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

1. $f$ is flat,

2. $K$ is bounded below,

3. $f^*Rf_*H^ q(K) \to H^ q(K)$ 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

1. $f$ is flat,

2. $K$ is bounded below,

3. $H^ q(K) \to Rf_*f^*H^ q(K)$ 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

1. $f$ is flat,

2. $f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$,

3. $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ 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. 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

1. $f$ is flat,

2. $f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$,

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

4. $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,

5. $\mathcal{C}', \mathcal{O}', \mathcal{A}'$ 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. 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

1. $f$ is flat,

2. $f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$,

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

4. $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,

5. $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ and $\mathcal{A}$ satisfy the assumption of Situation 21.25.5.

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. 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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