The Stacks project

Proposition 103.8.4. Let $\mathcal{X}$ be an algebraic stack. Then $\mathit{QC}(\mathcal{X})$ is canonically equivalent to $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. By Sheaves on Stacks, Lemma 95.26.6 pullback by the comparison morphism $\epsilon : \mathcal{X}_{affine, fppf} \to \mathcal{X}_{affine}$ identifies $\mathit{QC}(\mathcal{X})$ with a full subcategory $Q_\mathcal {X} \subset D(\mathcal{X}_{affine, fppf}, \mathcal{O})$. Using the equivalence of ringed topoi in Sheaves on Stacks, Equation (95.24.3.1) we may and do view $Q_\mathcal {X}$ as a full subcategory of $D(\mathcal{X}_{fppf}, \mathcal{O})$.

Similarly by Lemma 103.5.4 and Remark 103.5.5 we find that $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ may be viewed as the left orthogonal $\mathcal{A}$ of the left admissible subcategory $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$.

To finish we will show that $Q_\mathcal {X}$ is equal to $\mathcal{A}$ as subcategories of $D(\mathcal{X}_{fppf}, \mathcal{O})$.

Step 1: $Q_\mathcal {X}$ is contained in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. An object $K$ of $Q_\mathcal {X}$ is characterized by the property that $K$, viewed as an object of $D(\mathcal{X}_{affine, fppf}, \mathcal{O})$ satisfies $R\epsilon _*K$ is an object of $\mathit{QC}(\mathcal{X}_{affine}, \mathcal{O})$. This in turn means exactly that for all morphisms $x \to x'$ of $\mathcal{X}_{affine}$ the map

\[ R\Gamma (x', K) \otimes _{\mathcal{O}(x')}^\mathbf {L} \mathcal{O}(x) \longrightarrow R\Gamma (x, K) \]

is an isomorphism, see footnote in statement of Cohomology on Sites, Lemma 21.43.12. Now, if $x' \to x$ lies over a flat morphism of affine schemes, then this means that

\[ H^ i(x', K) \otimes _{\mathcal{O}(x')} \mathcal{O}(x) \cong H^ i(x, K) \]

This clearly means that $H^ i(K)$ is a sheaf for the ├ętale topology (Sheaves on Stacks, Lemma 95.25.1) and that it has the flat base change property (small detail omitted).

Step 2: $Q_\mathcal {X}$ is contained in $\mathcal{A}$. To see this it suffices to show that for $K$ in $Q_\mathcal {X}$ we have $\mathop{\mathrm{Hom}}\nolimits (K, P) = 0$ for all $P$ in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. Consider the object

\[ H = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(K, P) \]

Let $x$ be an object of $\mathcal{X}$ which lies over an affine scheme $U = p(x)$. By Cohomology on Sites, Lemma 21.35.1 we have the first equality in

\[ R\Gamma (x, H) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {X}}(K|_{\mathcal{X}/x}, P|_{\mathcal{X}/x}) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}}(K|_{\mathcal{X}_{affine}/x}, P|_{\mathcal{X}_{affine}/x}) \]

The second equality stems from the fact that the topos of the site $\mathcal{X}/x$ is equivalent to the topos of the site $\mathcal{X}_{affine}/x$, see Sheaves on Stacks, Equation (95.24.3.1). We may write $K = \epsilon ^*N$ for some $N$ in $\mathit{QC}(\mathcal{O})$. Then by Cohomology on Sites, Lemma 21.43.13 we see that

\[ R\Gamma (x, H) = R\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}(x))}(R\Gamma (x, N), R\Gamma (x, P)) \]

By Lemma 103.8.1 we see that $R\Gamma (x, P) = 0$ if $U \to \mathcal{X}$ is flat and hence $R\Gamma (x, H) = 0$ under the same hypothesis. By Lemma 103.8.2 we conclude that $R\Gamma (\mathcal{X}, H) = 0$ and therefore $\mathop{\mathrm{Hom}}\nolimits (K, P) = 0$.

Step 3: $\mathcal{A}$ is contained in $Q_\mathcal {X}$. Let $K$ be an object of $\mathcal{A}$ and let $x \to x'$ be a morphism of $\mathcal{X}_{affine}$. We have to show that

\[ R\Gamma (x', K) \otimes _{\mathcal{O}(x')}^\mathbf {L} \mathcal{O}(x) \longrightarrow R\Gamma (x, K) \]

is a quasi-isomorphism, see footnote in statement of Cohomology on Sites, Lemma 21.43.12. By the proof of Lemma 103.5.4 and the discussion in Remark 103.5.5 we see that $\mathcal{A}$ is the image of the restriction of $Lg_!$ to $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Thus we may assume $K = Lg_!M$ for some $M$ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Then the desired equality follow from Lemma 103.8.3. $\square$


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