Suppose $X$ is a smooth variety over $\mathbb{C}$. Let $K^{b}(X)$ be the homotopy category of bounded complex of coherent sheaves, and $D^{b}(X))$ be the derived category of bounded complex of coherent sheaves. One can define:
$$Hom^{\cdot}(A^{\cdot}, B^{\cdot}): K^{b}(X) \to K(Ab)$$ as
$$Hom^{i}(A^{\cdot},B^{\cdot}):= \oplus Hom(A^k,B^{k+i}),$$
$$d(f):=d_B \circ f – (-1)^{i} f \circ d_A.$$
Then we can pass to the derived categories, and define the right derived functor:
$$RHom^{\cdot}(A^{\cdot},): D^{b}(X ) \to D(Ab).$$
My questions is:
How to compute $RHom(F,k(x))$ in the derived category? Here $F$ is a sheaf on $X$(viewed as a complex in $D(X)$ concentrated in degree $0$), and $k(x)$ is a skyscraper sheaf on $x \in X$. In particular how to compute $RHom(k(x),k(x))$?
I think by definition, one has to do an injective resolution to $k(x)$, and perform the computation in the homotopy category.
Best Answer
For computations you it is better to replace the first argument by a locally free resolution. This allows to compute the local $R{\mathcal H}om$. Then you can use the local-to-global spectral sequence to compute the global $RHom$. In your particular example, as everything happens in a neighborhood of $x$ and since $x$ is a locally complete intersection in $x$, you can find a vector bundle $E$ and it section in a neighborhood of $x$ such that its zero locus is $x$. Then the Koszul complex $$ 0 \to \Lambda^n E^* \to \dots \to \Lambda^2 E^* \to E^* \to O_X \to 0 $$ is a locally free resolution of the skyscraper. This shows that $$ {\mathcal E}xt^i(k(x),k(x)) = \Lambda^i E_x $$ and hence $$ Ext^i(k(x),k(x)) = \Lambda^i E_x $$ as well. Moreover, one has $E_x \cong T_xX$, so the final answer is $$ RHom(k(x),k(x)) = \oplus_{i=0}^n \Lambda^iT_xX[-i]. $$
EDIT. To compute local Ext's you just apply ${\mathcal H}om$ to the Koszul resolution. You will get a complex $$ 0 \to k(x) \to E\otimes k(x) \to \Lambda^2E \otimes k(x) \to \dots \to \Lambda^nE \otimes k(x) \to 0. $$ Since the map in the Koszul complex are given by wedging with the section of $E$ which vanish at $x$, it follows that in the above complex all maps are zero, so its cohomology sheaves are just its terms.
In this particular case you see that all local Ext's are sheaves supported at a point, so they don't have higher cohomology and the local-to-global spectral sequence degenerates in the second term automatically. In general it does not degenerate so quickly, so you may have to calculate the higher differentials.