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\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*}

12.14 Truncation of complexes

Let $\mathcal{A}$ be an abelian category. Let $A_\bullet $ be a chain complex. There are several ways to truncate the complex $A_\bullet $.

  1. The “stupid” truncation $\sigma _{\leq n}$ is the subcomplex $\sigma _{\leq n} A_\bullet $ defined by the rule $(\sigma _{\leq n} A_\bullet )_ i = 0$ if $i > n$ and $(\sigma _{\leq n} A_\bullet )_ i = A_ i$ if $i \leq n$. In a picture

    \[ \xymatrix{ \sigma _{\leq n}A_\bullet \ar[d] & \ldots \ar[r] & 0 \ar[r] \ar[d] & A_ n \ar[r] \ar[d] & A_{n - 1} \ar[r] \ar[d] & \ldots \\ A_\bullet & \ldots \ar[r] & A_{n + 1} \ar[r] & A_ n \ar[r] & A_{n - 1} \ar[r] & \ldots } \]

    Note the property $\sigma _{\leq n}A_\bullet / \sigma _{\leq n - 1}A_\bullet = A_ n[-n]$.

  2. The “stupid” truncation $\sigma _{\geq n}$ is the quotient complex $\sigma _{\geq n} A_\bullet $ defined by the rule $(\sigma _{\geq n} A_\bullet )_ i = A_ i$ if $i \geq n$ and $(\sigma _{\geq n} A_\bullet )_ i = 0$ if $i < n$. In a picture

    \[ \xymatrix{ A_\bullet \ar[d] & \ldots \ar[r] & A_{n + 1} \ar[r] \ar[d] & A_ n \ar[r] \ar[d] & A_{n - 1} \ar[r] \ar[d] & \ldots \\ \sigma _{\geq n}A_\bullet & \ldots \ar[r] & A_{n + 1} \ar[r] & A_ n \ar[r] & 0 \ar[r] & \ldots } \]

    The map of complexes $\sigma _{\geq n}A_\bullet \to \sigma _{\geq n + 1}A_\bullet $ is surjective with kernel $A_ n[-n]$.

  3. The canonical truncation $\tau _{\geq n}A_\bullet $ is defined by the picture

    \[ \xymatrix{ \tau _{\geq n}A_\bullet \ar[d] & \ldots \ar[r] & A_{n + 1} \ar[r] \ar[d] & \mathop{\mathrm{Ker}}(d_ n) \ar[r] \ar[d] & 0 \ar[r] \ar[d] & \ldots \\ A_\bullet & \ldots \ar[r] & A_{n + 1} \ar[r] & A_ n \ar[r] & A_{n - 1} \ar[r] & \ldots } \]

    Note that these complexes have the property that

    \[ H_ i(\tau _{\geq n}A_\bullet ) = \left\{ \begin{matrix} H_ i(A_\bullet ) & \text{if} & i \geq n \\ 0 & \text{if} & i < n \end{matrix} \right. \]
  4. The canonical truncation $\tau _{\leq n}A_\bullet $ is defined by the picture

    \[ \xymatrix{ A_\bullet \ar[d] & \ldots \ar[r] & A_{n + 1} \ar[r] \ar[d] & A_ n \ar[r] \ar[d] & A_{n - 1} \ar[r] \ar[d] & \ldots \\ \tau _{\leq n}A_\bullet & \ldots \ar[r] & 0 \ar[r] & \mathop{\mathrm{Coker}}(d_{n + 1}) \ar[r] & A_{n - 1} \ar[r] & \ldots } \]

    Note that these complexes have the property that

    \[ H_ i(\tau _{\leq n}A_\bullet ) = \left\{ \begin{matrix} H_ i(A_\bullet ) & \text{if} & i \leq n \\ 0 & \text{if} & i > n \end{matrix} \right. \]

Let $\mathcal{A}$ be an abelian category. Let $A^\bullet $ be a cochain complex. There are four ways to truncate the complex $A^\bullet $.

  1. The “stupid” truncation $\sigma _{\geq n}$ is the subcomplex $\sigma _{\geq n} A^\bullet $ defined by the rule $(\sigma _{\geq n} A^\bullet )^ i = 0$ if $i < n$ and $(\sigma _{\geq n} A^\bullet )^ i = A_ i$ if $i \geq n$. In a picture

    \[ \xymatrix{ \sigma _{\geq n}A^\bullet \ar[d] & \ldots \ar[r] & 0 \ar[r] \ar[d] & A^ n \ar[r] \ar[d] & A^{n + 1} \ar[r] \ar[d] & \ldots \\ A^\bullet & \ldots \ar[r] & A^{n - 1} \ar[r] & A^ n \ar[r] & A^{n + 1} \ar[r] & \ldots } \]

    Note the property $\sigma _{\geq n}A^\bullet / \sigma _{\geq n + 1}A^\bullet = A^ n[-n]$.

  2. The “stupid” truncation $\sigma _{\leq n}$ is the quotient complex $\sigma _{\leq n} A^\bullet $ defined by the rule $(\sigma _{\leq n} A^\bullet )^ i = 0$ if $i > n$ and $(\sigma _{\leq n} A^\bullet )^ i = A^ i$ if $i \leq n$. In a picture

    \[ \xymatrix{ A^\bullet \ar[d] & \ldots \ar[r] & A^{n - 1} \ar[r] \ar[d] & A^ n \ar[r] \ar[d] & A^{n + 1} \ar[r] \ar[d] & \ldots \\ \sigma _{\leq n}A^\bullet & \ldots \ar[r] & A^{n - 1} \ar[r] & A^ n \ar[r] & 0 \ar[r] & \ldots \\ } \]

    The map of complexes $\sigma _{\leq n}A^\bullet \to \sigma _{\leq n - 1}A^\bullet $ is surjective with kernel $A^ n[-n]$.

  3. The canonical truncation $\tau _{\leq n}A^\bullet $ is defined by the picture

    \[ \xymatrix{ \tau _{\leq n}A^\bullet \ar[d] & \ldots \ar[r] & A^{n - 1} \ar[r] \ar[d] & \mathop{\mathrm{Ker}}(d^ n) \ar[r] \ar[d] & 0 \ar[r] \ar[d] & \ldots \\ A^\bullet & \ldots \ar[r] & A^{n - 1} \ar[r] & A^ n \ar[r] & A^{n + 1} \ar[r] & \ldots } \]

    Note that these complexes have the property that

    \[ H^ i(\tau _{\leq n}A^\bullet ) = \left\{ \begin{matrix} H^ i(A^\bullet ) & \text{if} & i \leq n \\ 0 & \text{if} & i > n \end{matrix} \right. \]
  4. The canonical truncation $\tau _{\geq n}A^\bullet $ is defined by the picture

    \[ \xymatrix{ A^\bullet \ar[d] & \ldots \ar[r] & A^{n - 1} \ar[r] \ar[d] & A^ n \ar[r] \ar[d] & A^{n + 1} \ar[r] \ar[d] & \ldots \\ \tau _{\geq n}A^\bullet & \ldots \ar[r] & 0 \ar[r] & \mathop{\mathrm{Coker}}(d^{n - 1}) \ar[r] & A^{n + 1} \ar[r] & \ldots } \]

    Note that these complexes have the property that

    \[ H^ i(\tau _{\geq n}A^\bullet ) = \left\{ \begin{matrix} 0 & \text{if} & i < n \\ H^ i(A^\bullet ) & \text{if} & i \geq n \end{matrix} \right. \]

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