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

Problems

Problem 2

Part A

Compute the homology groups at [ilmath]\mathbb{Q}^5[/ilmath] and [ilmath]\mathbb{Z}^5[/ilmath] of the the following complexes:

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

Part B

Let:

[ilmath]\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 }[/ilmath]

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

• If [ilmath]f_1,f_2,f_4[/ilmath] and [ilmath]f_5[/ilmath] are isomorphism then so is [ilmath]f_3[/ilmath]