Section 21.28 (0D7N): Formalities on cohomological descent

21.28 Formalities on cohomological descent

In this section we discuss only to what extent a morphism of ringed topoi determines an embedding from the derived category downstairs to the derived category upstairs. Here is a typical result.

Lemma 21.28.1. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Consider the full subcategory $D' \subset D(\mathcal{O}_\mathcal {D})$ consisting of objects $K$ such that

$K \longrightarrow Rf_*Lf^*K$

is an isomorphism. Then $D'$ is a saturated triangulated strictly full subcategory of $D(\mathcal{O}_\mathcal {D})$ and the functor $Lf^* : D' \to D(\mathcal{O}_\mathcal {C})$ is fully faithful.

Proof. See Derived Categories, Definition 13.6.1 for the definition of saturated in this setting. See Derived Categories, Lemma 13.4.16 for a discussion of triangulated subcategories. The canonical map of the lemma is the unit of the adjoint pair of functors $(Lf^*, Rf_*)$ , see Lemma 21.19.1. Having said this the proof that $D'$ is a saturated triangulated subcategory is omitted; it follows formally from the fact that $Lf^*$ and $Rf_*$ are exact functors of triangulated categories. The final part follows formally from fact that $Lf^*$ and $Rf_*$ are adjoint; compare with Categories, Lemma 4.24.4. $\square$

Lemma 21.28.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Consider the full subcategory $D' \subset D(\mathcal{O}_\mathcal {C})$ consisting of objects $K$ such that

$Lf^*Rf_*K \longrightarrow K$

is an isomorphism. Then $D'$ is a saturated triangulated strictly full subcategory of $D(\mathcal{O}_\mathcal {C})$ and the functor $Rf_* : D' \to D(\mathcal{O}_\mathcal {D})$ is fully faithful.

Proof. See Derived Categories, Definition 13.6.1 for the definition of saturated in this setting. See Derived Categories, Lemma 13.4.16 for a discussion of triangulated subcategories. The canonical map of the lemma is the counit of the adjoint pair of functors $(Lf^*, Rf_*)$ , see Lemma 21.19.1. Having said this the proof that $D'$ is a saturated triangulated subcategory is omitted; it follows formally from the fact that $Lf^*$ and $Rf_*$ are exact functors of triangulated categories. The final part follows formally from fact that $Lf^*$ and $Rf_*$ are adjoint; compare with Categories, Lemma 4.24.4. $\square$

Lemma 21.28.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $K$ be an object of $D(\mathcal{O}_\mathcal {C})$ . Assume

$f$ is flat,
$K$ is bounded below,
$f^*Rf_*H^ q(K) \to H^ q(K)$ is an isomorphism.

Then $f^*Rf_*K \to K$ is an isomorphism.

Proof. Observe that $f^*Rf_*K \to K$ is an isomorphism if and only if it is an isomorphism on cohomology sheaves $H^ j$ . Observe that $H^ j(f^*Rf_*K) = f^*H^ j(Rf_*K) = f^*H^ j(Rf_*\tau _{\leq j}K) = H^ j(f^*Rf_*\tau _{\leq j}K)$ . Hence we may assume that $K$ is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory $D' \subset D(\mathcal{O}_\mathcal {C})$ of Lemma 21.28.2. Hence $K$ is in it too. $\square$

Lemma 21.28.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $K$ be an object of $D(\mathcal{O}_\mathcal {D})$ . Assume

$f$ is flat,
$K$ is bounded below,
$H^ q(K) \to Rf_*f^*H^ q(K)$ is an isomorphism.

Then $K \to Rf_*f^*K$ is an isomorphism.

Proof. Observe that $K \to Rf_*f^*K$ is an isomorphism if and only if it is an isomorphism on cohomology sheaves $H^ j$ . Observe that $H^ j(Rf_*f^*K) = H^ j(Rf_*\tau _{\leq j}f^*K) = H^ j(Rf_*f^*\tau _{\leq j}K)$ . Hence we may assume that $K$ is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory $D' \subset D(\mathcal{O}_\mathcal {D})$ of Lemma 21.28.1. Hence $K$ is in it too. $\square$

Lemma 21.28.5. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume

$f$ is flat,
$f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$ ,
$\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism for $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ .

Then $f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal {A}^+(\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\mathcal {A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')$ .

Proof. By assumptions (2) and (3) and Lemmas 21.28.4 and 21.28.1 we see that $f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal {A}^+(\mathcal{O})$ is fully faithful. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ . Then we can write $\mathcal{F} = f^*\mathcal{F}'$ . Then $Rf_*\mathcal{F} = Rf_* f^*\mathcal{F}' = \mathcal{F}'$ . In particular, we have $R^ pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ . Thus for any $K \in D^+_\mathcal {A}(\mathcal{O})$ we see, using the spectral sequence $E_2^{p, q} = R^ pf_*H^ q(K)$ converging to $R^{p + q}f_*K$ , that $Rf_*K$ is in $D^+_{\mathcal{A}'}(\mathcal{O}')$ . Of course, it also follows from Lemmas 21.28.3 and 21.28.2 that $Rf_* : D_\mathcal {A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')$ is fully faithful. Since $f^*$ and $Rf_*$ are adjoint we then get the result of the lemma, for example by Categories, Lemma 4.24.4. $\square$

Lemma 21.28.6.reference Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume

$f$ is flat,
$f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$ ,
$\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism for $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ ,
$\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,
$\mathcal{C}', \mathcal{O}', \mathcal{A}'$ satisfy the assumption of Situation 21.25.1.

Then $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal {A}(\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\mathcal {A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$ .

Proof. Since $f^*$ is exact, it is clear that $f^*$ defines a functor $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal {A}(\mathcal{O})$ as in the statement of the lemma and that moreover this functor commutes with the truncation functors $\tau _{\geq -n}$ . We already know that $f^*$ and $Rf_*$ are quasi-inverse equivalence on the corresponding bounded below categories, see Lemma 21.28.5. By Lemma 21.25.4 with $N = 0$ we see that $Rf_*$ indeed defines a functor $Rf_* : D_\mathcal {A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$ and that moreover this functor commutes with the truncation functors $\tau _{\geq -n}$ . Thus for $K$ in $D_\mathcal {A}(\mathcal{O})$ the map $f^*Rf_*K \to K$ is an isomorphism as this is true on trunctions. Similarly, for $K'$ in $D_{\mathcal{A}'}(\mathcal{O}')$ the map $K' \to Rf_*f^*K'$ is an isomorphism as this is true on trunctions. This finishes the proof. $\square$

Lemma 21.28.7.reference Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ be a morphism of ringed sites. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume

$f$ is flat,
$f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$ ,
$\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism for $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ ,
$\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,
$f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ and $\mathcal{A}$ satisfy the assumption of Situation 21.25.5.

Proof. The proof of this lemma is exactly the same as the proof of Lemma 21.28.6 except the reference to Lemma 21.25.4 is replaced by a reference to Lemma 21.25.6. $\square$

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21.28 Formalities on cohomological descent

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