Let $V, W$ be algebraic varieties and $F: V \rightarrow W$ be an affine morphism: my definition of affine is that there exists an open cover $\{ U_i \}_i$ of $W$ with each $U_i$ (isomorphic to) an affine variety, such that $F^{-1}(U_i)$ is (isomorphic to) an affine variety for any $i$.
I'm trying to prove that for any $U \subseteq W$ open and affine, $F^{-1}(U)$ is affine. The first step in the proof I'm following is to assume that $W$ itself is affine. I don't understand why we can do this.
I know something about how the preimages of affine subvarieties look through morphisms of affine varieties, but have no clue about such preimages through morphisms of algebraic varieties. So what guarantees that $F^{-1}(U)$ is an algebraic variety and that $F|_{F^{-1}(U)} : F^{-1}(U) \rightarrow U$ is still an affine morphism of algebraic varieties? Or is this even needed?
Question: "So what guarantees that F−1(U) is an algebraic variety and that F|F−1(U):F−1(U)→U is still an affine morphism of algebraic varieties? Or is this even needed?"
Answer: Let "algebraic variety" be the one defined in Hartshorne, Chapter I: This means $k$ is an algebraically closed field and $V,W$ are quasi projective varieties of finite type over $k$ (in the sense of HH.I.3). If $F: V \rightarrow W$ is a morphism of algebraic varieties and if $U\subseteq W$ is an open subset, it follows $F^{-1}(U) \subseteq V$ is an open subset, and any open subset of $V$ has canonically the structure of an algebraic variety. Hence $F^{-1}(U)$ is canonically an algebraic variety. I believe you will need a result in HH.II.2 - Exercise 2.17 which is a criterion for a "scheme" to be affine to conclude.
You must formulate and prove a similar result for algebraic varieties in the sense of Chapter I. Let $(X, \mathcal{O}_X)$ be an algebraic variety in the sense of HH.I
You must prove that $X$ is an affine algebraic variety iff there are elements
$$f_1,..,f_l \in H^0(X, \mathcal{O}_X):=A$$
such that the open subsets $X_{f_i}$ are affine algebraic varieties and the ideal $I:=(f_1,..,f_l)\subseteq A$ generate the unit ideal,