I have a question about the definition of a complex analytic space from its Wikipedia article.
So locally, we are looking at an open set $U$ in $\mathbb C^n$, and considering the sheaf $\mathcal O_U$ of holomorphic functions on $U$. If $X$ is the closed set in $U$ defined as the vanishing set of finitely many holomorphic functions $f_1, … , f_k$ on $U$, Wikipedia says we define a sheaf of rings $\mathcal O_X$ on $X$ to be the "restriction of $\mathcal O_U/(f_1, … , f_k)$ to $X$."
I am not quite sure what this means. I can think of one possible way to interpret this, but it may not be correct. Maybe one can define a sheaf of rings $\mathscr F$ on $U$ as the sheaf associated to the presheaf $W \mapsto \mathcal O_U(W)/(f_1|W, … , f_k|W)$, and then take $\mathcal O_X$ to be the inverse image sheaf $i^{-1} \mathscr F$ under the inclusion map $i: X \rightarrow U$. Is this the correct way to define $\mathcal O_X$?
Best Answer
One approach less formal than you're taking is the following. The structure sheaf is, among other things, specifying a collection of actual functions $f: W \to \mathbb{C}$ for some open subset $W$ of $X$. Since $X$ has the induced topology, $W = W' \cap X$ for an open subset $W'$ of $U$. For any two $g, h \in \mathcal{O}_U(W')$ with $g - h \in (f_1, \dots, f_k)\mathcal{O}_U(W')$, it's easy to check that $g(x) = h(x)$ for all $x \in W$, and so we can define $g|_W : W \to \mathbb{C}$ as $g|_W(x) = g(x)$... or $h(x)$. By construction, $g|_W$ corresponds to the coset/element $g + (f_1, \dots, f_k)$ of $(\mathcal{O}_U/(f_1, \dots, f_k)))(W')$. Therefore, it makes sense to view $\mathcal{O}_U/(f_1, \dots, f_k)$ as a sheaf of functions defined on $X$, and if we unwind how we are thinking of these sections as functions on $X$, a sensible name for this is the restriction of the sheaf to $X$.
It's useful to work out why this is the same as the "inverse image" definition you gave.