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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?

flag	asked Dec 10 at 14:01 Martin Brandenburg 29.1k●2●45●129

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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 at 15:16

Thank you Angelo. Please add this as an answer, then I will accept it. – Martin Brandenburg Dec 11 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 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 at 21:09

I don't think that the corresponding derived formula holds either. – Angelo Dec 12 at 7:44

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Angelo · Accepted Answer · 2012-12-11 15:50:41Z UTC

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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.

edited Dec 11 at 15:50

answered Dec 11 at 15:38

Angelo
19.5k●3●47●75

Projection formula for immersions

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