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 ?
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2$\begingroup$ Have you compared with the counterexamples in the Stacks project concerning schematic closure? math.columbia.edu/algebraic_geometry/stacks-git/morphisms.pdf $\endgroup$– Martin BrandenburgCommented Oct 24, 2010 at 22:30
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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.