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12.15 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

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

Comments (5)

Comment #6828 by Seongsu Jeon on

I think the homology of canonical truncation (4) should be if , and the cochain version as well.

Comment #6969 by on

Can you try your comment again? In your comment there is no condition on so it doesn't make sense to me. Also, what the text says seems right to me.

Comment #6989 by Seongsu Jeon on

What I meant was . But, I understood the statement is true. Thanks!

Comment #7861 by Sándor on

There should be a variant of the "upper" canonical truncation that starts with im d^n instead of coker d^{n-1}. That one inherits the cohomology of the original complex starting at n+1 (accordingly Hatshorne in R&D denotes this with an index ">n" instead of "\geq n"). The advantage of that definition is that then one has a short exact sequence of the canonical truncations: 0 --> \sigma_{\leq n} A^\cdot --> A^\cdot --> \sigma_{>n} A^\cdot -->0. A similar ses works with the above definitions of the stupid truncation (though one still needs a shift, but that's not a big deal), but it does not work with the current definition of the upper canonical truncation.


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