The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

21.30 Comparing cohomology

We develop some general theory which will help us compare cohomology in different topologies. Given $\mathcal{C}$, $\tau $, and $\tau '$ as in Section 21.27 and a morphism $f : X \to Y$ in $\mathcal{C}$ we obtain a commutative diagram of morphisms of topoi

21.30.0.1
\begin{equation} \label{sites-cohomology-equation-commutative-epsilon} \vcenter { \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_\tau /X) \ar[r]_{f_\tau } \ar[d]_{\epsilon _ X} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_\tau /Y) \ar[d]^{\epsilon _ Y} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\tau '}/X) \ar[r]^{f_{\tau '}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\tau '}/X) } } \end{equation}

Here the morphism $\epsilon _ X$, resp. $\epsilon _ Y$ is the comparison morphism of Section 21.27 for the category $\mathcal{C}/X$ endowed with the two topologies $\tau $ and $\tau '$. The morphisms $f_\tau $ and $f_{\tau '}$ are “relocalization” morphisms (Sites, Lemma 7.25.8). The commutativity of the diagram is a special case of Sites, Lemma 7.28.1 (applied with $\mathcal{C} = \mathcal{C}_\tau /Y$, $\mathcal{D} = \mathcal{C}_{\tau '}/Y$, $u = \text{id}$, $U = X$, and $V = X$). We also get $\epsilon _{X, *} \circ f_\tau ^{-1} = f_{\tau '}^{-1} \circ \epsilon _{Y, *}$ either from the lemma or because it is obvious.

Situation 21.30.1. With $\mathcal{C}$, $\tau $, and $\tau '$ as in Section 21.27. Assume we are given a subset $\mathcal{P} \subset \text{Arrows}(\mathcal{C})$ and for every object $X$ of $\mathcal{C}$ we are given a weak Serre subcategory $\mathcal{A}'_ X \subset \textit{Ab}(\mathcal{C}_{\tau '}/X)$. We make the following assumption:

  1. given $f : X \to Y$ in $\mathcal{P}$ and $Y' \to Y$ general, then $X \times _ Y Y'$ exists and $X \times _ Y Y' \to Y'$ is in $\mathcal{P}$,

  2. $f_{\tau '}^{-1}$ sends $\mathcal{A}'_ Y$ into $\mathcal{A}'_ X$ for any morphism $f : X \to Y$ of $\mathcal{C}$,

  3. given $X$ in $\mathcal{C}$ and $\mathcal{F}'$ in $\mathcal{A}'_ X$, then $\mathcal{F}'$ satisfies the sheaf condition for $\tau $-coverings, i.e., $\mathcal{F}' = \epsilon _{X, *}\epsilon _ X^{-1}\mathcal{F}'$,

  4. if $f : X \to Y$ in $\mathcal{P}$ and $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}'_ X)$, then $R^ if_{\tau ', *}\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}'_ Y)$ for $i \geq 0$.

  5. if $\{ U_ i \to U\} _{i \in I}$ is a $\tau $-covering, then there exist

    1. a $\tau '$-covering $\{ V_ j \to U\} _{j \in J}$,

    2. a $\tau $-covering $\{ f_ j : W_ j \to V_ j\} $ consisting of a single $f_ j \in \mathcal{P}$, and

    3. a $\tau '$-covering $\{ W_{jk} \to W_ j\} _{k \in K_ j}$

    such that $\{ W_{jk} \to U\} _{j \in J, k \in K_ j}$ is a refinement of $\{ U_ i \to U\} _{i \in I}$.

Lemma 21.30.2. In Situation 21.30.1 for $X$ in $\mathcal{C}$ denote $\mathcal{A}_ X$ the objects of $\textit{Ab}(\mathcal{C}_\tau /X)$ of the form $\epsilon _ X^{-1}\mathcal{F}'$ with $\mathcal{F}'$ in $\mathcal{A}'_ X$. Then

  1. for $\mathcal{F}$ in $\textit{Ab}(\mathcal{C}_\tau /X)$ we have $\mathcal{F} \in \mathcal{A}_ X \Leftrightarrow \epsilon _{X, *}\mathcal{F} \in \mathcal{A}'_ X$, and

  2. $f_\tau ^{-1}$ sends $\mathcal{A}_ Y$ into $\mathcal{A}_ X$ for any morphism $f : X \to Y$ of $\mathcal{C}$.

Proof. Part (1) follows from (3) and part (2) follows from (2) and the commutativity of (21.30.0.1) which gives $\epsilon _ X^{-1} \circ f_{\tau '}^{-1} = f_\tau ^{-1} \circ \epsilon _ Y^{-1}$. $\square$

Our next goal is to prove Lemmas 21.30.10 and 21.30.9. We will do this by an induction argument using the following induction hypothesis.

$(V_ n)$ For $X$ in $\mathcal{C}$ and $\mathcal{F}$ in $\mathcal{A}_ X$ we have $R^ i\epsilon _{X, *}\mathcal{F} = 0$ for $1 \leq i \leq n$.

Lemma 21.30.3. In Situation 21.30.1 assume $(V_ n)$ holds. For $f : X \to Y$ in $\mathcal{P}$ and $\mathcal{F}$ in $\mathcal{A}_ X$ we have $R^ if_{\tau ', *}\epsilon _{X, *}\mathcal{F} = \epsilon _{Y, *}R^ if_{\tau , *}\mathcal{F}$ for $i \leq n$.

Proof. We will use the commutative diagram (21.30.0.1) without further mention. In particular have

\[ Rf_{\tau ', *}R\epsilon _{X, *}\mathcal{F} = R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F} \]

Assumption $(V_ n)$ tells us that $\epsilon _{X, *}\mathcal{F} \to R\epsilon _{X, *}\mathcal{F}$ is an isomorphism in degrees $\leq n$. Hence $Rf_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to Rf_{\tau ', *}R\epsilon _{X, *}\mathcal{F}$ is an isomorphism in degrees $\leq n$. We conclude that

\[ R^ if_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to H^ i(R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F}) \]

is an isomorphism for $i \leq n$. We will prove the lemma by looking at the second page of the spectral sequence of Lemma 21.15.7 for $R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F}$. Here is a picture:

\[ \begin{matrix} \ldots & \ldots & \ldots & \ldots \\ \epsilon _{Y, *}R^2f_{\tau , *}\mathcal{F} & R^1\epsilon _{Y, *}R^2f_{\tau , *}\mathcal{F} & R^2\epsilon _{Y, *}R^2f_{\tau , *}\mathcal{F} & \ldots \\ \epsilon _{Y, *}R^1f_{\tau , *}\mathcal{F} & R^1\epsilon _{Y, *}R^1f_{\tau , *}\mathcal{F} & R^2\epsilon _{Y, *}R^1f_{\tau , *}\mathcal{F} & \ldots \\ \epsilon _{Y, *}f_{\tau , *}\mathcal{F} & R^1\epsilon _{Y, *}f_{\tau , *}\mathcal{F} & R^2\epsilon _{Y, *}f_{\tau , *}\mathcal{F} & \ldots \end{matrix} \]

Let $(C_ m)$ be the hypothesis: $R^ if_{\tau ', *}\epsilon _{X, *}\mathcal{F} = \epsilon _{Y, *}R^ if_{\tau , *}\mathcal{F}$ for $i \leq m$. Observe that $(C_0)$ holds. We will show that $(C_{m - 1}) \Rightarrow (C_ m)$ for $m < n$. Namely, if $(C_{m - 1})$ holds, then for $n \geq p > 0$ and $q \leq m - 1$ we have

\begin{align*} R^ p\epsilon _{Y, *}R^ qf_{\tau , *}\mathcal{F} & = R^ p\epsilon _{Y, *} \epsilon _ Y^{-1} \epsilon _{Y, *} R^ qf_{\tau , *}\mathcal{F} \\ & = R^ p\epsilon _{Y, *} \epsilon _ Y^{-1}R^ qf_{\tau ', *}\epsilon _{X, *}\mathcal{F} = 0 \end{align*}

First equality as $\epsilon _ Y^{-1}\epsilon _{Y, *} = \text{id}$, the second by $(C_{m - 1})$, and the final by by $(V_ n)$ because $\epsilon _ Y^{-1}R^ qf_{\tau ', *}\epsilon _{X, *}\mathcal{F}$ is in $\mathcal{A}_ Y$ by (4). Looking at the spectral sequence we see that $E_2^{0, m} = \epsilon _{Y, *}R^ mf_{\tau , *}\mathcal{F}$ is the only nonzero term $E_2^{p, q}$ with $p + q = m$. Recall that $\text{d}_ r^{p, q} : E_ r^{p, q} \to E_ r^{p + r, q - r + 1}$. Hence there are no nonzero differentials $\text{d}_ r^{p, q}$, $r \geq 2$ either emanating or entering this spot. We conclude that $H^ m(R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F}) = \epsilon _{Y, *}R^ mf_{\tau , *}\mathcal{F}$ which implies $(C_ m)$ by the discussion above.

Finally, assume $(C_{n - 1})$. The same analysis shows that $E_2^{0, n} = \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F}$ is the only nonzero term $E_2^{p, q}$ with $p + q = n$. We do still have no nonzero differentials entering this spot, but there can be a nonzero differential emanating it. Namely, the map $d_{n + 1}^{0, n} : \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F} \to R^{n + 1}\epsilon _{Y, *}f_{\tau , *}\mathcal{F}$. We conclude that there is an exact sequence

\[ 0 \to R^ nf_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F} \to R^{n + 1}\epsilon _{Y, *}f_{\tau , *}\mathcal{F} \]

By (4) and (3) the sheaf $R^ nf_{\tau ', *}\epsilon _{X, *}\mathcal{F}$ satisfies the sheaf property for $\tau $-coverings as does $\epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F}$ (use the description of $\epsilon _*$ in Section 21.27). However, the $\tau $-sheafification of the $\tau '$-sheaf $R^{n + 1}\epsilon _{Y, *}f_{\tau , *}\mathcal{F}$ is zero (by locality of cohomology; use Lemmas 21.8.3 and 21.8.4). Thus $R^ nf_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F}$ has to be an isomorphism and the proof is complete. $\square$

If $E'$, resp. $E$ is an object of $D(\mathcal{C}_{\tau '}/X)$, resp. $D(\mathcal{C}_\tau /X)$ then we will write $H^ n_{\tau '}(U, E')$, resp. $H^ n_\tau (U, E)$ for the cohomology of $E'$, resp. $E$ over an object $U$ of $\mathcal{C}/X$.

Lemma 21.30.4. In Situation 21.30.1 if $(V_ n)$ holds, then for $X$ in $\mathcal{C}$ and $L \in D(\mathcal{C}_{\tau '}/X)$ with $H^ i(L) = 0$ for $i < 0$ and $H^ i(L)$ in $\mathcal{A}'_ X$ for $0 \leq i \leq n$ we have $H^ n_{\tau '}(X, L) = H^ n_\tau (X, \epsilon _ X^{-1}L)$.

Proof. By Lemma 21.21.5 we have $H^ n_\tau (X, \epsilon _ X^{-1}L) = H^ n_{\tau '}(X, R\epsilon _{X, *}\epsilon _ X^{-1}L)$. There is a spectral sequence

\[ E_2^{p, q} = R^ p\epsilon _{X, *}\epsilon _ X^{-1}H^ q(L) \]

converging to $H^{p + q}(R\epsilon _{X, *}\epsilon _ X^{-1}L)$. By $(V_ n)$ we have the vanishing of $E_2^{p, q}$ for $0 < p \leq n$ and $0 \leq q \leq n$. Thus $E_2^{0, q} = \epsilon _{X, *}\epsilon _ X^{-1}H^ q(L) = H^ q(L)$ are the only nonzero terms $E_2^{p, q}$ with $p + q \leq n$. It follows that the map

\[ L \longrightarrow R\epsilon _{X, *}\epsilon _ X^{-1}L \]

is an isomorphism in degrees $\leq n$ (small detail omitted). Hence we find that $H^ i_{\tau '}(X, L) = H^ i_{\tau '}(X, R\epsilon _{X, *}\epsilon _ X^{-1}L)$ for $i \leq n$. Thus the lemma is proved. $\square$

Lemma 21.30.5. In Situation 21.30.1 if $(V_ n)$ holds, then for $X$ in $\mathcal{C}$ and $\mathcal{F}$ in $\mathcal{A}_ X$ the map $H^{n + 1}_{\tau '}(X, \epsilon _{X, *}\mathcal{F}) \to H^{n + 1}_\tau (X, \mathcal{F})$ is injective with image those classes which become trivial on a $\tau '$-covering of $X$.

Proof. Recall that $\epsilon _ X^{-1}\epsilon _{X, *}\mathcal{F} = \mathcal{F}$ hence the map is given by pulling back cohomology classes by $\epsilon _ X$. The Leray spectral sequence (Lemma 21.15.5)

\[ E_2^{p, q} = H^ p_{\tau '}(X, R^ q\epsilon _{X, *}\mathcal{F}) \Rightarrow H^{p + q}_\tau (X, \mathcal{F}) \]

combined with the assumed vanishing gives an exact sequence

\[ 0 \to H^{n + 1}_{\tau '}(X, \epsilon _{X, *}\mathcal{F}) \to H^{n + 1}_\tau (X, \mathcal{F}) \to H^0_{\tau '}(X, R^{n + 1}\epsilon _{X, *}\mathcal{F}) \]

This is a restatement of the lemma. $\square$

Lemma 21.30.6. In Situation 21.30.1 let $f : X \to Y$ be in $\mathcal{P}$ such that $\{ X \to Y\} $ is a $\tau $-covering. Let $\mathcal{F}'$ be in $\mathcal{A}'_ Y$. If $n \geq 0$ and

\[ \theta \in \text{Equalizer}\left( \xymatrix{ H^{n + 1}_{\tau '}(X, \mathcal{F}') \ar@<1ex>[r] \ar@<-1ex>[r] & H^{n + 1}_{\tau '}(X \times _ Y X, \mathcal{F}') } \right) \]

then there exists a $\tau '$-covering $\{ Y_ i \to Y\} $ such that $\theta $ restricts to zero in $H^{n + 1}_{\tau '}(Y_ i \times _ Y X, \mathcal{F}')$.

Proof. Observe that $X \times _ Y X$ exists by (1). For $Z$ in $\mathcal{C}/Y$ denote $\mathcal{F}'|_ Z$ the restriction of $\mathcal{F}'$ to $\mathcal{C}_{\tau '}/Z$. Recall that $H^{n + 1}_{\tau '}(X, \mathcal{F}') = H^{n + 1}(\mathcal{C}_{\tau '}/X, \mathcal{F}'|_ X)$, see Lemma 21.8.1. The lemma asserts that the image $\overline{\theta } \in H^0(Y, R^{n + 1}f_{\tau ', *}\mathcal{F}'|_ X)$ of $\theta $ is zero. Consider the cartesian diagram

\[ \xymatrix{ X \times _ Y X \ar[d]_{\text{pr}_1} \ar[r]_{\text{pr}_2} & X \ar[d]^ f \\ X \ar[r]^ f & Y } \]

By trivial base change (Lemma 21.22.1) we have

\[ f_{\tau '}^{-1}R^{n + 1}f_{\tau ', *}(\mathcal{F}'|_ X) = R^{n + 1}\text{pr}_{1, \tau ', *}(\mathcal{F}'|_{X \times _ Y X}) \]

If $\text{pr}_1^{-1}\theta = \text{pr}_2^{-1}\theta $, then the section $f_{\tau '}^{-1}\overline{\theta }$ of $f_{\tau '}^{-1}R^{n + 1}f_{\tau ', *}(\mathcal{F}'|_ X)$ is zero, because it is clear that $\text{pr}_1^{-1}\theta $ maps to the zero element in $H^0(X, R^{n + 1}\text{pr}_{1, \tau ', *}(\mathcal{F}'|_{X \times _ Y X}))$. By (2) we have $\mathcal{F}'|_ X$ in $\mathcal{A}'_ X$. Thus $\mathcal{G}' = R^{n + 1}f_{\tau ', *}(\mathcal{F}'|_ X)$ is an object of $\mathcal{A}'_ Y$ by (4). Thus $\mathcal{G}'$ satisfies the sheaf property for $\tau $-coverings by (3). Since $\{ X \to Y\} $ is a $\tau $-covering we conclude that restriction $\mathcal{G}'(Y) \to \mathcal{G}'(X)$ is injective. It follows that $\overline{\theta }$ is zero. $\square$

Proof. Let $X$ in $\mathcal{C}$ and $\mathcal{F}$ in $\mathcal{A}_ X$. Let $\xi \in H^{n + 1}_\tau (U, \mathcal{F})$ for some $U/X$. We have to show that $\xi $ restricts to zero on the members of a $\tau '$-covering of $U$. See Lemma 21.8.4. It follows from this that we may replace $U$ by the members of a $\tau '$-covering of $U$.

By locality of cohomology (Lemma 21.8.3) we can choose a $\tau $-covering $\{ U_ i \to U\} $ such that $\xi $ restricts to zero on $U_ i$. Choose $\{ V_ j \to V\} $, $\{ f_ j : W_ j \to V_ j\} $, and $\{ W_{jk} \to W_ j\} $ as in (5). After replacing both $U$ by $V_ j$ and $\mathcal{F}$ by its restriction to $\mathcal{C}_\tau /V_ j$, which is allowed by (1), we reduce to the case discussed in the next paragraph.

Here $f : X \to Y$ is an element of $\mathcal{P}$ such that $\{ X \to Y\} $ is a $\tau $-covering, $\mathcal{F}$ is an object of $\mathcal{A}_ Y$, and $\xi \in H^{n + 1}_\tau (Y, \mathcal{F})$ is such that there exists a $\tau '$-covering $\{ X_ i \to X\} _{i \in I}$ such that $\xi $ restricts to zero on $X_ i$ for all $i \in I$. Problem: show that $\xi $ restricts to zero on a $\tau '$-covering of $Y$.

By Lemma 21.30.5 there exists a unique $\tau '$-cohomology class $\theta \in H^{n + 1}_{\tau '}(X, \epsilon _{X, *}\mathcal{F})$ whose image is $\xi |_ X$. Since $\xi |_ X$ pulls back to the same class on $X \times _ Y X$ via the two projections, we find that the same is true for $\theta $ (by uniqueness). By Lemma 21.30.6 we see that after replacing $Y$ by the members of a $\tau '$-covering, we may assume that $\theta = 0$. Consequently, we may assume that $\xi |_ X$ is zero.

Let $f : X \to Y$ be an element of $\mathcal{P}$ such that $\{ X \to Y\} $ is a $\tau $-covering, $\mathcal{F}$ is an object of $\mathcal{A}_ Y$, and $\xi \in H^{n + 1}_\tau (Y, \mathcal{F})$ maps to zero in $H^{n + 1}_\tau (X, \mathcal{F})$. Problem: show that $\xi $ restricts to zero on a $\tau '$-covering of $Y$.

The assumptions tell us $\xi $ maps to zero under the map

\[ \mathcal{F} \longrightarrow Rf_{\tau , *}f_\tau ^{-1}\mathcal{F} \]

Use Lemma 21.21.5. A simple argument using the distinguished triangle of truncations (Derived Categories, Remark 13.12.4) shows that $\xi $ maps to zero under the map

\[ \mathcal{F} \longrightarrow \tau _{\leq n}Rf_{\tau , *}f_\tau ^{-1}\mathcal{F} \]

We will compare this with the map $\epsilon _{Y, *}\mathcal{F} \to K$ where

\[ K = \tau _{\leq n}Rf_{\tau ', *}f_{\tau '}^{-1}\epsilon _{Y, *}\mathcal{F} = \tau _{\leq n}Rf_{\tau ', *}\epsilon _{X, *}f_{\tau }^{-1}\mathcal{F} \]

The equality $\epsilon _{X, *} f_\tau ^{-1} = f_{\tau '}^{-1} \epsilon _{Y, *}$ is a property of (21.30.0.1). Consider the map

\[ Rf_{\tau ', *}\epsilon _{X, *}f_{\tau }^{-1}\mathcal{F} \longrightarrow Rf_{\tau ', *}R\epsilon _{X, *}f_{\tau }^{-1}\mathcal{F} = R\epsilon _{Y, *}Rf_{\tau , *}f_\tau ^{-1}\mathcal{F} \]

used in the proof of Lemma 21.30.3 which induces by adjunction a map

\[ \epsilon _ Y^{-1} Rf_{\tau ', *}\epsilon _{X, *}f_{\tau }^{-1}\mathcal{F} \to Rf_{\tau , *}f_\tau ^{-1}\mathcal{F} \]

Taking trunctions we find a map

\[ \epsilon _ Y^{-1}K \longrightarrow \tau _{\leq n}Rf_{\tau , *}f_\tau ^{-1}\mathcal{F} \]

which is an isomorphism by Lemma 21.30.3; the lemma applies because $f_\tau ^{-1}\mathcal{F}$ is in $\mathcal{A}_ X$ by Lemma 21.30.2. Choose a distinguished triangle

\[ \epsilon _{Y, *}\mathcal{F} \to K \to L \to \epsilon _{Y, *}\mathcal{F}[1] \]

The map $\mathcal{F} \to f_{\tau , *}f_\tau ^{-1}\mathcal{F}$ is injective as $\{ X \to Y\} $ is a $\tau $-covering. Thus $\epsilon _{Y, *}\mathcal{F} \to \epsilon _{Y, *}f_{\tau , *}f_\tau ^{-1}\mathcal{F} = f_{\tau ', *}f_{\tau '}^{-1}\epsilon _{Y, *}\mathcal{F}$ is injective too. Hence $L$ only has nonzero cohomology sheaves in degrees $0, \ldots , n$. As $f_{\tau ', *}f_{\tau '}^{-1}\epsilon _{Y, *}\mathcal{F}$ is in $\mathcal{A}'_ Y$ by (2) and (4) we conclude that

\[ H^0(L) = \mathop{\mathrm{Coker}}(\epsilon _{Y, *}\mathcal{F} \to f_{\tau ', *}f_{\tau '}^{-1}\epsilon _{Y, *}\mathcal{F}) \]

is in the weak Serre subcategory $\mathcal{A}'_ Y$. For $1 \leq i \leq n$ we see that $H^ i(L) = R^ if_{\tau ', *}f_{\tau '}^{-1}\epsilon _{Y, *}\mathcal{F}$ is in $\mathcal{A}'_ Y$ by (2) and (4). Pulling back the distinguished triangle above by $\epsilon _ Y$ we get the distinguished triangle

\[ \mathcal{F} \to \tau _{\leq n}Rf_{\tau , *}f_\tau ^{-1}\mathcal{F} \to \epsilon _ Y^{-1}L \to \mathcal{F}[1] \]

Since $\xi $ maps to zero in the middle term we find that $\xi $ is the image of an element $\xi ' \in H^ n_\tau (Y, \epsilon _ Y^{-1}L)$. By Lemma 21.30.4 we have

\[ H^ n_{\tau '}(Y, L) = H^ n_\tau (Y, \epsilon _ Y^{-1}L), \]

Thus we may lift $\xi '$ to an element of $H^ n_{\tau '}(Y, L)$ and take the boundary into $H^{n + 1}_{\tau '}(Y, \epsilon _{Y, *}\mathcal{F})$ to see that $\xi $ is in the image of the canonical map $H^{n + 1}_{\tau '}(Y, \epsilon _{Y, *}\mathcal{F}) \to H^{n + 1}_\tau (Y, \mathcal{F})$. By locality of cohomology for $H^{n + 1}_{\tau '}(Y,\epsilon _{Y, *}\mathcal{F})$, see Lemma 21.8.3, we conclude. $\square$

Lemma 21.30.8. In Situation 21.30.1 we have that $(V_ n)$ is true for all $n$. Moreover:

  1. For $X$ in $\mathcal{C}$ and $K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X)$ the map $K' \to R\epsilon _{X, *}(\epsilon _ X^{-1}K')$ is an isomorphism.

  2. For $f : X \to Y$ in $\mathcal{P}$ and $K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X)$ we have $Rf_{\tau ', *}K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/Y)$ and $\epsilon _ Y^{-1}(Rf_{\tau ', *}K') = Rf_{\tau , *}(\epsilon _ X^{-1}K')$.

Proof. Observe that $(V_0)$ holds as it is the empty condition. Then we get $(V_ n)$ for all $n$ by Lemma 21.30.7.

Proof of (1). The object $K = \epsilon _ X^{-1}K'$ has cohomology sheaves $H^ i(K) = \epsilon _ X^{-1}H^ i(K')$ in $\mathcal{A}_ X$. Hence the spectral sequence

\[ E_2^{p, q} = R^ p\epsilon _{X, *} H^ q(K) \Rightarrow H^{p + q}(R\epsilon _{X, *}K) \]

degenerates by $(V_ n)$ for all $n$ and we find

\[ H^ n(R\epsilon _{X, *}K) = \epsilon _{X, *}H^ n(K) = \epsilon _{X, *}\epsilon _ X^{-1}H^ i(K') = H^ i(K'). \]

again because $H^ i(K')$ is in $\mathcal{A}'_ X$. Thus the canonical map $K' \to R\epsilon _{X, *}(\epsilon _ X^{-1}K')$ is an isomorphism.

Proof of (2). Using the spectral sequence

\[ E_2^{p, q} = R^ pf_{\tau ', *}H^ q(K') \Rightarrow R^{p + q}f_{\tau ', *}K' \]

the fact that $R^ pf_{\tau ', *}H^ q(K')$ is in $\mathcal{A}'_ Y$ by (4), the fact that $\mathcal{A}'_ Y$ is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau '}/Y)$, and Homology, Lemma 12.21.11 we conclude that $Rf_{\tau ', *}K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X)$. To finish the proof we have to show the base change map

\[ \epsilon _ Y^{-1}(Rf_{\tau ', *}K') \longrightarrow Rf_{\tau , *}(\epsilon _ X^{-1}K') \]

is an isomorphism. Comparing the spectral sequence above to the spectral sequence

\[ E_2^{p, q} = R^ pf_{\tau , *}H^ q(\epsilon _ X^{-1}K') \Rightarrow R^{p + q}f_{\tau , *}\epsilon _ X^{-1}K' \]

we reduce this to the case where $K'$ has a single nonzero cohomology sheaf $\mathcal{F}'$ in $\mathcal{A}'_ X$; details omitted. Then Lemma 21.30.3 gives $\epsilon _ Y^{-1}R^ if_{\tau ', *}\mathcal{F}' = R^ if_{\tau , *}\epsilon _ X^{-1}\mathcal{F}'$ for all $i$ and the proof is complete. $\square$

Lemma 21.30.9. In Situation 21.30.1. For any $X$ in $\mathcal{C}$ the category $\mathcal{A}_ X \subset \textit{Ab}(\mathcal{C}_\tau /X)$ is a weak Serre subcategory and the functor

\[ R\epsilon _{X, *} : D^+_{\mathcal{A}_ X}(\mathcal{C}_\tau /X) \longrightarrow D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X) \]

is an equivalence with quasi-inverse given by $\epsilon _ X^{-1}$.

Proof. We need to check the conditions listed in Homology, Lemma 12.9.3 for $\mathcal{A}_ X$. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map in $\mathcal{A}_ X$, then $\epsilon _{X, *}\varphi : \epsilon _{X, *}\mathcal{F} \to \epsilon _{X, *}\mathcal{G}$ is a map in $\mathcal{A}'_ X$. Hence $\mathop{\mathrm{Ker}}(\epsilon _{X, *}\varphi )$ and $\mathop{\mathrm{Coker}}(\epsilon _{X, *}\varphi )$ are objects of $\mathcal{A}'_ X$ as this is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau '}/X)$. Applying $\epsilon _ X^{-1}$ we obtain an exact sequence

\[ 0 \to \epsilon _ X^{-1}\mathop{\mathrm{Ker}}(\epsilon _{X, *}\varphi ) \to \mathcal{F} \to \mathcal{G} \to \epsilon _ X^{-1}\mathop{\mathrm{Coker}}(\epsilon _{X, *}\varphi ) \to 0 \]

and we see that $\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are in $\mathcal{A}_ X$. Finally, suppose that

\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]

is a short exact sequence in $\textit{Ab}(\mathcal{C}_\tau /X)$ with $\mathcal{F}_1$ and $\mathcal{F}_3$ in $\mathcal{A}_ X$. Then applying $\epsilon _{X, *}$ we obtain an exact sequence

\[ 0 \to \epsilon _{X, *}\mathcal{F}_1 \to \epsilon _{X, *}\mathcal{F}_2 \to \epsilon _{X, *}\mathcal{F}_3 \to R^1\epsilon _{X, *}\mathcal{F}_1 = 0 \]

Vanishing by Lemma 21.30.8. Hence $\epsilon _{X, *}\mathcal{F}_2$ is in $\mathcal{A}'_ X$ as this is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau '}/X)$. Pulling back by $\epsilon _ X$ we conclude that $\mathcal{F}_2$ is in $\mathcal{A}_ X$.

Thus $\mathcal{A}_ X$ is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_\tau /X)$ and it makes sense to consider the category $D^+_{\mathcal{A}_ X}(\mathcal{C}_\tau /X)$. Observe that $\epsilon _ X^{-1} : \mathcal{A}'_ X \to \mathcal{A}_ X$ is an equivalence and that $\mathcal{F}' \to R\epsilon _{X, *}\epsilon _ X^{-1}\mathcal{F}'$ is an isomorphism for $\mathcal{F}'$ in $\mathcal{A}'_ X$ since we have $(V_ n)$ for all $n$ by Lemma 21.30.8. Thus we conclude by Lemma 21.28.5. $\square$

Lemma 21.30.10. In Situation 21.30.1. Let $X$ be in $\mathcal{C}$.

  1. for $\mathcal{F}'$ in $\mathcal{A}'_ X$ we have $H^ n_{\tau '}(X, \mathcal{F}') = H^ n_\tau (X, \epsilon _ X^{-1}\mathcal{F}')$,

  2. for $K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X)$ we have $H^ n_{\tau '}(X, K') = H^ n_\tau (X, \epsilon _ X^{-1}K')$.


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