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 ?

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. Am I missing something here ?