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*}

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. $\square$

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 0F18. Beware of the difference between the letter 'O' and the digit '0'.