I would like to ask for a reference of the following fact: the irreducible components of the closure of a locally closed subset are the closures of the irreducible components of the subset.
Thank you very much.
2026-03-28 16:58:14.1774717094
Irreducible components of the closure of a locally closed subset
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I am not an expert on that topic, but isnt it by definition? I mean you have your locally closed W= U $\bigcap$V with U open and V closed. By taking V=$\overline{W}=\bigcup V_i$ a decomposition in irredicuble subsets. Then you have that $W=\bigcup V_i\cap U$ and $\overline{V_i\cap U}=V_i$.
If I am wrong, I am almost sure that in the Harthsorne you can find something.