This problem is from Gathmann's notes problem 3.23 https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/alggeom-2014.pdf
Let $X ⊂ \mathbb{A}^n$ be an affine variety, and let $a ∈ X$. Show that $\mathscr{O}_{X,a} \cong \mathscr{O}_{\mathbb{A}^n,a}/I(X)\mathscr{O}_{\mathbb{A}^n,a}$,
where $I(X)\mathscr{O}_{\mathbb{A}^n,a}$ denotes the ideal in $\mathscr{O}_{\mathbb{A}^n,a}$ generated by all quotients $\frac{f}{
1}$
for $f ∈ I(X)$.
My first idea is to use the first isomorphism theorem. So I want a homomorphism $φ:\mathscr{O}_{\mathbb{A}^n,a} \to \mathscr{O}_{X,a}$ such that $\varphi$ is surjective and $\ker(\varphi)=I(X)\mathscr{O}_{\mathbb{A}^n,a}$. But I am really having trouble understanding the definition of $\mathscr{O}_{\mathbb{A}^n,a}$ and what the elements looks like. Any help is appreciated, thank you.
Best Answer
The elements of $\mathscr{O}_{\mathbb{A}^n,a}$ are just rational functions $\frac{g}{f}$, where $f, g$ are polynomials of $n$ variables (i.e. elements of $\mathscr{O}_{\mathbb{A}^n}$) such that $f(a) \neq 0$.
This of course is nothing but a copy of Lemma 3.21 in your linked notes.
Intuitively, you should think about $\mathbb{A}^n$ as the $n$-dimensional complex space $\mathbb{C}^n$.
A polynomial $f\in \mathbb{C}[X_1, \cdots, X_n]$ then defines a function on $\mathbb{C}^n$: it sends a point $a = (a_1, \cdots, a_n)$ to $f(a_1, \cdots, a_n)$.
Now if $f, g\in \mathbb{C}[X_1, \cdots, X_n]$ are two polynomials, in general you cannot say that $\frac{g}{f}$ defines a function on $\mathbb{C}^n$, simply because the value of $f$ may vanish at some points.
But for a given point $a$, if we have $f(a) \neq 0$, then it's clear that $f$ does not vanish in a small neighborhood of $a$ (with respect to the usual topology of $\mathbb{C}^n$), hence on that small neighborhood we may define $\frac{g}{f}$ without problem.
And if we gather all functions that can be defined as $\frac{g}{f}$ in a small neighborhood of the given point $a$, we get the local ring $\mathscr{O}_{\mathbb{A}^n,a}$.