Let \(f:X\rightarrow Y\) be an affine morphism and \(\mathcal{F}\) is a quasi coherent sheaf of \(\mathcal{O}_X\) modules. Then, \(f_*\mathcal{F}\) is a quasi coherent sheaf of \(\mathcal{O}_Y\) modules.
We have the following result :
Let \(X\) be a scheme. Then, an \(\mathcal{O}_X\) module \(\mathcal{F}\) is quasi coherent iff for every open affine subset \(U=\text{Spec}(A)\) of \(X\), there is an \(A\) module \(M\) such that \(\mathcal{F}|_U=\tilde{M}\).
Let \(U\subseteq Y\) be an open affine subset say \(U=\text{Spec}(A)\). As \(f\) is affine, \(f^{-1}(U)\) is affine open, say \(f^{-1}(U)=\text{Spec}(B)\subseteq X\).
As \(\mathcal{F}\) is quasi coherent sheaf of \(\mathcal{O}_X\) modules and \(f^{-1}(U)=\text{Spec}(B)\) is open affine subset of \(X\), there exists a \(B\) module \(M\) such that \(\mathcal{F}|_{f^{-1}(U)}\cong \widetilde{M}\). As \(f^{-1}(U)=\text{Spec}(B)\) we have \(f:\text{Spec}(B)\rightarrow \text{Spec}(A)\) which also gives a ring morphsim \(A\rightarrow B\). The isomorphism \(\mathcal{F}|_{f^{-1}(U)}\cong \widetilde{M}\) implies
\(f_*(\mathcal{F}|_{f^{-1}(U)})\cong f_*\widetilde{M}\).
As \(f_*(\mathcal{F}|_{f^{-1}(U)})\cong f_*\mathcal{F}|_U\) we have\(f_*\mathcal{F}|_U\cong f_*\widetilde{M}\).
From previous observation we see that \(f_*\widetilde{M}\cong (_A M)\) where \(_A M\) is \(M\) considered as a \(A\) module under ring morphism \(A\rightarrow B\) defined above. So, we have\(f_*\mathcal{F}|_U\cong \widetilde{_A M}\).
So, given an open affine \(U=\text{Spec}(A)\subseteq X\) there exists an \(A\) module \(_A M\) such that\((f_*\mathcal{F})|_U\cong \widetilde{_A M}\).
Thus, \(f_*\mathcal{F}\) is a quasi coherent sheaf of \(\mathcal{O}_Y\) modules.