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 quasicompact (I mean I cannot prove that this sheaf of ideals is quasicoherent). Am I missing something here ?

2$\begingroup$ Have you compared with the counterexamples in the Stacks project concerning schematic closure? math.columbia.edu/algebraic_geometry/stacksgit/morphisms.pdf $\endgroup$ – Martin Brandenburg Oct 24 '10 at 22:30
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 quasicoherent, 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 quasicompact.