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Let $i : Y \to X$ be a quasi-compact immersion of schemes and let $M$ be a quasi-coherent sheaf on $X$. There is a canonical homomorphism

$M \otimes i_* \mathcal{O}_Y \to i_* i^* M.$

Question: Is it always an isomorphism?

Clearly this question is local on $X$. The class of $M$ satisfying the condition is closed under finite direct sums and contains $\mathcal{O}_X$. It follows that it contains all sheaves which are locally free of finite rank.

It is true in general if $i$ is an affine morphism (for example, when $i$ is a closed immersion). So what happens for open immersions?

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Set $X = \mathbb A^2_k$, $Y = X \smallsetminus \{(0,0)\}$, and suppose that $M$ is non-zero and supported at the origin. – Angelo Dec 10 '12 at 15:16
Thank you Angelo. Please add this as an answer, then I will accept it. – Martin Brandenburg Dec 11 '12 at 6:56
Martin, how do you prove this for affine morphisms? Wouldn't $X=\mathbb A^1$, $Y=X\setminus\{0\}$, $M$ supported on $X\setminus Y$ give a counter-example? – Sándor Kovács Dec 11 '12 at 18:06
isn't a good approach to take everything to be derived (where the formula always holds) and then see for which class of morphisms derived = underived? – Jacob Bell Dec 11 '12 at 21:09
I don't think that the corresponding derived formula holds either. – Angelo Dec 12 '12 at 7:44
up vote 2 down vote accepted

Set $X = \mathbb A^2_k$, $Y = X \smallsetminus \{(0,0)\}$, and suppose that $M$ is non-zero and supported at the origin.

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