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*}

21.29 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

  1. every object of $\mathcal{A}$ is a sheaf for the $\tau $-topology,

  2. $\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\tau )$,

  3. every object of $\mathcal{C}$ has a $\tau '$-covering whose members are elements of $\mathcal{B}$, and

  4. 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 21.8.4. 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 12.9.3) 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 12.9.1. 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 12.9.1.

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 13.13. 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

  1. 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 $,

  2. 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 bounded below complex ${\mathcal{K}'}^\bullet $ of sheaves of $\mathcal{O}_{\tau '}$-modules representing $K'$. We will show by induction on $n$ that we may assume for $p \leq n$ we have $(\mathcal{K}')^ p = \epsilon _*\mathcal{J}^ p$ for some injective sheaf $\mathcal{J}^ p$ of $\mathcal{O}_{\tau }$-modules. The assertion is true for $n \ll 0$ because $(\mathcal{K}')^\bullet $ is bounded below.

Induction step. Assume we have $(\mathcal{K}')^ p = \epsilon _*\mathcal{J}^ p$ for some injective sheaves $\mathcal{J}^ p$ of $\mathcal{O}_\tau $-modules for $p \leq n$. Denote $\mathcal{J}^\bullet $ the bounded complex of injective $\mathcal{O}_\tau $-modules made from these sheaves and the maps between them. Consider the short exact sequence of complexes

\[ 0 \to \sigma _{\geq n + 1}(\mathcal{K}')^\bullet \to (\mathcal{K}')^\bullet \to \epsilon _*\mathcal{J}^\bullet \to 0 \]

where $\sigma _{\geq n + 1}$ denotes the “stupid” truncation. By assumption (2) the object $\epsilon _*\mathcal{J}^\bullet $ of $D(\mathcal{O}_{\tau '})$ satisfies (P). By Lemma 21.26.2 we conclude that $\sigma _{\geq n + 1}(\mathcal{K}')^\bullet $ satisfies (P). We conclude that for $\alpha \in A$ the sequence

\[ \begin{matrix} 0 \\ \downarrow \\ H^{n + 1}_{\tau '}(X_\alpha , \sigma _{\geq n + 1}(\mathcal{K}')^\bullet ) \\ \downarrow \\ H^{n + 1}_{\tau '}(Z_\alpha , \sigma _{\geq n + 1}(\mathcal{K}')^\bullet ) \oplus H^{n + 1}_{\tau '}(Y_\alpha , \sigma _{\geq n + 1}(\mathcal{K}')^\bullet ) \\ \downarrow \\ H^{n + 1}_{\tau '}(E_\alpha , \sigma _{\geq n + 1}(\mathcal{K}')^\bullet ) \end{matrix} \]

is exact by the distinguished triangle of Lemma 21.26.1 and the fact that $\sigma _{\geq n + 1}(\mathcal{K}')^\bullet $ has vanishing cohomology over $E_\alpha $ in degrees $< n + 1$. We conclude that

\[ \mathcal{F}' = \mathop{\mathrm{Ker}}((\mathcal{K}')^{n + 1} \to (\mathcal{K}')^{n + 2}) \]

is a $\tau $-sheaf by assumption (1) because the cohomology groups above evaluate to $\mathcal{F}'(X_\alpha )$, $\mathcal{F}'(Z_\alpha ) \oplus \mathcal{F}'(Y_\alpha )$, and $\mathcal{F}'(E_\alpha )$. Thus we may choose an injective $\mathcal{O}_\tau $-module $\mathcal{J}^{n + 1}$ and an injection $\mathcal{F}' \to \epsilon _*\mathcal{J}^{n + 1}$. Since $\epsilon _*\mathcal{J}^{n + 1}$ is also an injective $\mathcal{O}_{\tau '}$-module (Lemma 21.15.2) we can extend $\mathcal{F}' \to \epsilon _*\mathcal{J}^{n + 1}$ to a map $(\mathcal{K}')^{n + 1} \to \epsilon _*\mathcal{J}^{n + 1}$. Then the complex $(\mathcal{K}')^\bullet $ is quasi-isomorphic to the complex

\[ \ldots \to \epsilon _*\mathcal{J}^ n \to \epsilon _*\mathcal{J}^{n + 1} \to \frac{\epsilon _*\mathcal{J}^{n + 1} \oplus (\mathcal{K}')^{n + 2}}{(\mathcal{K}')^{n + 1}} \to (\mathcal{K}')^{n + 3} \to \ldots \]

This finishes the induction step.

The induction procedure described above actually produces a sequence of quasi-isomorphisms of complexes

\[ (\mathcal{K}')^\bullet \to (\mathcal{K}'_{n_0})^\bullet \to (\mathcal{K}'_{n_0 + 1})^\bullet \to (\mathcal{K}'_{n_0 + 2})^\bullet \to \ldots \]

where $(\mathcal{K}'_ n)^\bullet \to (\mathcal{K}'_{n + 1})^\bullet $ is an isomorphism in degrees $\leq n$ and such that $(\mathcal{K}'_ n)^ p = \epsilon _*\mathcal{J}^ p$ for $p \leq n$. Taking the “limit” of these maps therefore gives a quasi-isomorphism $(\mathcal{K}')^\bullet \to \epsilon _*\mathcal{J}^\bullet $ which proves the lemma. $\square$

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

  1. $h_ X^\# = h_ Y^\# \amalg _{h_ E^\# } h_ Z^\# $, and

  2. $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 21.21.9. 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F17. Beware of the difference between the letter 'O' and the digit '0'.