# Prove that $V(f_1,\ldots,f_k)=Supp(\mathcal{O}_U/\mathcal{J})$

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!