View source for Exercises:Rings and Modules - 2016 - 1/Problem 2

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==Problems==
===Problem 2===
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====Part A====
Compute the [[homology groups]] at {{M|\mathbb{Q}^5}} and {{M|\mathbb{Z}^5}} of the the following complexes:
# {{M|\xymatrix{ \mathbb{Q}^3 \ar[r]^{f_1} & \mathbb{Q}^5 \ar[r]^{f_2} & \mathbb{Q}^2 } }} with {{M|1=f_1:=\begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{pmatrix} }} and {{M|1=f_2:=\begin{pmatrix} 1 & 1 & 0 & -1 & -1 \\ 2 & 2 & 0 & -2 & -2\end{pmatrix} }}
# {{M|\xymatrix{ \mathbb{Z}^3 \ar[r]^{f_1} & \mathbb{Z}^5 \ar[r]^{f_2} & \mathbb{Z}^2 } }} with {{M|1=f_1:=\begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{pmatrix} }} and {{M|1=f_2:=\begin{pmatrix} 1 & 1 & 0 & -1 & -1 \\ 2 & 2 & 0 & -2 & -2\end{pmatrix} }}
=====Solution=====
====Part B====
Let:
<div style="overflow:hidden;"><m>\xymatrix{
 M_1 \ar[d]_{f_1} \ar[r] & M_2 \ar[r] \ar[d]_{f_2} & M_3 \ar[r] \ar[d]_{f_3} & M_4 \ar[r] \ar[d]_{f_4} & M_5 \ar[d]_{f_5} \\ N_1 \ar[r] & N_2 \ar[r] & N_3 \ar[r] & N_4 \ar[r]  & N_5
}</m></div>
be a [[commutative diagram]] of {{M|R}}-[[modules]] in which the rows are [[exact sequences]]. Show the [[Five lemma]]:
* If {{M|f_1,f_2,f_4}} and {{M|f_5}} are {{link|isomorphism|u-module}} then so is {{M|f_3}}
=====Solution=====
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==Notes==
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==References==
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