Let $U\subset \mathbb{C}^n$ be open and $f_1,\ldots,f_k\in \mathcal{O}(U)$, where $\mathcal{O}$ is the sheaf of holomorphic functions in $\mathbb{C}^n$. Show that $V(f_1,\ldots,f_k)=Supp(\mathcal{O}_U/\mathcal{J})$, where $\mathcal{J}=f_1\mathcal{O}_U+\cdots+f_k\mathcal{O}_U$ is an ideal sheaf of $\mathcal{O}_U$.

My attempt: $Supp(\mathcal{O}_U/\mathcal{J})=\{p\in U:\mathcal{J}_p\neq \mathcal{O}_{U,p}\}$, so it suffices to show that $f_i(p)=0$ for all $i$ if and only if $\mathcal{J}_p\neq \mathcal{O}_{U,p}.$ The 'only if' part is trivial since there exists $g\in \mathcal{O}_{U,p}$ such that $g(p)\neq 0$. Conversely, suppose $f_i(p)\neq 0$ for some $i$. Then $f_i$ has a multiplicative inverse near $p$, so for any $h\in \mathcal{O}_{U,p}$, $h=(hf_i^{-1})f_i\in \mathcal{J}_p$, so $\mathcal{J}_p=\mathcal{O}_{U,p}$.

Am I correct? Any notice for errors are welcome!


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