Any object X whose underlying space is locally noetherian is quasi-separated. Y is also quasi-separated. This proves the assertion. Then g is quasi-separated. D f is quasi-compact.
Y are two morphisms and g is quasi-compact and surjective.
For simplicity we shall write X' for X Y' and f' for f Y' . X be the canonical projection. Y is quasi-compact. First we prove the assertion under the assumption that Y is separated. Y is quasi-compact.