The Stacks project

Lemma 98.5.6. In Situation 98.5.1 assume that $B \to S$ is locally of finite presentation. Then $p : \mathcal{C}\! \mathit{oh}_{X/B} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition 97.11.1).

Proof. Write $B(T)$ for the discrete category whose objects are the $S$-morphisms $T \to B$. Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a filtered limit of affine schemes over $S$. Assigning to an object $(T, h, \mathcal{F})$ of $\mathcal{C}\! \mathit{oh}_{X/B, T}$ the object $h$ of $B(T)$ gives us a commutative diagram of fibre categories

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{C}\! \mathit{oh}_{X/B, T_ i} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}_{X/B, T} \ar[d] \\ \mathop{\mathrm{colim}}\nolimits B(T_ i) \ar[r] & B(T) } \]

We have to show the top horizontal arrow is an equivalence. Since we have assumed that $B$ is locally of finite presentation over $S$ we see from Limits of Spaces, Remark 69.3.11 that the bottom horizontal arrow is an equivalence. This means that we may assume $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a filtered limit of affine schemes over $B$. Denote $g_ i : T_ i \to B$ and $g : T \to B$ the corresponding morphisms. Set $X_ i = T_ i \times _{g_ i, B} X$ and $X_ T = T \times _{g, B} X$. Observe that $X_ T = \mathop{\mathrm{colim}}\nolimits X_ i$ and that the algebraic spaces $X_ i$ and $X_ T$ are quasi-separated and quasi-compact (as they are of finite presentation over the affines $T_ i$ and $T$). By Limits of Spaces, Lemma 69.7.2 we see that

\[ \mathop{\mathrm{colim}}\nolimits \textit{FP}(X_ i) = \textit{FP}(X_ T). \]

where $\textit{FP}(W)$ is short hand for the category of finitely presented $\mathcal{O}_ W$-modules. The results of Limits of Spaces, Lemmas 69.6.12 and 69.12.3 tell us the same thing is true if we replace $\textit{FP}(X_ i)$ and $\textit{FP}(X_ T)$ by the full subcategory of objects flat over $T_ i$ and $T$ with scheme theoretic support proper over $T_ i$ and $T$. This proves the lemma. $\square$

Comments (2)

Comment #2735 by Emanuel Reinecke on

Typo in the fifth sentence: assumed

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