variety have more rational functions in the above sense. So, localization is the corresponding operation on rings. If $f$ defined on $\Bbb{A}^1_k$ does not vanish on an open set $V$, then the functions on $V$ contain $\frac{1}{f}$. One can rephrase this by saying that we localize at the multiplicatively closed subset of $k[x]$ of functions not vanishing on $V$.
Alekos Robotis
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