Exercises:Rings and Modules - 2016 - 1/Problem 2

Problems

Problem 2

Part A

Compute the homology groups at $\mathbb{Q}^5$ and $\mathbb{Z}^5$ of the the following complexes:

1. $\xymatrix{ \mathbb{Q}^3 \ar[r]^{f_1} & \mathbb{Q}^5 \ar[r]^{f_2} & \mathbb{Q}^2 }$ with $f_1:=\begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{pmatrix}$ and $f_2:=\begin{pmatrix} 1 & 1 & 0 & -1 & -1 \\ 2 & 2 & 0 & -2 & -2\end{pmatrix}$
2. $\xymatrix{ \mathbb{Z}^3 \ar[r]^{f_1} & \mathbb{Z}^5 \ar[r]^{f_2} & \mathbb{Z}^2 }$ with $f_1:=\begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{pmatrix}$ and $f_2:=\begin{pmatrix} 1 & 1 & 0 & -1 & -1 \\ 2 & 2 & 0 & -2 & -2\end{pmatrix}$

Part B

Let:

$\xymatrix{ M_1 \ar[d]_{f_1} \ar[r] & M_2 \ar[r] \ar[d]_{f_2} & M_3 \ar[r] \ar[d]_{f_3} & M_4 \ar[r] \ar[d]_{f_4} & M_5 \ar[d]_{f_5} \\ N_1 \ar[r] & N_2 \ar[r] & N_3 \ar[r] & N_4 \ar[r] & N_5 }$

be a commutative diagram of $R$-modules in which the rows are exact sequences. Show the Five lemma:

• If $f_1,f_2,f_4$ and $f_5$ are isomorphism then so is $f_3$