Theorem 99.5.12 (Algebraicity of the stack of coherent sheaves; flat case). Let S be a scheme. Let f : X \to B be a morphism of algebraic spaces over S. Assume that f is of finite presentation, separated, and flat1. Then \mathcal{C}\! \mathit{oh}_{X/B} is an algebraic stack over S.
Proof. Set \mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}. We have seen that \mathcal{X} is a stack in groupoids over (\mathit{Sch}/S)_{fppf} with diagonal representable by algebraic spaces (Lemmas 99.5.4 and 99.5.3). Hence it suffices to find a scheme W and a surjective and smooth morphism W \to \mathcal{X}.
Let B' be a scheme and let B' \to B be a surjective étale morphism. Set X' = B' \times _ B X and denote f' : X' \to B' the projection. Then \mathcal{X}' = \mathcal{C}\! \mathit{oh}_{X'/B'} is equal to the 2-fibre product of \mathcal{X} with the category fibred in sets associated to B' over the category fibred in sets associated to B (Remark 99.5.5). By the material in Algebraic Stacks, Section 94.10 the morphism \mathcal{X}' \to \mathcal{X} is surjective and étale. Hence it suffices to prove the result for \mathcal{X}'. In other words, we may assume B is a scheme.
Assume B is a scheme. In this case we may replace S by B, see Algebraic Stacks, Section 94.19. Thus we may assume S = B.
Assume S = B. Choose an affine open covering S = \bigcup U_ i. Denote \mathcal{X}_ i the restriction of \mathcal{X} to (\mathit{Sch}/U_ i)_{fppf}. If we can find schemes W_ i over U_ i and surjective smooth morphisms W_ i \to \mathcal{X}_ i, then we set W = \coprod W_ i and we obtain a surjective smooth morphism W \to \mathcal{X}. Thus we may assume S = B is affine.
Assume S = B is affine, say S = \mathop{\mathrm{Spec}}(\Lambda ). Write \Lambda = \mathop{\mathrm{colim}}\nolimits \Lambda _ i as a filtered colimit with each \Lambda _ i of finite type over \mathbf{Z}. For some i we can find a morphism of algebraic spaces X_ i \to \mathop{\mathrm{Spec}}(\Lambda _ i) which is of finite presentation, separated, and flat and whose base change to \Lambda is X. See Limits of Spaces, Lemmas 70.7.1, 70.6.9, and 70.6.12. If we show that \mathcal{C}\! \mathit{oh}_{X_ i/\mathop{\mathrm{Spec}}(\Lambda _ i)} is an algebraic stack, then it follows by base change (Remark 99.5.5 and Algebraic Stacks, Section 94.19) that \mathcal{X} is an algebraic stack. Thus we may assume that \Lambda is a finite type \mathbf{Z}-algebra.
Assume S = B = \mathop{\mathrm{Spec}}(\Lambda ) is affine of finite type over \mathbf{Z}. In this case we will verify conditions (1), (2), (3), (4), and (5) of Artin's Axioms, Lemma 98.17.1 to conclude that \mathcal{X} is an algebraic stack. Note that \Lambda is a G-ring, see More on Algebra, Proposition 15.50.12. Hence all local rings of S are G-rings. Thus (5) holds. By Lemma 99.5.11 we have that \mathcal{X} satisfies openness of versality, hence (4) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin's Axioms, Section 98.14. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 99.5.4, 99.5.6, 99.5.7, 99.5.9. Condition (3) follows from Lemma 99.5.10. Finally, condition (1) is Lemma 99.5.3. This finishes the proof of the theorem. \square
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