The Stacks project

Lemma 37.65.2. Let $Z \to S$ and $X \to S$ be morphisms of schemes. If $Z \to S$ is finite locally free and $X \to S$ is affine, then $\mathit{Mor}_ S(Z, X)$ is representable by a scheme affine over $S$.

Proof. Choose an affine open covering $S = \bigcup U_ i$ such that $\Gamma (Z \times _ S U_ i, \mathcal{O}_{Z \times _ S U_ i})$ is finite free over $\mathcal{O}_ S(U_ i)$. Let $F_ i \subset \mathit{Mor}_ S(Z, X)$ be the subfunctor which assigns to $T/S$ the empty set if $T \to S$ does not factor through $U_ i$ and $\mathit{Mor}_ S(Z, X)(T)$ otherwise. Then the collection of these subfunctors satisfy the conditions (2)(a), (2)(b), (2)(c) of Schemes, Lemma 26.15.4 which proves the lemma. Condition (2)(a) follows from Lemma 37.65.1 and the other two follow from straightforward arguments. $\square$


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