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The Stacks project

Lemma 99.5.6. In Situation 99.5.1 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 98.11.1).

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 70.3.11 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 70.7.2 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 70.6.12 and 70.12.3 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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