The Stacks project

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

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

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

is an isomorphism for $i \leq n$. We will prove the lemma by looking at the second page of the spectral sequence of Lemma 21.14.7 for $R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F}$. Here is a picture:

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

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

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 21.7.3 and 21.7.4). 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. $\square$

Proof. By Lemma 21.20.5 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

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

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. $\square$

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 21.14.5)

combined with the assumed vanishing gives an exact sequence

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

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 21.7.1. 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

By trivial base change (Lemma 21.21.1) we have

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 21.7.4. It follows from this that we may replace $U$ by the members of a $\tau '$-covering of $U$.

By locality of cohomology (Lemma 21.7.3) 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

Use Lemma 21.20.5. A simple argument using the distinguished triangle of truncations (Derived Categories, Remark 13.12.4) shows that $\xi $ maps to zero under the map

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

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

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

Taking trunctions we find a map

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

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

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

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

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 21.7.3, we conclude. $\square$

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

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

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

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 12.24.11 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

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

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. $\square$

Proof. We need to check the conditions listed in Homology, Lemma 12.10.3 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

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

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

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. $\square$

Proof. This follows from Lemma 21.30.8 by Remark 21.14.4. $\square$