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. $\square$


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