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The Stacks project

Lemma 29.6.6. Let

\xymatrix{ X_1 \ar[d] \ar[r]_{f_1} & Y_1 \ar[d] \\ X_2 \ar[r]^{f_2} & Y_2 }

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

\xymatrix{ X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\ X_2 \ar[r] & Z_2 \ar[r] & Y_2 }

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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