# 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!
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991417527198792, "perplexity": 30.452036854373496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256600.32/warc/CC-MAIN-20190522002845-20190522024845-00200.warc.gz"}