How to complete $f^*f_*G\to G$ and $F\to f_*f^*F$ into a distinguished triangles for a double branched covering $f:X\to M$?

Question

My main question is that for any proper morphism of smooth projective varieties $f:X\to Y$, is there some general canonical distinguished triangles about $F\to f_*f^*F$ and $f^*f_*G\to G$? May be we can use $f^!$ to fix them (maybe some six functor arguments?)? Note that the functors we consider are all derived functor.

Here is a special case for closed embedding of smooth divisors which is not so difficult: How to complete....

Here is one of situations I found and I don't know why (in the paper CALABI–YAU AND FRACTIONAL CALABI–YAU CATEGORIES of A. Kuznetsov)?

【Q】 Consider the map $f:X\to M$ is a double covering branched in a divisor in the linear system $D\in |\mathscr{L}^{\otimes 2}|$. We may let this is a cyclic cover determined by $D$ and $\mathscr{L}$ (it is if $\mathrm{Pic}(M)$ has no non-trivial $2$-torsion). Let $\tau$ be the covering involution. Then we have canonical distinguished triangles: $$F\to f_* f^*F\to F\otimes\mathscr{L}^{-1}\to,\quad \tau^*G\otimes\mathscr{L}_X^{-1}[1]\to f^*f_*G\to G\to.$$

Note that in this case $f^!F = f^*F\otimes\mathscr{L}_X$. But I have no idea for both cases!

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Thank you for your any help!!!

Answer 1

The main question can be answered as follows. Note that $f_*$ and $f^*$ are both Fourier-Mukai functors with the FM kernels given by the structure sheaf of the graph $$ \mathcal{O}_{\Gamma(f)} \in \mathbf{D}^b(X \times Y). $$ Consequently, the compositions of these functors are given by the convolution-products of the FM kernels. On the other hand, the identity functor is given by the structure sheaf of the appropriate diagonal. Moreover, one can check that the unit and counit of adjunction are induced by appropriate morphisms of the corresponding FM kernels (from/to the convolution of the graphs to/from the diagonal). Therefore, you can take the cones of these morphisms of the FM kernels, and the corresponding FM functors applied to $F$ will extend the distinguished triangles.

For instance, consider the case where $f \colon X \to Y$ is a double covering and we consider the counit of adjunction. Since $f$ is flat, it is easy to check that the convolution of the FM kernels is given by the structure sheaf of the fiber product, so one needs to compute $$ \mathrm{Cone}(\mathcal{O}_{X \times_Y X} \to \mathcal{O}_\Delta), $$ where $\Delta$ is the diagonal. To do this note that $$ X \times_Y X = \Delta \cup \Gamma_\tau $$ (the union of the diagonal and the graph of the covering involution), hence there is an exact sequence $$ 0 \to \mathcal{O}_{\Gamma_\tau}(-R) \to \mathcal{O}_{X \times_Y X} \to \mathcal{O}_{\Delta} \to 0, $$ where $R = \Delta \cap \Gamma_\tau$ is the fixed locus of $\tau$, i.e., ramification divisor of $f$. It shows that the required cone is isomorphic to $\mathcal{O}_{\Gamma_\tau}(-R)[1]$. It remains to note that the corresponding FM functor is the composition of $\tau^*$, twist by $-R$, and shift.