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Let $f\colon X \rightarrow Y$, $g\colon Y' \rightarrow Y$, be two morphisms of schemes. Let $X' = X\times_Y Y'$, and let $f'\colon X' \rightarrow Y'$ be the projection. We are interested in the relation between the scheme theoretic image of $f'$ and that of $f$ (in the sense of Hartshorne, exercise II, 3.11). Namely, when does $f'(X') = g^{-1}(f(X))$ hold?

Note that the equality holds for set theoretic images (see this question), so I am asking whether (or under what conditions) the scheme structure is also preserved.

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is $g^{-1}(f(X))$ a closed subscheme of $Y'$? what is its scheme structure and how is the closed embedding defined? – Dima Sustretov Feb 11 '14 at 15:49

Proposition 4.20 in Algebraic Geometry I by Görtz and Wedhorn will give you the answer. It's a valuable book in any case, I can recommend it very much.

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It will take a while for me to get access to that book... Could you summarize what the answer is? – quim Jan 21 '14 at 12:25
Check your email. :) – David Hornshaw Jan 22 '14 at 16:24

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