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open subscheme

An open subscheme of \(\left(X, \mathscr{O}_{X}\right)\) is a scheme \(\left(U, \mathscr{O}_{U}\right)\) whose underlying space is an open subset of \(X\) and whose structure sheaf is isomorphic to \(\left.\mathscr{O}_{X}\right|_{U}\).

Warning: this definition does not work for a closed subscheme.