# Projection formula for immersions

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?

• 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
• 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
• @Angelo: (although at this point I doubt you'd see this) the derived projection formula should always hold, Prop 5.6 in Hartshorne RD. And if Martin doesn't like the word noetherian, one can probably stretch the assumptions in that proposition by using unbounded complexes and the work of Spaltenstein. – Jacob Bell Mar 8 '13 at 9:13

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