Projection formula for immersions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:48:34Z http://mathoverflow.net/feeds/question/115980 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115980/projection-formula-for-immersions Projection formula for immersions Martin Brandenburg 2012-12-10T14:01:24Z 2012-12-11T15:50:41Z <p>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</p> <p><code>$M \otimes i_* \mathcal{O}_Y \to i_* i^* M.$</code></p> <p><strong>Question</strong>: Is it always an isomorphism?</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/115980/projection-formula-for-immersions/116098#116098 Answer by Angelo for Projection formula for immersions Angelo 2012-12-11T15:38:00Z 2012-12-11T15:50:41Z <p>Set $X = \mathbb A^2_k$, <code>$Y = X \smallsetminus \{(0,0)\}$</code>, and suppose that $M$ is non-zero and supported at the origin.</p>