## 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

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

Comment #6828 by Seongsu Jeon on

I think the homology of canonical truncation (4) should be $H_i(\tau_{\leq n} A_{\bullet}) = 0$ if $n=0$, and the cochain version as well.

Comment #6969 by on

Can you try your comment again? In your comment there is no condition on $i$ 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 $i = n$. 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.

Comment #8079 by on

Not going to do this. The thing with the short exact sequence is explained in Remark 13.12.4; as you can see we just use $K/\tau_{\leq a}K$ for the thing that fits into the ses.

Other people have mentioned they prefer to use superscripts, so use $\tau^{\leq n}$ and $\tau^{\geq n}$ when using truncation functors for complexes with upper numbering (so for cochain complexes). If you read this and have an opinion, please leave a comment.

Comment #8194 by Sándor on

OK, this does produce the essentially equivalent distinguished triangle, but with the other convention you get an actual ses. I know, you might say we get it with $K/\tau_{\leq a}K$. True, but a bit ugly and it forces the reader to figure out what $K/\tau_{\leq a}K$ is. But it's your decision, so I am not arguing... :) As far as superscripts are concerned, I would prefer subscripts. Also, just for fun: Have you noticed that Hartshorne uses $\sigma$ for the canonical truncation and $\tau$ for the "stupid" one in R&D? (I'm OK with it this way).

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