# Vanishing of pushforwards under Fourier-Mukai

In the book Fourier-Mukai and Nahm transforms in geometry and mathematical physics page 85 corollary 3.5 there is the following claim:

First some notation.

Let $$X$$ be an abelian variety of dimension $$g$$ with dual $$Y$$. We have the poincare bundle giving a FM functor $$S := D^b(X) \to D^b(Y)$$ and using $$P^*$$ we have $$\hat{S}: D^b(Y) \to D^b(X)$$.

The composition is $$[-g]$$.

Then the claim is that if we take some sheaf $$E$$ on $$X$$, and consider the sheaf $$F$$ on $$Y$$ defined as $$H^0(S(E))$$ (this is inner cohomology, not pushforward to a point), then $$F$$ satisfies that $$\hat{S}(F)$$ is a complex whose cohomologies (Again inner) vanish except at the index $$g$$.

It is claimed this follows from the spectral sequence of the composition of $$S,\hat{S}$$ composing to $$[-g]$$, but I don't understand how. We have the spectral sequence $$\hat{S}^i(S^j(E))$$ which converges to $$E$$ when $$i+j=g$$ and $$0$$ otherwise. But there are still many arrows inside touching the column $$S^0(E)$$ so how does the corollary follow?

Attached is a picture