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?

`$Y = X \smallsetminus \{(0,0)\}$`

, and suppose that $M$ is non-zero and supported at the origin. $\endgroup$ – Angelo Dec 10 '12 at 15:16`$Y=X\setminus\{0\}$`

, $M$ supported on $X\setminus Y$ give a counter-example? $\endgroup$ – Sándor Kovács Dec 11 '12 at 18:06