I'm reading the book about moduli spaces by Huybrechts and Lehn, and i'm stuck understanding a proof, it is Theorem 6.1.8.:


Given a K3-surface $X$ and a 2-dimensional space $M$, coherent and torsion free sheafs $F$ on $X$ and $E$ on $M\times X$. We have projections $p,q$ from $M\times X$ to $M$ and $X$ resp.


They claim the class $a:=ch(p\_{\\*}\mathcal{H}om(q^{\\*}F,E))$ as an element in $H^{\\*}(M,\mathbb{Q})$, where $p\_{\\*}\mathcal{H}om(q^{\\*}F,E)=\sum\limits\_{i=0}^2 (-1)^i \mathcal{E}xt^i\_p(q^{\\*}F,E)$, only depends on the classes of $ch(q^{\\*}F)$ and $ch(E)$ as elements in $H^{\\*}(M\times X,\mathbb{Q})$, where $\mathcal{E}xt\_p^i(q^{\\*}F,E)=R^i(p\_{\\*}\mathcal{H}om(q^{\\*}F,E))$.


So using Grothendieck-Riemann-Roch as suggested shows:
$ch(\sum\limits\_{i=0}^2 (-1)^i \mathcal{E}xt^i\_p(q^{\\*}F,E))td(M)=p\_{\\*}(ch(\mathcal{H}om(q^{\\*}F,E)td(M\times X))$


Here i am stuck. Why does this show that $a$ only depends on $ch(E)$ and $ch(q^{\\*}F)$. I think one has to show that $ch(\mathcal{H}om(q^{\\*}F,E))$ only depends on this classes, but i can't see why.



Chern character of Hom-sheaves
If you apply ${\mathcal H}om(-,E)$ to a resolution of a sheaf $G$, you obtain a complex, the cohomology of which are ${\mathcal E}xt^i(G,E)$, hence by additivity of the Chern character, the alternating sum of Chern characters of the terms of the complex equals the alternating sum of Chern characters of the Ext sheaves. 


