Illustration of convergence

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Illustration of convergence

Consider simple case, filtration of a complex $\HH(C^{\bullet})$

$\begin{displaymath} \hdots=C_{-2}=C_{-1}=C_0\supset C_1\supset C_2 \supset 0=\hdots \end{displaymath}$

$\begin{diagram}[small] \hdots & = & C_{-2} & = & C_{-1} & = & C_0 & \supset & C_... ...& = & C_0 & \supset & C_1 & \supset & C_2 & = & 0 & = & \hdots \\ \end{diagram}$
Here

$\begin{displaymath} B=\hdots\oplus 0 \oplus 0 \oplus C_0/C_1 \oplus C_1/C_2 \oplus C_2 \oplus 0 \oplus \hdots \end{displaymath}$

Taking homology we get sequences

$\begin{displaymath} \HH(C^{\bullet})=\HH(C_0)\leftarrow \HH(C_1)\leftarrow\HH(C_2)\leftarrow 0\leftarrow\hdots \end{displaymath}$

$\begin{displaymath} A_1:=\bigoplus_{p\in\bZ}\HH(C_p) \end{displaymath}$

$\begin{displaymath} \HH(C^{\bullet})=\HH(C_0)\supset i_*\HH(C_1)\leftarrow i_*\HH(C_2)\leftarrow 0\leftarrow\hdots \end{displaymath}$

$\begin{displaymath} A_2:=\bigoplus_{p\in\bZ}i_*\HH(C_p) \end{displaymath}$

$\begin{displaymath} \HH(C^{\bullet})=\HH(C_0)\supset i_*\HH(C_1)\supset i_*i_*\HH(C_2)\leftarrow 0\leftarrow\hdots \end{displaymath}$

$\begin{displaymath} A_3:=\bigoplus_{p\in\bZ}i_*i_*\HH(C_p). \end{displaymath}$

When we reach the stage in wich all maps become inclusions, process is stationary i.e.

$\begin{displaymath} A_3=A_4=\hdots \end{displaymath}$

$\begin{diagram}[small] A_3 &&\rTo^{i}&& A_3 \\ & \luTo<{k}&& \ldTo>{j}& \\ && B_3 & = & \HH(A_3) \\ \end{diagram}$
where

is inclusion, $\im k=\ker i=0$ so

. This means that also

$\begin{displaymath} B_3=B_4=\hdots \end{displaymath}$

since

Pawel Witkowski 2006-03-14