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Let $ X $ be a smooth projective variety (over $ \mathbb{C} $) of dimension $ n $ and $ x : \operatorname{Spec} \mathbb{C} \rightarrow X $ a point. How can I compute the complex $ \operatorname{\mathbf{R}Hom}^{\bullet} ( \mathbb{C}(x), \mathbb{C}(x) ) $ for the skyscraper sheaf $ \mathbb{C}(x) $ at $ x $? In general, if $ i : Y \rightarrow X $ is a variety of codimension $ d $, how can I compute $ \operatorname{\mathbf{R}Hom}^{\bullet} ( i_* \mathcal{O}_Y , i_* \mathcal{O}_Y ) $? To make the second question better, I probably need to add that $ i $ is a regular embedding.

I recieved some hints from my advisor on the first question. Because $ X $ is smooth, the point $ x $ can be cut out by $ n $ equations on an affine open $ U $ around $ x $ - so that a 'Koszul complex' exists locally. But I don't see how this gives an answer on $ X $, this just seems to compute it locally.

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    $\begingroup$ For any closed immersion $i\colon Y\to X$ any complexes of sheaves $F$ and $G$ on $Y$, we have $RHom(F,G)\cong RHom(i_*F,i_*G)$ in the derived category. This holds because $i_*$ is fully faithful on derived categories, which follows from the fact that both $i_*$ and $i^*$ are exact on ordinary sheaves and $i^*i_*F\cong F$ for any sheaf $F$. This is purely topological and has nothing to do with any regularity conditions. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 8, 2023 at 5:26
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    $\begingroup$ @D.-C.Cisinski: I guess the OP is about the derived category of coherent sheaves, and on it the functor $i_*$ is not fully faithful. $\endgroup$
    – Sasha
    Commented Feb 8, 2023 at 6:09
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    $\begingroup$ @Cranium Clamp: I think you can find an answer in the book of Daniel Huybrechts. $\endgroup$
    – Sasha
    Commented Feb 8, 2023 at 6:10

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