# Why is the scheme of isomorphisms of sheaves affine over the base?

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

• I actually don't know how to prove this but I would expect this to be linked somehow to the fact that $Gl_n$ is an affine group scheme embedded inside $Mat_n$. (Sorry for the vague answer I'll try to see if I can write down some true stuff). Nov 4, 2021 at 13:29
• @TommasoScognamiglio Discussing a bit with my professor, I realized that actually Starr and de Jong only claim the affine-ness in the case where $F$ and $G$ are locally free, and then I think this is exactly what you mean: Locally it should be $Gl_n \times X$. Nov 4, 2021 at 15:24
• I believe that EGA IV Section 8.8 only works if $F$ is flat. So if you want the Hom in each direction to be representable I believe you need to assume that both $F$ and $G$ are locally free. Nov 4, 2021 at 21:03
• The EGA reference, I think, is in EGA III-2. It is about representing Hom sheaves of coherent sheaves by a sheaf, not a scheme. And indeed there is a flatness assumption on $\mathscr{F}$. In any case, taking $X=S$, $\mathscr{F}=\mathscr{O}_X$, and $\mathscr{G}=$ a skyscraper sheaf at a closed point, I am sure there is no scheme $H$ as in the question. Nov 5, 2021 at 7:09
• @DavidRoberts $X_H = X \times_S H$. Similarly, $F_H$ and $G_H$ are the pull-backs of $F$ and $G$ to $X_H$. Nov 5, 2021 at 9:02

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 $$S$$.