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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

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    $\begingroup$ 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). $\endgroup$
    – Tommaso Scognamiglio
    Commented Nov 4, 2021 at 13:29
  • $\begingroup$ @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 $Gl_n \times X$. $\endgroup$
    – red_trumpet
    Commented Nov 4, 2021 at 15:24
  • $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented Nov 4, 2021 at 21:03
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    $\begingroup$ 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. $\endgroup$
    – Laurent Moret-Bailly
    Commented Nov 5, 2021 at 7:09
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    $\begingroup$ @DavidRoberts $X_H = X \times_S H$. Similarly, $F_H$ and $G_H$ are the pull-backs of $F$ and $G$ to $X_H$. $\endgroup$
    – red_trumpet
    Commented Nov 5, 2021 at 9:02

2 Answers 2

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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 $S$.

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This result is true quite generally. One reference is:

https://stacks.math.columbia.edu/tag/08K9

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    $\begingroup$ @LaurentMoret-Bailly's example shows that it is not true without some hypothesis. $\endgroup$
    – LSpice
    Commented Dec 7, 2021 at 21:19

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