TITLE: Fourier-Mukai transform - a first example
QUESTION [7 upvotes]: Suppose $X$ and $Y$ are schemes of finite type over a field, and let $f: X\rightarrow Y$ be a morphism. Let $\Gamma_f$ be the closed subscheme of $X\times Y$, then the first example of the Fourier-Mukai transform says that
$$f_*()=p_Y{_*}(p_X^*()\bigotimes_{\mathcal{O}_{X\times Y}}\mathcal{O}_{\Gamma_f}),$$ similarly there is an expression for $f^*$. Has anyone checked this before? I am having some difficulties (at least on some commutative algebra) in verifying them. Does anyone know why those formulas are true?

REPLY [3 votes]: You should use the fact that $p_X^*(-)\otimes_{\mathcal{O}_{X\times Y}}\mathcal{O}_{\Gamma_f}\cong i_*(p_X|_{\Gamma_f})^*(-)$ where $i:\Gamma_f\to X\times Y$ is the inclusion. Then the statment becomes much easier.