Lemma 29.6.6. Let
be a commutative diagram of schemes. Let Z_ i \subset Y_ i, i = 1, 2 be the scheme theoretic image of f_ i. Then the morphism Y_1 \to Y_2 induces a morphism Z_1 \to Z_2 and a commutative diagram
Lemma 29.6.6. Let
be a commutative diagram of schemes. Let Z_ i \subset Y_ i, i = 1, 2 be the scheme theoretic image of f_ i. Then the morphism Y_1 \to Y_2 induces a morphism Z_1 \to Z_2 and a commutative diagram
Proof. The scheme theoretic inverse image of Z_2 in Y_1 is a closed subscheme of Y_1 through which f_1 factors. Hence Z_1 is contained in this. This proves the lemma. \square
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