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?



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


