The Stacks project

Lemma 101.4.3. Let $\mathcal{X}$ be an algebraic stack.

  1. Let $\mathcal{F}^\bullet $ be an object of $D_{\mathcal{M}_\mathcal {X}}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. With $g$ as in Cohomology of Stacks, Lemma 100.11.2 for the lisse-étale site we have

    1. $g^{-1}\mathcal{F}^\bullet $ is in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$,

    2. $g^{-1}\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet $ is in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$,

    3. $Lg_!\mathcal{H}^\bullet $ is in $D_{\mathcal{M}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ for $\mathcal{H}^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$, and

    4. the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors

      \[ \xymatrix{ D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \ar@<1ex>[l]^-{Lg_!} } \]
  2. Let $\mathcal{F}^\bullet $ be an object of $D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$. With $g$ as in Cohomology of Stacks, Lemma 100.11.2 for the flat-fppf site we have

    1. $g^{-1}\mathcal{F}^\bullet $ is in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$,

    2. $g^{-1}\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet $ is in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$,

    3. $Lg_!\mathcal{H}^\bullet $ is in $D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$ for $\mathcal{H}^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$, and

    4. the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors

      \[ \xymatrix{ D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{Lg_!} } \]

Proof. The functor $g^{-1}$ is exact, hence (1)(a), (2)(a), (1)(b), and (2)(b) follow from Cohomology of Stacks, Lemmas 100.12.3 and 100.11.5.

Proof of (1)(c) and (2)(c). The construction of $Lg_!$ in Lemma 101.3.1 (via Cohomology on Sites, Lemma 21.36.2 which in turn uses Derived Categories, Proposition 13.29.2) shows that $Lg_!$ on any object $\mathcal{H}^\bullet $ of $D(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ is computed as

\[ Lg_!\mathcal{H}^\bullet = \mathop{\mathrm{colim}}\nolimits g_!\mathcal{K}_ n^\bullet = g_! \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \]

(termwise colimits) where the quasi-isomorphism $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{H}^\bullet $ induces quasi-isomorphisms $\mathcal{K}_ n^\bullet \to \tau _{\leq n} \mathcal{H}^\bullet $. Since $\mathcal{M}_\mathcal {X} \subset \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ (resp. $\mathcal{M}_\mathcal {X} \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$) is preserved under colimits we see that it suffices to prove (c) on bounded above complexes $\mathcal{H}^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ (resp. $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$). In this case to show that $H^ n(Lg_!\mathcal{H}^\bullet )$ is in $\mathcal{M}_\mathcal {X}$ we can argue by induction on the integer $m$ such that $\mathcal{H}^ i = 0$ for $i > m$. If $m < n$, then $H^ n(Lg_!\mathcal{H}^\bullet ) = 0$ and the result holds. In general consider the distinguished triangle

\[ \tau _{\leq m - 1}\mathcal{H}^\bullet \to \mathcal{H}^\bullet \to H^ m(\mathcal{H}^\bullet )[-m] \to \ldots \]

(Derived Categories, Remark 13.12.4) and apply the functor $Lg_!$. Since $\mathcal{M}_\mathcal {X}$ is a weak Serre subcategory of the module category it suffices to prove (c) for two out of three. We have the result for $Lg_!\tau _{\leq m - 1}\mathcal{H}^\bullet $ by induction and we have the result for $Lg_!H^ m(\mathcal{H}^\bullet )[-m]$ by Lemma 101.3.3. Whence (c) holds.

Let us prove (2)(d). By (2)(a) and (2)(b) the functor $g^{-1} = g^*$ induces a functor

\[ c : D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \]

see Derived Categories, Lemma 13.6.8. Thus we have the following diagram of triangulated categories

\[ \xymatrix{ D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X}) \ar[rd]^{g^{-1}} \ar[rr]_ q & & D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar[ld]^ c \\ & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \ar@<1ex>[lu]^{Lg_!} } \]

where $q$ is the quotient functor, the inner triangle is commutative, and $g^{-1}Lg_! = \text{id}$. For any object of $E$ of $D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$ the map $a : Lg_!g^{-1}E \to E$ maps to a quasi-isomorphism in $D(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Hence the cone on $a$ maps to zero under $g^{-1}$ and by (2)(b) we see that $q(a)$ is an isomorphism. Thus $q \circ Lg_!$ is a quasi-inverse to $c$.

In the case of the lisse-étale site exactly the same argument as above proves that

\[ D_{\mathcal{M}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) / D_{\mathcal{P}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \]

is equivalent to $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$. Applying the last equivalence of Lemma 101.4.2 finishes the proof. $\square$


Comments (2)

Comment #1148 by Olaf Schnürer on

It seems in (1) should be in so that is defined. Similarly in (1)(b) should write , and in (1)(d) one may replace the left hand side by the equivalent quotient category from the previous tag.

In (2)(a) the index should be "flat, fppf".

Please note that my last name contains the letter h.

Comment #1169 by on

Thanks very much. I made the changes you suggested but I've left statement (1)(d) alone as (in the proof you can see the replacement using the previous lemma). Also fixed spelling of your name. See here.


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