New answers tagged adjoint-functors
6
votes
Accepted
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
The cone of $i^*i_*\mathcal{F} \to \mathcal{F}$ is isomorphic to $\mathcal{F} \otimes \mathcal{O}_X(-X)[2]$.
EDIT. Let me write an argument for a sheaf $F$. Consider the distinguished triangle
$$
i^*...
- 31.8k
2
votes
Accepted
What is the cone of $\mathcal{F}\to i_*i^*\mathcal{F}$ for a divisor $i: D\hookrightarrow X$?
For any Cartier divisor $i \colon D \hookrightarrow X$ and any $F \in D^b(X)$ there is an exact triangle
$$
F \otimes \mathcal{O}_X(-D) \to F \to i_*i^*F.
$$
It can be obtained by tensoring the exact ...
- 31.8k
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