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is there an example schemes such that underlying spaces are not homeomorphic but sheaves are isomorphic? Maybe if there exist, I want to see that example

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    Dear QiL. I no have such a example. The reason that this question Hartshorne book "Algebraic geometry" Proposition II.2.6 http://math.stackexchange.com/questions/240477/question-of-hartshorne-books-proposion-ii-2-6. In question, $V$ is hmeomorphic to the set of the closed point of $X$. But, It say that sheaf on $X$ is isomorphic to sheaf on $V$. Right?2012-11-27

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Let $Y$ be the spectrum of $k$, $X=\mathbb P^1_k$ and $f: X\to Y$ be the canonical morphism. Then $f_*O_X$ is a sheaf supported in one point (that of $Y$), so it can be identified with $(f_*O_X)(Y)=\Gamma(X, O_X)=k$. Therefore $O_Y\to f_*O_X$ is an isomorphism, but $X$ is not isomorphic to $Y$.