Fourier-Mukai kernels of mutations?

Question

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of cones, but the "correct" way to define them is via Fourier-Mukai kernels.

What is the expression for these kernels?

I couldn't find it in the literature.

Answer 1

If $p,q$ are the projections, the kernels are given by

$$ cone( p^* E^\vee \otimes q^* E \to O_\Delta ) $$

and

$$ cone( O_\Delta \to p^* D(E) \otimes q^* E )[-1] $$

where

$$ D(E) = \underline{Hom}(E,\omega[\dim])$$

I was confusing global hom with local hom and this messed me up.