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 $f_*()$ = $p_Y{_*}$ $(p_X^*()$ $\otimes_{\mathcal{O}{X\times Y}}$ $\mathcal{O}{\Gamma_f})$; $f_*()=p_Y{_*}(p_X^*()\bigotimes_{\mathcal{O}_{X\times Y}}\mathcal{O}_{\Gamma_f}),$$ similarly there is an expression for f^*. Has anyone checked this before? I am having some difficulties (at least on some commutative algebra) in verifying them. Does anyone know why those formulas are true? 3 added 251 characters in body Suppose X and Y are schemes of finite type over a field, and let f: X\rightarrow Y be a morphism. Let \Gamma_f be the closed subscheme of X\times Y, then the first example of the Fourier-Mukai transform says that f_*()=p_Y{_*}(p_X^*()\otimes_{\mathcal{O}_{X\times f_*() = p_Y{_*} (p_X^*() \otimes_{\mathcal{O}{X\times Y}} \mathcal{O}{\Gamma_f}); similarly there is an expression for f^*. Has anyone checked this before? I am having some difficulties (at least on some commutative algebra) in verifying them. Does anyone know why those formulas are true? 2 deleted 340 characters in body; added 58 characters in body Suppose X and Y are schemes of finite type over a field, and let f: X\rightarrow Y be a morphism. Let \Gamma_f be the closed subscheme of X\times Y, then the first example of the Fourier-Mukai transform says that f_*()=p_Y{_*}(p_X^*()\otimes_{\mathcal{O}{X\times f_*()=p_Y{_*}(p_X^*()\otimes_{\mathcal{O}_{X\times Y}}$$f*()=p_Y{_*}(p_X^*()\otimes_{\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 (on the commutative algebra at least) in verifying them. Does anyone know why those formulas are true?