Suppose $S$ a noetherian base scheme, $X \to S$ is projective and $F, G$ are coherent $\mathcal O\_X$-modules. Then by EGA (7.7.8) and (7.7.9) there exists a scheme $H = \underline{\operatorname{Hom}}\_X(F, G)$, affine over $S$, which represents the functor
$$(f: T \to S) \mapsto \operatorname{Hom}\_{X\_T}(f^\*F, f^\*G).$$
On $X\_H$ there is a universal homomorphism $\varphi: F\_H \to G\_H$. Considering the exact sequence
$$0 \to \ker \varphi \to F\_H \xrightarrow{\varphi} G\_H \to \operatorname{coker} \varphi \to 0,$$
one can check that the open set $$\underline{\operatorname{Isom}}\_X(F, G):= H \setminus (\operatorname{supp}(\ker \varphi) \cup \operatorname{supp}(\operatorname{coker} \varphi)) \subset H$$ represents the functor
$$(f: T \to S) \mapsto \operatorname{Isom}\_{X\_T}(f^\*F, f^\*G).$$
In [1, Section 2.1], Jason Starr and Johan de Jong write that $\underline{\operatorname{Isom}}\_X(F, G)$ is affine over the base $S$. Why is that true? In general, opens of affine schemes are not affine.


[1] Jason Starr, Johan de Jong; *Almost proper GIT stacks and Discriminant Avoidance*



Why is the scheme of isomorphisms of sheaves affine over the base?
So I realized Jason Starr and Johan de Jong only claim that $H = \underline{\operatorname{Hom}}\_S(F, G)$ is affine if $F$ and $G$ are locally free. In that case, if $U = \operatorname{Spec}(A) \subset H$ is such that $F$ and $G$ are free of rank $n$ on $U$, we get
$$H\_U = Gl\_n(A) = \operatorname{Spec}A[X\_{ij}|i,j = 1, \dotsc, n][\frac{1}{\det}],$$
so $H$ is affine over $S$.


