Theorem 96.6.1 (Algebraicity of the stack of coherent sheaves; general case). Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\mathcal{C}\! \mathit{oh}_{X/B}$ is an algebraic stack over $S$.

**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 96.5.4 and 96.5.3). 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 96.5.5). By the material in Algebraic Stacks, Section 91.10 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 91.19. 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 67.7.1 and 67.6.9. 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 96.5.5 and Algebraic Stacks, Section 91.19) 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 95.17.1 to conclude that $\mathcal{X}$ is an algebraic stack. Note that $\Lambda $ is a G-ring, see More on Algebra, Proposition 15.49.12. 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 95.14. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 96.5.4, 96.5.6, 96.5.7, 96.5.9. Condition (3) is Lemma 96.5.10. Condition (1) is Lemma 96.5.3.

It remains to show condition (4) which is openness of versality. To see this we will use Artin's Axioms, Lemma 95.19.3. 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 95.19.3. This is Flatness on Spaces, Theorem 74.12.8 and the proof is complete. $\square$

## 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.

## Comments (0)