Scheme theoretic closure of a locallly closed subscheme - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:21:35Z http://mathoverflow.net/feeds/question/43426 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43426/scheme-theoretic-closure-of-a-locallly-closed-subscheme Scheme theoretic closure of a locallly closed subscheme brunoh 2010-10-24T20:30:06Z 2010-10-26T14:28:26Z <p>In the book "The Geometry of Schemes" of Eisenbud and Harris, page 26, it is said that the scheme theoretic closure of a closed subscheme Z of an open subscheme U is the closed subscheme of X defined by the sheaf of ideals consisting of regular functions whose restrictions to U vanish on Z. I cannot verify this assertion when the open immersion of U in X is not quasi-compact (I mean I cannot prove that this sheaf of ideals is quasi-coherent). Am I missing something here ? </p> http://mathoverflow.net/questions/43426/scheme-theoretic-closure-of-a-locallly-closed-subscheme/43671#43671 Answer by brunoh for Scheme theoretic closure of a locallly closed subscheme brunoh 2010-10-26T14:28:26Z 2010-10-26T14:28:26Z <p>It seems indeed that example 2.10 in the Stacks project morphisms of schemes chapter provides a counter example, where the sheaf of ideals of regular functions whose restrictions to U vanish on Z is not quasi-coherent, because if it were part (3) of lemma 4.3 would be fulfilled. Thank you again for this hint, and if I am not mistaken, I have answered my own question, and an errata should be added for this book on page 26, where the open immersion of U in X should be supposed quasi-compact.</p>