The Stacks project

98.6 The stack of coherent sheaves in the non-flat case

In Theorem 98.5.12 the assumption that $f : X \to B$ is flat is not necessary. In this section we give a different proof which avoids the flatness assumption and avoids checking openness of versality by using the results in Flatness on Spaces, Section 76.12 and Artin's Axioms, Section 97.20.

For a different approach to this problem the reader may wish to consult [ArtinI] and follow the method discussed in the papers [olsson-starr], [lieblich_remarks], [olsson_proper], [Hall-Rydh], [Hall-Rydh-Hilbert], [rydh_representability]. Some of these papers deal with the more general case of the stack of coherent sheaves on an algebraic stack over an algebraic stack and others deal with similar problems in the case of Hilbert stacks or Quot functors. Our strategy will be to show algebraicity of some cases of Hilbert stacks and Quot functors as a consequence of the algebraicity of the stack of coherent sheaves.

Proof. Only the last step of the proof is different from the proof in the flat case, but we repeat all the arguments here to make sure everything works.

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 98.5.4 and 98.5.3). 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 ├ętale 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 98.5.5). By the material in Algebraic Stacks, Section 93.10 the morphism $\mathcal{X}' \to \mathcal{X}$ is surjective and ├ętale. 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 93.19. 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 separated and of finite presentation and whose base change to $\Lambda $ is $X$. See Limits of Spaces, Lemmas 69.7.1 and 69.6.9. 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 98.5.5 and Algebraic Stacks, Section 93.19) 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 97.17.1 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. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin's Axioms, Section 97.14. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 98.5.4, 98.5.6, 98.5.7, 98.5.9. Condition (3) is Lemma 98.5.10. Condition (1) is Lemma 98.5.3.

It remains to show condition (4) which is openness of versality. To see this we will use Artin's Axioms, Lemma 97.20.3. We have already seen that $\mathcal{X}$ has diagonal representable by algebraic spaces, has (RS*), and is limit preserving (see lemmas used above). Hence we only need to see that $\mathcal{X}$ satisfies the strong formal effectiveness formulated in Artin's Axioms, Lemma 97.20.3. This is Flatness on Spaces, Theorem 76.12.8 and the proof is complete. $\square$

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