The Stacks project

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. $\square$

[1] This assumption is not necessary. See Section 99.6.

Comments (4)

Comment #4571 by Ariyan Javanpeykar on

Should "algebraicity of stack coherent sheaves" be "algebraicity of the stack of coherent sheaves"?


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08WC. Beware of the difference between the letter 'O' and the digit '0'.