21.29 Comparing two topologies, II
Let $\mathcal{C}$ be a category. Let $\text{Cov}(\mathcal{C}) \supset \text{Cov}'(\mathcal{C})$ be two ways to endow $\mathcal{C}$ with the structure of a site. Denote $\tau $ the topology corresponding to $\text{Cov}(\mathcal{C})$ and $\tau '$ the topology corresponding to $\text{Cov}'(\mathcal{C})$. Then the identity functor on $\mathcal{C}$ defines a morphism of sites
\[ \epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau '} \]
where $\epsilon _*$ is the identity functor on underlying presheaves and where $\epsilon ^{-1}$ is the $\tau $-sheafification of a $\tau '$-sheaf (hence clearly exact). Let $\mathcal{O}$ be a sheaf of rings for the $\tau $-topology. Then $\mathcal{O}$ is also a sheaf for the $\tau '$-topology and $\epsilon $ becomes a morphism of ringed sites
\[ \epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \longrightarrow (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '}) \]
For more discussion, see Section 21.27.
Lemma 21.29.1. With $\epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '})$ as above. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Let $\mathcal{A} \subset \textit{PMod}(\mathcal{O})$ be a full subcategory. Assume
every object of $\mathcal{A}$ is a sheaf for the $\tau $-topology,
$\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\tau )$,
every object of $\mathcal{C}$ has a $\tau '$-covering whose members are elements of $\mathcal{B}$, and
for every $U \in \mathcal{B}$ we have $H^ p_\tau (U, \mathcal{F}) = 0$, $p > 0$ for all $\mathcal{F} \in \mathcal{A}$.
Then $\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\tau '})$ and there is an equivalence of triangulated categories $D_\mathcal {A}(\mathcal{O}_\tau ) = D_\mathcal {A}(\mathcal{O}_{\tau '})$ given by $\epsilon ^*$ and $R\epsilon _*$.
Proof.
Since $\epsilon ^{-1}\mathcal{O}_{\tau '} = \mathcal{O}_\tau $ we see that $\epsilon $ is a flat morphism of ringed sites and that in fact $\epsilon ^{-1} = \epsilon ^*$ on sheaves of modules. By property (1) we can think of every object of $\mathcal{A}$ as a sheaf of $\mathcal{O}_\tau $-modules and as a sheaf of $\mathcal{O}_{\tau '}$-modules. In other words, we have fully faithful inclusion functors
\[ \mathcal{A} \to \textit{Mod}(\mathcal{O}_\tau ) \to \textit{Mod}(\mathcal{O}_{\tau '}) \]
To avoid confusion we will denote $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}_{\tau '})$ the image of $\mathcal{A}$. Then it is clear that $\epsilon _* : \mathcal{A} \to \mathcal{A}'$ and $\epsilon ^* : \mathcal{A}' \to \mathcal{A}$ are quasi-inverse equivalences (see discussion preceding the lemma and use that objects of $\mathcal{A}'$ are sheaves in the $\tau $ topology).
Conditions (3) and (4) imply that $R^ p\epsilon _*\mathcal{F} = 0$ for $p > 0$ and $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. This is true because $R^ p\epsilon _*$ is the sheaf associated to the presheave $U \mapsto H^ p_\tau (U, \mathcal{F})$, see Lemma 21.7.4. Thus any exact complex in $\mathcal{A}$ (which is the same thing as an exact complex in $\textit{Mod}(\mathcal{O}_\tau )$ whose terms are in $\mathcal{A}$, see Homology, Lemma 12.10.3) remains exact upon applying the functor $\epsilon _*$.
Consider an exact sequence
\[ \mathcal{F}'_0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to \mathcal{F}'_4 \]
in $\textit{Mod}(\mathcal{O}_{\tau '})$ with $\mathcal{F}'_0, \mathcal{F}'_1, \mathcal{F}'_3, \mathcal{F}'_4$ in $\mathcal{A}'$. Apply the exact functor $\epsilon ^*$ to get an exact sequence
\[ \epsilon ^*\mathcal{F}'_0 \to \epsilon ^*\mathcal{F}'_1 \to \epsilon ^*\mathcal{F}'_2 \to \epsilon ^*\mathcal{F}'_3 \to \epsilon ^*\mathcal{F}'_4 \]
in $\textit{Mod}(\mathcal{O}_\tau )$. Since $\mathcal{A}$ is a weak Serre subcategory and since $\epsilon ^*\mathcal{F}'_0, \epsilon ^*\mathcal{F}'_1, \epsilon ^*\mathcal{F}'_3, \epsilon ^*\mathcal{F}'_4$ are in $\mathcal{A}$, we conclude that $\epsilon ^*\mathcal{F}_2$ is in $\mathcal{A}$ by Homology, Definition 12.10.1. Consider the map of sequences
\[ \xymatrix{ \mathcal{F}'_0 \ar[r] \ar[d] & \mathcal{F}'_1 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F}'_3 \ar[r] \ar[d] & \mathcal{F}'_4 \ar[d] \\ \epsilon _*\epsilon ^*\mathcal{F}'_0 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_1 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_2 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_3 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_4 } \]
The lower row is exact by the discussion in the preceding paragraph. The vertical arrows with index $0$, $1$, $3$, $4$ are isomorphisms by the discussion in the first paragraph. By the $5$ lemma (Homology, Lemma 12.5.20) we find that $\mathcal{F}'_2 \cong \epsilon _*\epsilon ^*\mathcal{F}'_2$ and hence $\mathcal{F}'_2$ is in $\mathcal{A}'$. In this way we see that $\mathcal{A}'$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\tau '})$, see Homology, Definition 12.10.1.
At this point it makes sense to talk about the derived categories $D_\mathcal {A}(\mathcal{O}_\tau )$ and $D_{\mathcal{A}'}(\mathcal{O}_{\tau '})$, see Derived Categories, Section 13.17. To finish the proof we show that conditions (1) – (5) of Lemma 21.28.7 apply. We have already seen (1), (2), (3) above. Note that since every object has a $\tau '$-covering by objects of $\mathcal{B}$, a fortiori every object has a $\tau $-covering by objects of $\mathcal{B}$. Hence condition (4) of Lemma 21.28.7 is satisfied. Similarly, condition (5) is satisfied as well.
$\square$
Lemma 21.29.2. With $\epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '})$ as above. Let $A$ be a set and for $\alpha \in A$ let
\[ \xymatrix{ E_\alpha \ar[d] \ar[r] & Y_\alpha \ar[d] \\ Z_\alpha \ar[r] & X_\alpha } \]
be a commutative diagram in the category $\mathcal{C}$. Assume that
a $\tau '$-sheaf $\mathcal{F}'$ is a $\tau $-sheaf if $\mathcal{F}'(X_\alpha ) = \mathcal{F}'(Z_\alpha ) \times _{\mathcal{F}'(E_\alpha )} \mathcal{F}'(Y_\alpha )$ for all $\alpha $,
for $K'$ in $D(\mathcal{O}_{\tau '})$ in the essential image of $R\epsilon _*$ the maps $c^{K'}_{X_\alpha , Z_\alpha , Y_\alpha , E_\alpha }$ of Lemma 21.26.1 are isomorphisms for all $\alpha $.
Then $K' \in D^+(\mathcal{O}_{\tau '})$ is in the essential image of $R\epsilon _*$ if and only if the maps $c^{K'}_{X_\alpha , Z_\alpha , Y_\alpha , E_\alpha }$ are isomorphisms for all $\alpha $.
Proof.
The “only if” direction is implied by assumption (2). On the other hand, if $K'$ has a unique nonzero cohomology sheaf, then the “if” direction follows from assumption (1). In general we will use an induction argument to prove the “if” direction. Let us say an object $K'$ of $D^+(\mathcal{O}_{\tau '})$ satisfies (P) if the maps $c^{K'}_{X_\alpha , Z_\alpha , Y_\alpha , E_\alpha }$ are isomorphisms for all $\alpha \in A$.
Namely, let $K'$ be an object of $D^+(\mathcal{O}_{\tau '})$ satisfying (P). Choose a distinguished triangle
\[ K' \to R\epsilon _*\epsilon ^{-1}K' \to M' \to K'[1] \]
in $D^+(\mathcal{O}_{\tau '})$ where the first arrow is the adjuntion map. By (2) and Lemma 21.26.2 we see that $M'$ has (P). On the other hand, applying $\epsilon ^{-1}$ and using that $\epsilon ^{-1}R\epsilon _* = \text{id}$ by Section 21.27 we find that $\epsilon ^{-1}M' = 0$. In the next paragraph we will show $M' = 0$ which finishes the proof.
Let $K'$ be an object of $D^+(\mathcal{O}_{\tau '})$ satisfying (P) with $\epsilon ^{-1}K' = 0$. We will show $K' = 0$. Namely, given $n \in \mathbf{Z}$ such that $H^ i(K') = 0$ for $i < n$ we will show that $H^ n(K') = 0$. For $\alpha \in A$ we have a distinguished triangle
\[ R\Gamma _{\tau '}(X_\alpha , K') \to R\Gamma _{\tau '}(Z_\alpha , K') \oplus R\Gamma _{\tau '}(Y_\alpha , K') \to R\Gamma _{\tau '}(E_\alpha , K') \to R\Gamma _{\tau '}(X_\alpha , K')[1] \]
by Lemma 21.26.1. Taking cohomology in degree $n$ and using the assumed vanishing of cohomology sheaves of $K'$ we obtain an exact sequence
\[ 0 \to H^ n_{\tau '}(X_\alpha , K') \to H^ n_{\tau '}(Z_\alpha , K') \oplus H^ n_{\tau '}(Y_\alpha , K') \to H^ n_{\tau '}(E_\alpha , K') \]
which is the same as the exact sequence
\[ 0 \to \Gamma (X_\alpha , H^ n(K')) \to \Gamma (Z_\alpha , H^ n(K')) \oplus \Gamma (Y_\alpha , H^ n(K')) \to \Gamma (E_\alpha , H^ n(K')) \]
We conclude that $H^ n(K')$ is a a $\tau $-sheaf by assumption (1). However, since the $\tau $-sheafification $\epsilon ^{-1}H^ n(K') = H^ n(\epsilon ^{-1}K')$ is $0$ as $\epsilon ^{-1}K' = 0$ we conclude that $H^ n(K') = 0$ as desired.
$\square$
Lemma 21.29.3. With $\epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '})$ as above. Let
\[ \xymatrix{ E \ar[d] \ar[r] & Y \ar[d] \\ Z \ar[r] & X } \]
be a commutative diagram in the category $\mathcal{C}$ such that
$h_ X^\# = h_ Y^\# \amalg _{h_ E^\# } h_ Z^\# $, and
$h_ E^\# \to h_ Y^\# $ is injective
where ${}^\# $ denotes $\tau $-sheafification. Then for $K' \in D(\mathcal{O}_{\tau '})$ in the essential image of $R\epsilon _*$ the map $c^{K'}_{X, Z, Y, E}$ of Lemma 21.26.1 (using the $\tau '$-topology) is an isomorphism.
Proof.
This helper lemma is an almost immediate consequence of Lemma 21.26.3 and we strongly urge the reader skip the proof. Say $K' = R\epsilon _*K$. Choose a K-injective complex of $\mathcal{O}_\tau $-modules $\mathcal{J}^\bullet $ representing $K$. Then $\epsilon _*\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}_{\tau '}$-modules representing $K'$, see Lemma 21.20.10. Next,
\[ 0 \to \mathcal{J}^\bullet (X) \xrightarrow {\alpha } \mathcal{J}^\bullet (Z) \oplus \mathcal{J}^\bullet (Y) \xrightarrow {\beta } \mathcal{J}^\bullet (E) \to 0 \]
is a short exact sequence of complexes of abelian groups, see Lemma 21.26.3 and its proof. Since this is the same as the sequence of complexes of abelian groups which is used to define $c^{K'}_{X, Z, Y, E}$, we conclude.
$\square$
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