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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 Sang, it doesn't make sense to say that sheaves on different spaces (especially not isomorphic ones!) are isomorphic. – Georges Elencwajg Nov 26 '12 at 12:45
A possible way to formulate the question is: if we have a morphism $f :X\to Y$ of schemes such that $O_Y\to f_*O_X$ is an isomorphism of sheaves on $Y$, are $X, Y$ isomorphic ? The answer is no. Can you find an example @SangCheolLee ? – user18119 Nov 26 '12 at 12:52
Dear QiL. I no have such a example. The reason that this question Hartshorne book "Algebraic geometry" Proposition 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? – Sang Cheol Lee Nov 27 '12 at 5:19
up vote 2 down vote accepted

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 $Y$.

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